course


May 2021 
Amazing professor. Took lin alg with him during pandemic. All exams are take home. First two exams are extremely straight forward. Final requires some work but he gave us multiple days. Great experience. Definitely recommend taking lin alg with him!
Apr 2021 
Professor Ahn is very knowledgeable and approachable. Take his course! Generally speaking, linear algebra is an easy course if you only want to focus on calculation. So for nonmath major students, simply watch the lecture and do the assignment will help a lot. However, if you really want to understand the underlying intuition behind each thing, professor Ahn is great. He is very patient and knowledgeable when you ask about more detailed things about LA, (e.g, how to compose a matrix of reflection or projection from orthonormal vectors, the relationship between rank of a matrix and algebraic multiplicity), which is not required but provides intuition behind it. He also held review sessions before exams, which basically covers whatever appeals in the exam. If you love Math and don't want to be bored with calculation, take his course! You will learn more intuition behind LA.
Jan 2021 
Sack of trash is what this class is. Her lectures are very confusing whether you're watching them live or recorded. Her flow is so choppy and inconsistent. She takes notation lightly which will confuse firsttime learners of the dotproduct. She assumes prior knowledge of linear algebra that makes the presentation of the material confusing. If you are an avid reader of the textbook then you'll perhaps be fine but if you're expecting quality lectures to understand the material, you're in the wrong section. She made our second midterm harder to intentionally bring down the average, wtf? Are we here to learn or compete for a grade? Additionally, she tested us on stuff that we didn't even learn yet. I did better in my higherlevel courses which just goes to show that she intentionally made this course more difficult than it should be. BTW, she miscalculated our final grades which goes to show how seriously she takes this. I don't like writing bad reviews, but this is the worst professor I've ever had. Be warned!
Jan 2021 
Gus is a super fair teacher. His exams are almost exactly like the practice exams he hands out. Clear lecturer (sometimes I got confused but he has a bunch of lecture notes on Courseworks that are super helpful and clear). overall  a great experience!
Jan 2021 
Pretty good! Nice guy, manageable homework. Lectures are really dense so you can't really be halfpaying attention, and you DEFINITELY need to read the textbook to fill in the gaps (he uses language like "Fact" or "Remark" for Theorems and you gotta know them so read the textbook) but a pretty good class for such difficult material. The one thing that hurt was he promised a generous curve and then did not deliver any curve. When asked about it by many students he basically said "yeah, bummer" and didn't change anything. But overall not a bad class, especially in online format. Great TAs too.
Jan 2021 
This is a nononsense linear algebra course. It will teach you exactly what you need to know, with no unnecessary BS but without glossing over anything important. Sacca is a great prof. One of the best I've had at Columbia, and easily the bestadapted to online learning (in pandemic times). Her explanations were incredibly clear, lectures were straightforward, and she was incredibly helpful in office hours and when people asked questions. She will never baby you, and will never just feed you an answer  she'll make you reason it out for yourself  but she will help you along if you are stuck, and her goal is never to make you feel bad, just to learn. Sacca clearly just loves math and wants you to learn it well. If you have the opportunity to take this class with her, I highly recommend it.
Jan 2021 
Excellent lecturer, gives very organized lecture notes, and just an allaround approachable, nice person. He teaches linear algebra well but also gives examples of reallife applications that help get through the dry parts of the course content. It's a hard subject but do the homework (aka don't slader it) because he chooses really meaningful problems to help grasp the concepts. The exams are fair with no surprises as long as you do the practice tests.
Dec 2020 
Kyle is a funny guy, and explains core linear algebra concepts well. The homework is generous. But the tests... are not easy. He breaks it down thus: 50% mechanical (e.g. finding the determinant) 25% conceptual/True or False 25% hard Unless you *really* understand linear algebra, after midterm 1, the last 25% will be very difficult to obtain. I was not prepared for these. I totally understand this exam formatting, but I do not feel that he adequately prepares you for them. It's the classic "In class, I'll do the most straightforward version, but on the test, I'll give you the most convoluted version" scenario.
Dec 2020 
Having taken linear algebra at another university and having been confused out of my mind, I've found Kyle Hayden to be a very good linear algebra professor who can explain concepts and connections between concepts in a clear and intuitive way that stimulates the students' abilities to understand the whole of linear algebra better. Overall, Hayden seems invested in the students and cares about his teaching. He had a few polls on how to structure the class (when to hold office hours, whether to use handwritten or pretyped notes in lecture, etc), offered corrections for a midterm, and grades exams himself instead of delegating to the TAs. Also, he is incredibly patient with answering questions and answers them well (even ones at a middle school algebra level), very chill and understanding (perhaps too understanding: one class started with "a lot of you cheated on the midterm, but I know it's been a tough semester for all of us. So don't worry about your grades, and don't let this be a drag on your semester. In fact, don't let this be a drag on your day" and jumped right into the math), and understanding of student's circumstances with online learning. Homework was from the book and rarely included super difficult questions. Midterm difficulty was fair but timing was challenging. Final difficulty was far harder but timing was extremely generous with a likely curve. I do think the pacing of the course is a bit awkward. The first third of the class was super easy: about half of it was review from high school algebra. But the second third felt more rushed, resulting in Hayden deciding to cut some content out. He also tells corny math jokes sometimes which is nice with such an early class time (in my time zone at least).
Sep 2020 
Prof. Dowlin is literally the best math professor ever. He gives us riddles every now and then and his notes are SO clear and easy to follow. I love Linear Algebra because of him! Homeworks weren't too difficult, exams were medium difficulty
Aug 2020 
hot
Jul 2020 
Carl was an extremely organized and knowledgeable instructor. He is a natural teacher and taking linear algebra with him was such a breeze, and it's definitely my favorite math class I have taken thus far. He doesn't overwhelm you with information but gives you just enough to keep your gears turning, and his lectures and homework problems definitely prepare you for the tests. He also follows the book so you can catch up if you miss something, and the book itself is very helpful so I would recommend using it as your primary study source in addition to lecture notes and extra practice problems. 10/10
May 2020 
Nathan Dowlin is literally the best math professor I've had during my time at Columbia. He is clear, never treats any question as stupid, and doesn't force you to sit through any unnecessary material. Once we switched over to remote lessons, I felt like he was the professor who best handled the transition out of the professors I had this semester which is pretty ironic considering how I'm a CS major with professors who supposedly know how to work a computer.
May 2020 
Nate is one of the best math professors I have ever had. He is enthusiastic, approachable, humorous, and quite gifted at teaching. No doubt a lot of professors are superb scholars, but Nate knows how to unpack the materials in a studentfriendly way because he is able to see the problems from our perspective (unlike some professors who understand the materials very well but just don't know how to effectively deliver the lecture). During lecture, we would go over important definitions, theorems, and proofs. Nate would also suggest you reading/flipping through the textbook by yourself after class (which is a good way to have a more indepth view of additional proofs and build a more systematic understanding of the materials across different chapters). Nate would also answer questions in and after class, sometimes giving us fun riddles as well. Nate had two different time slots for OH this semester, and I find them really helpful. He not only answered my questions but also encouraged me to dive deeper into the concepts. We have weekly problem sets that should take about 46 hours to complete each week, mainly consisted of book problems and the professor's own proof problems. The proof part might be a bit tricky but he shares useful tactics during lecture. We had two midterms and one final, all with useful practice exams to study beforehand and review sessions as well. Definitely consider taking his class! He is a truly wonderful professor.
May 2020 
This guy never replies to my email.
May 2020 
Probably one of the best math professors I've had at Columbia. Very clear, concise and reasonably paced lectures and homework. For Linear Algebra we were assigned pages from the textbook and some proof problems he made. He was always very nice and open during OH and super willing to help you understand and do the problem (unlike other professors who make you "figure it out yourself" and then you spend hours not knowing how to do it). His tests are also very reasonablejust make sure you do the practice exams beforehand. The only harder part of tests or homeworks were the proofs, but he does give tricks to do them in class and helps out a lot in his OH. We had 2 midterms and 1 final.
Apr 2020 
I took linear algebra with Nick and i really liked it! He explains things very clearly and makes jokes about the material to make class a bit more interesting :) Overall, Nick Salter is amazing. I'm taking his complex because I liked his linear so much!
Jan 2020 
Great lecturer and a very nice person. Almost fell in love with linear algebra despite being a humanities major. Definitely take him for Linear or any other math class in general if you can!
Nov 2019 
Michael Thaddeus is amazing! I would say it is slightly more work than other linear classes but totally worth it for engaging lectures, super helpful notes to copy on the blackboard, homeworks that reinforce concepts and stretch you a bit, and not bad exams. Exception: don't take it with him if you hate (informal) proofs.
Mar 2019 
Lotta people seem to like this man, hes really boring  if you need to learn linear for CS dont take him
Jan 2019 
I took Linear with Schrader and I enjoyed it very much. He not only knows the material (obviously) but he explains everything clearly and with much detail. There are a ton of proofs in Linear Algebra but all of them are broken down in class – you can always refer to the book for further review. I found the first midterm to be the easiest and close to what he gave as a practice midterm. Midterm 2 and the final were not really as close to what he gave as practice exams in my opinion. The Math help room, of course, is amazing if you are stuck. Also, Schrader's office hours were beneficial and he does take the time to make sure you understand whatever concept you are struggling with.
Dec 2018 
First off, I would absolutely recommend Bayer, Schrader, or Elliot if you do not have previous experience with proof structuring. This ate a couple people. There is a problem set every week, and you should expect to see the help room jammed on Thursday afternoons, almost comically considering that the vast majority of people are there for one class. For context on grading, he nearly forgot to put up an Arange on the board after our second midterm, which might have been a little closer to a freudian slip than I'm comfortable with! He also always had a Drange. The dents in the car: I would strongly recommend reading through the previous reviews to the bottom. Some students have commented that he hides his grading profile, and gives little to no context on where the class is at. This seems very accurate to me. The problem sets and midterms were given little context by the classes, or even the book for that matter. While the final was more manageable, it really did feel like an exercise in survival despite the amount you studied. I think Thaddeus would be an awesome analysis teacher, but before most students have that level of experience, he really throws sticks in the wheel. He's clearly not used to giving computational examples to proxy proofs and made a painful number of mistakes on the board every class. TL;DR: Don't take him if you don't have a ton of time to wrangle together normal context. Good guy, interesting guy, funny guy, grades like a train.
May 2018 
AMAZING PROFESSOR. Hands down the best math professor I've had here. He's clear, methodical, and really lays out the material in a comprehensive way, doing examples as we go along. While some professors seem to go through the material like it's a piece of cake and don't see why someone wouldn't understand something, Professor Dowlin knows how to teach students in a way we can understand. He always stops for questions and he knows his stuff, so he actually answers the questions thoroughly (what a concept). He writes clearly and at a nice speed on the board and takes little breaks to explain, so you're not frantically trying to catch up writing while missing what he says. He started the first day of class with a Harry Potter quote and always has riddles and hilarious snarky little comments that make class enjoyablewhat more can you ask for! I will say his class is challenging he wants you to really understand the material instead of plugging and chugging, but you get so much out of it. If I could take his class again, I totally would. He's also really friendly and gives great advice. While some professors are intimidating, he's really open and wants to help you. Don't be fooled by his young looksHe's a master in his field (or at least a wizard in disguise.)
May 2018 
The class is very clear and well structured. Lectures cover the textbook quite well, while also providing supplementary applications, that are more theoretical and interesting. The professor does a good job reiterating material, and his lectures are well supplemented by available Open Courseware videos. On the topic of exams: The midterm and final exams are supplemented with practice exams, that are very good preparation for the real tests. The problem sets are of reasonable length, and grading is fair and relatively fast. Students who dedicate time to preparing for the exams, by completing the practice test, doing the problem sets, and reviewing notes or textbook will excel. Altogether, this is a no surprise course, that does a good job covering the material.
May 2018 
I have decided to take this course with Guillaume after reading the positive reviews about him here and even though I ended up getting an A in the course, I do not recommend taking this class with him. Guillaume is a nice guy overall, he makes jokes in class here and there but I couldn't understand anything from his classes. He writes and talks extremely fast and barely solves problems on the board, which makes the material seem very abstract and hard to grasp, even though it isn't. I ended up studying everything on my own, reading from the textbook and watching videos online. Midway through the semester, the attendance dropped to about 10%20%. His exams are difficult and very technical. On the first midterm the mean was 32/50 I got a 31. On the second midterm the mean was 30/50 I got a 40, and on the final, the mean was 60/100 I got an 85. Like I said I ended up getting an A in the course. HW's are also pretty hard and are not similar to his exams. If you want to study for his exams, you should read the chapters in the textbook, solve the True/False questions, and solve the practice exams he hands out.
Dec 2017 
Monotonic and disinteresting. Most of the class stops showing up. Too proof oriented. Would not recommend taking his class if you actually want to learn Linear Algebra. All in all, take bayer or someone.
Nov 2017 
DO NOT TAKE THIS CLASS going to lectures is a WASTE of time he has a thick french? european? accent and his handwriting is awful so you never know what's going on he doesn't communicate with TA's, so they never know what's going to be on the exams, he doesn't communicate with students, we had to beg him to tell us which chapters would be covered on which exams nothing he teaches correlates to the exams, i think it's because he doesn't make them himself unhelpful, unhelpful, unhelpful
Nov 2017 
I feel as though I am fairly qualified to write a review for Barraquand as I have taken 3 semesters with him so far: Calc III, Linear and ODE. Over all, I think Guillaume is a great teacher in comparison to the rest of the math department. That being said, he is not the best teacher ever. GB is not the best teacher in terms of actual lectures. He sometimes goes on tangents of proofs which are albeit interesting not really relevant to the class. That being said, when discussing actual content he is fairly clear on what to expect and how to approach most problems. In linear particular he was a little confusing, but that was perhaps due to the subject matter rather than himself. His classes are somewhat dry but enjoyable, he cracks a lot of jokes and the class is fun. I think GB is an extremely fair teacher. All his HWs are very fair and his grading gives a lot of partial credit. His test, while difficult ask about the entire scope of the topic. They are very well made and really show if you understand the material. His curves are very generous, the last commenter was accurate in terms of grading. Moreover, since the tests are difficult and the averages are usually in the 50s60s, getting a good grade is a lot easier then meets the eye. I would gladly recommend him to any of my friends.
May 2017 
TLDR; Meh teacher. Generous grader. Good tests and homework are okay. You'll learn the fundamentals if you study. Take him if you're an engineer looking for a pretty easy A. Long Version: I'm really conflicted about my experience. I got an A in the class (barely missed the A+ cut off), but I don't feel like I learned much. Barraquand does interesting proofs in the class that are decently rigorous, but he doesn't really show how the concepts of linear algebra are applied or the beauty of it. For example, isomorphisms are apparently really important, but he just mumbled some definitions without much geometric interpretation. Have no idea how to prove it or why I should care even though it seems cool. So I didn't learn how to do real proofs or learn how linear algebra is useful. Thus his class is a bit boring and attendance really dropped to 20% after the first few classes. Be prepared to study a lot and understand the material by yourself. His tests are actually really fair, interesting, and do a decent job of testing your grasp on the fundamentals. There are usually some true/false questions that aren't very hard if you think about them (kinda fun in reality), a few easy computational problems, and then some very long derivations/proofs. The derivation and proofs would be hard if he didn't walk you through them (e.g writing something like "hint, use a laplace expansion on the first row). His practice tests are similar/slightly harder than the actual midterm/final. The final would have been annoying if it weren't for his generous grading and partial credit. I was really bad at multiplying matrices so I had to use variables for some final results. Still scored well above the average despite that fact I screwed up most of the matrix multiplication. Homework was alright. Not too hard if you think about them. Grading was really erratic though. I always checked my homework with a friend, and we had the exact same answers (yes I showed my work). We often got very different grades.
May 2017 
HORRIBLE professor I cannot stress this enough DO NOT TAKE THIS CLASS His lectures are rambling and make no sense, his handwriting is horrible, he looses track of the point of his lectures. Honestly going to class was a waste of time, but learning the material on your own was also impossible The midterms were 4 problems each, with very little correlation between what we did in class/on homework and what was on the exam Averages of the second midterm and the final were both ~50%
Mar 2017 
Not sure where the negative reviews are coming from. This guy is great, honestly one of the nicest math teachers I've ever had at Columbia (I'm a math major and thus have had a lot of exposure to professors in the department throughout the years). He's super understanding, helpful, and friendly, and made linear algebra a really great experience. He gives practice exams for every exam and is really approachable if you have questions.
Jan 2017 
Gabriele was a pleasure to have as a teacher; I had a wonderful experience with him in Lin Alg. He was very approachable at hours and always happy to help. He's a very nice, laidback guy. He told us he would be very generous with the curve and a lax grader, and he held that promise.
Jan 2017 
If you want to learn tricks to solve problems then take Linear Algebra with Bayer. If you want to learn actual Linear Algebra then do yourself a favor and suck it up and take this course with someone else. If you are taking this course as a prereq. for another course, then don't make the same mistake that I did. Take it with someone because you will not know what you will need to know if you take it with Bayer.
Dec 2016 
Professor Bayer is clearly an extremely intelligent guy. However, this class is not for you if you're not willing to self study the material. Bayer jumps right into linear algebra and doesn't really explain the basic concepts very clearly in lecture. If you're smart, learn them beforehand. If not, cram them the night before the midterm. The midterms are very straightforward and the study guides Bayer puts together are very helpful. There are generally over fifty questions with worked out solutions that'll give you a pretty good hint as to what you'll see on the exam. Exam grades are usually very good and it's not hard to get an Aseeing as Bayer gives out as many as he possibly can (!!!). The midterms aren't cumulative, but the material builds on itself so it essentially is. In terms of homework there hardly is any. Bayer will assign roughly 10 or 20 questions due at each midterm and 4 or 5 due before the final. The questions come out of the book and usually don't look anything like what you'd see on an exam so it's pretty easy to do the day before it's due.
Dec 2016 
DO NOT TAKE. Simply the worst math class I have ever taken. The lecture's are boring, scattered, and are entirely untethered from the book. Bayer does not teach you concepts, only methods with no reasoning. There are no theorems, no explanations. Just him working through problems trying to convey a method. The TA's themselves frequently do not know how to explain the methods he uses other than "that's the bayer way of doing it." You are left entirely confused, and have to study by essentially trying to copy the method from his answer sheets until you get the method. If you are taking this as a quant major, it isn't worth it  you will need linear algebra down the road, and this will not teach you it. If you are taking this as a nonmajor looking to try out an easier, useful noncalc math class, you will also be hugely disappointed. Although it is true that this class is wellcurved (typically, almost a 50% Arange), it is so incomprehensible that you will really struggle. In fact, the Arange fact is somewhat misleading, because an extraordinarily large number of students drop out by the midterm because of his terrible teaching style, leaving only those quant majors who are able to get by.
Jan 2015 
Alex is easily one of the best math TA's I've had. He's quick to reply to emails, holds very helpful review sessions, and is eloquent in his explanations. Although Bayer provides a handful of past exams in addition to a practice exam and homework which are all similar to the midterms, Alex's practice exams are usually much more challenging than the actual exams and push your understanding of the material. This is extremely helpful especially when Bayer decides to change up a few problems in the exam, which often trips up students who simply memorize patterns and formulas. Overall a very approachable TA!
May 2014 
I had professor Maulik my entire freshman year, Calc III first semester, and Linear Algebra second semester. He was the definition of a good math professor. He is a decent lecturer (American accent and loud speaker for whom it concerns), peppering his lessons with semieasy access proofs, motivations for learning certain concepts, and a few sparse funny comments. It's not necessarily a class which you will love attending, but after having gone to most classes first semester and not nearly as many second semester, I would definitely say it's worth going. He assigned challenging but helpful written Psets every week, which sometimes took as little time as 2 hours, but a few as many as 9 (but that's probably just me). He didn't really have great review sessions or anything like that, but he leaves that up to the TAs (we had an amazing TA in Calc III). He's pretty fair when it comes to regrades, doesn't make you use anything as annoying as WebAssign, and is altogether a decent teacher who does not fall into the category of researcher who couldn't care less about his students. I also highly recommend going to office hours. Whenever you feel that he may be going too fast in class, or if you get behind and you feel the material is going over your head, if you go to him during office hours, he will really sit down with you and explain the concept as many ways as you need until you get it, and I think he appreciates the interest. All in all, with respect to everyone complaining about the calc teachers (and Bayer), I really feel like I have learned a lot of math this year, and I have Davesh to thank, so I highly recommend choosing him if he's available. P.S. For reference, if people wanna check my bias, I got an A first semester, and an A the next, and I did essentially all of the reading.
May 2014 
I have had many professors over the years in both undergrad and business school, and Davesh is the best professor I have had. When I think about the overall quality of a professor, I think in terms of a few categories: horsepower (i.e. pure math ability), teaching ability, and helpfulness. Horsepower Davesh is an incredibly talented mathematician. He got tenure at a relatively young age, and for those of you who like math competitions, Davesh was a Putnam winner while in college. That's the math equivalent of winning a Heisman Trophy, so for those of you who like math, that's pretty cool. While having a professor with more horsepower does not always make for a better teacher, Davesh's horsepower directly benefited the class. He never got lost in proofs; he didn't waste time; and he was able to show how all the material linked up in a very logical, thorough way. One of the prior reviewers mentioned a criticism along the lines that Davesh uses handheld notes at times  this makes the class more efficient and also he likes to use numerical examples to make things clear. Without notes, one would have to make up numerical examples on the fly which could be very messy computationally. I found the numerical examples incredibly helpful. He covers a lot of material in a lecture and the notes make things much more efficient. I don't like it when professors try to wing lectures and go offtopic. Teaching Ability While one can do fine in the course without fully understanding all the proofs, I like that Davesh took the time to take us through most proofs. He essentially proved everything we use, and an understanding of the proofs contributes to a deeper understanding of the material. Davesh was very concise and articulate in building up the motivations, the proofs, and then the numerical examples to bring it all home. Also, I liked that he spent the first 510 minutes of class consolidating the material from the prior class. One thing which helped me tremendously was skimming the relevant section prior to class (takes 1015 mins), and then reviewing it in detail after the lecture. There were a few points which I felt needed clarification in the text, and often I saw them addressed in my lecture notes which was nice. Helpfulness Davesh was very good about being accessible after class, during the 3 minute break in the middle of class, and during office hours. Whenever he had a conflict, he would schedule another time for office hours instead of canceling them. He provides practice problems prior to exams, and he definitely wants the class to do well. He also has a good sense of humor and is in tune with the ability of the class. Other comments Some people have said they don't like the text. I actually think the text is quite good as it goes through proofs while also providing lots of numerical examples. It strikes the right balance between theory and applications. Linear Algebra is a very interesting subject and I was able to appreciate it given my prior experiences. For example, the Linear Algebra approach to least squares is much more powerful and elegant than what is generally taught in a econometrics/statistics class. Instead of just thinking about plugging in numbers, one can now step back and think of projecting a vector onto a subspace. The undergrad stat classes focus primarily on linear least squares  linear algebra gives us a generalized tool (linear, polynomial, exponential, trig, etc). Also, the discrete dynamical systems was interesting for me in the context of valuing certain stocks where the businesses have the characteristics of those models.
May 2014 
Maulik is not a good lecturer; he goes through topics really fast and it's very hard to focus on his lectures. He tries hard to address the students' questions, and clarifies some of them, but his speed is the main problem. The best you can do in a lecture is to take notes throughout without having understood much because you haven't had enough time to digest the information. Without looking at the textbook (which isn't a very useful one), it's impossible to understand what the theories he does in class imply practically. The lectures are very theoretical, but the assignments, midterms and finals are the contrary. His curve isn't very good either; you have to perform more than a few points above average to get a B+. My ultimate advice would be to take this class with Bayer instead.
May 2014 
I echo the previous review. Prof. Maulik was efficient and taught the class quite well. Like most other math professors, he definitely could have been clearer at times, and often went a bit too quickly through proofs and material. But all in all he got the job done well and had a dry sense of humor that was refreshing. The one stumbling block that I would warn people about is more generally about Linear Algebra itself. This class is one of those math classes that a bunch of people do REALLY well in, so that test averages are usually in the mid 70's or high 80's, so if you're not one of those people who easily gets 100% (there were many in this class) in tests, this might be a struggle. All in all the material isn't impossible and usually just involves arithmetic and *surprise!* algebra. The only hard part is that sometimes it gets really abstract when talking about linear spaces and applying the rules of linear algebra to spaces that aren't matrices. But all in all Linear Algebra felt like a VERY useful class that, in itself might not be all that interesting, but provided us with tools that apply to a lot of different fields: economics, statistics, data analysis, etc. All in all, very good class and teaches you a lot of useful stuff, particularly when it comes to data and dynamical systems.
May 2014 
I would like to respectfully disagree with the April 1, 2014 review of Davesh Maulik. I had Maulik for Calculus III in the previous semester, and his teaching style has remained consistent. His voice is not terribly monotonous, and he does not check his lecture notes so frequently that it disrupts the lecture. The main concern is that he goes through examples and proofs very quickly. However, those examples are still relevant to what he teaches. As for the proofs, you don't really have to know the specific details since he doesn't test proofs on exams. If you don't really understand something, he responds to emails fairly quickly. The textbook is...okay (even though it has terrible reviews on Amazon). The author is really heavy on applications, which might be tiresome for students (some problems are quite lengthy). I think the lectures and the book supplement each other well. The rest of that review with regards to the assignments and midterm is fairly accurate, although I personally think that Maulik is a good professor. For more information on his teaching style, see the December 5, 2013 review  it's the same in the Linear Algebra class.
Apr 2014 
Terrible, just terrible. He is basically a robot, speaking with the same monotonous voice for 75 minutes. He holds a cheat sheet he made for every class while giving the lecture, and he has to check it every 2 minutes. He might be a better Calc teacher, but he is unable to give proper examples for Linear Algebra. He is making simple topics look hard by overloading the class with lengthy descriptions and minimal examples; sometimes I feel like I'm taking a humanities class (AND I'M AN ENGINEER). The textbook we're using is not the best one either, considering that Maulik fails at teaching, it is somewhat hard to actually learn the material. The assignments are not that hard, the midterms are easier than the homework, and the curve is somewhat general, but if you're taking this class to actually learn the material or you simply don't like not being able to be follow a lecture, pick another professor!
Feb 2014 
Disclaimer: I'm writing this after switching out of Bayer's class 23 weeks into this semester. After reading the CULPA reviews of Bayer, going to the first lecture, and briefly talking to him about getting into his (full) class, everything seemed fine. Sure, maybe he wouldn't teach with the rigor I'm used to in math classes, but heyit'll be easier to get an A, and he's a very nice and intelligent guy. However, come the first midterm, I realized what was terribly wrong with this class, and it's exactly the type of stuff people had pointed out below: Bayer DOES NOT TEACH THE FUNDAMENTALS OF LINEAR ALGEBRA (at least for the short time that I was in his class, i.e. up to the first midterm). Rather, he teaches you how to do specific types of problems, and you have to memorize how to do them for exams. He gives little, if any, geometric or mathematical underpinning to the concepts he teaches, and you just end up desperately memorizing algorithms for solving his problems and hoping he doesn't change up the problems slightly so that your method falters. In the words of a girl at a review session, "It's like constipation." As others have said, I can't imagine finishing that course with a real foundation in linear algebra, which is obviously bad if (1) you *need* to thoroughly understand it for some later class, and/or (2) understanding why things work is enjoyable/super useful to you. At this point you may be thinking, "Wow, that sucks. But can't I just read the textbook/watch Khan Academy to learn the fundamentals alongside Bayer's class?" That's what I thought, and the answer is: NOPE. To quote two TAs verbatim, "The textbook is...kind of useless." Bayer's lectures do not, in any discernible way, shape, or form, follow the Bretscher textbook that the department uses, so it's nighimpossible to read along with his lectures. (The same goes for Khan Academy.) Now, if you *have* to take Bayer for some reason...well, I'm sorry. But be comforted in these facts: (1) His tests and homeworks are *incredibly* formulaic, and he posts huge amounts of review materials on his page before each exam. If you memorize how to do his problems, you'll probably ace the exams. (2) He's a really nice and approachable guy, and is very clearly passionate about lecturing (evidenced by his lack of shoes/the layer of chalk dust that coats him by the end of lectures). In my opinion, however, his style of teaching just doesn't cut it if you want/need a thorough grasp of linear algebra.
Jan 2014 
Very clear lecturer. She was very straightforward, organized, and receptive to questions from the class. She made it clear the resources that existed on campus to help us (ie the help room and her office hours), and she genuinely wanted the class to do well. I really enjoyed having her as a professor. The subject matter is hard. It starts out pretty simple, but about a month in becomes conceptually hard to understand, though the calculations never get too crazy. That being said, she puts mostly conceptual questions on her exams, and the questions are often not straightforward and require you to think. Though her exams were hard, the average was usually fairly low (~60%), and her curve was generous. I wound up getting an A in the class, and while I was higher than average on the exams, I did not have over 85% on any exam. tl;dr: given the chance to do it over, I would take this class with her again. If you do the problem sets and study some, there is no reason you shouldn't be able to get a B+ or above.
Jan 2014 
Prof. Bayer is very good for some students, very bad for others. Don't think "Ah, silver nugget, let's go for that!" without reading these reviews. If you're looking for an easy, ridiculously generously curved introduction to Linear Algebra so that you can multiply matrices, compute a determinant and inverse, and fulfill a requirement this may well be the class for you. There's no homework (BTW, don't bother to buy the book), the tests are easy and straightforward, and you can happily skip most classes as long as you leave a few days before each of the four tests to study. He gives As (not Alevel, mind you, full As) to at least one third of the class by my count. Before every exam he gives more than enough practice problems, exams and problem sessions. Additionally, Prof. Bayer is very nice and seemed approachable, and gives back grades very quickly after tests. He's also funny and doesn't talk over your head, which is good news for notsomathy people. That being said, you will learn next to no actual linear algebra in this class. After 5 semesters at Columbia I consider this the most useless (though also one of the easiest) class I've taken. The best way of putting this is that the majority or vast majority of students in the class would probably fail, or do very badly, on a linear algebra exam of any other professor. That's because of two main things. First, he generally teaches by example, almost never explaining the general rule or principle. He explicitly states that his main purpose is for you to be able to compute quickly, not understand indepth concepts. The second reason is that the exams are predictable collections of predetermined questions, so you'll be studying just those question types you know will be on the exam, and that you can do by memorizing techniques rather than understanding the ideas. This makes the class terrible for anyone who really wants to understand Linear Algebra thoroughly, or even understand the basic ways in which it works. So all in all, do not take this class if you're a math major, someone who needs to know why things work the way they do and not just how, or if "handwaving" frustrates you in any way. But if you, like me, are taking this class just to get over with it and come out with a good grade, this should probably be right up your alley.
Dec 2013 
This is a Math class which requires no more skills than simple arithmetic and making sure you copy over a matrix correctly from the question. A 6th grade kid could memorize Bayer's practice exams and do fine in this class. Like another review says, he teaches only 'neat tricks' and completely leaves out any actual fundamentals of linear algebra. I just don't get how you teach a linear algebra class without ever mentioning a null space. You learn nothing in this class and the exams are super easy. They're so easy that everyone does well, so if you mess up once, there's no catching up. Bayer's grading philosophy is the OPPOSITE of every decent Math teacher I've ever had, He cares more about your final answer than your working. I received 2/5 on a question that had everything right but the answer. Also, there's no homework in this class, which is great because it lessens your weekly workload, but who teaches Math without making student practice regularly? It just doesn't work. And since all you need for the exams is to study the practice exams, this class gets condensed into 4 days: your 3 midterms and your final. Nothing else of significance occurs in that classroom, assuming that these exams are significant at all. This is a classic case of a Columbia professor who has a silver nugget on CULPA for being 'super nice' and an easy grader. This is not the definition of a good professor. If you're interested in learning anything at all about linear algebra, stay away from this class.
Dec 2013 
I'll start by giving you the summary of my experience: 1. Wei Ho is a great lecturer. 2. This class is absolute torture and avoid it like the plague if you can. Linear Algebra starts off incredibly easy, which is deceptive because you still have the illusion that it is a manageable class by the time the drop deadline rolls around. The first month or so was a piece of cake: matrix algebra, reduced row echelon form (if you don't know what it is don't worry, it's high school math), etc. But once the basics are down, this class takes a turn for the ugly and before you know it it's REALLY conceptual. Give me 5 equations and tell me to reduce them? Fine. But don't ask me what happens when you invert shapes in four dimensions and collapse it onto another dimension, because that means nothing to me. The emphasis on conceptual was the one thing about Wei Ho that really tripped me up. Other than that, Wei is a very clear lecturer and an incredibly approachable person. She is extremely patient and teaches the class in a very organized and methodical way that complements the textbook very well. In that sense, Wei is a great instructor: clear, effective, organized. The fact that she is very straightforward in class really does not translate well to her midterms and finals, all of which are designed to make you think conceptually about "the larger picture." None of the questions are straightforward, and there is at least one or two big questions about things we rushed through in class or in the problem sets. Truly, these exams were evil. I don't think any amount of preparation helps because at the end of the day, there are only so many ways you can "think outside the box" with Linear Algebra after you've done all the problem sets. My advice is to take this class if and only if (iff) you are very comfortable with math and you actually enjoy challenging yourself beyond the scope of the basic, Intro to Linear Algebra, material. Some people in my class actually seemed to enjoy this about the class. Then again, some people are crazy. If you, like me, want a class where you will learn basic Linear Algebra and not be asked to theorize about the wonders of multidimensional space, take another class with an easier professor. I hear that there's some cute professor who teaches it sometimes  hunt them down, or something.
Nov 2013 
Pro's: He teaches you lot's of neat tricks, and I kind of liked his personality. No homework. Con's: He only teaches tricks. None of the standard methods are taught. You will learn how to do 5 or 6 very specific types of problems for each midterm. Tests are too easy: lot's of 100's goes to a vicious curve. Lectures can be helpful because he goes over pretty much the exact problems that will be on the tests, but he does not tell you about the principles he applies to solve these problems. Lots of practice exams are on his website which accurately reflect real exams.
Jul 2013 
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May 2013 
Taking a class with Bayer has its advantages and disadvantages. He has a very particular teaching style which I find to be way too handsoff and disorganized to follow. After the first midterm, only roughly a third of the class attended lectures frequently. I attended every lecture and still felt lost without further assistance. I ended up getting a straight A in his class nonetheless but only because of the help of a tutor. I am usually an A or A student when it comes to math classes and one of my family members is a mathematician who has taught math at the university level so my strong reaction to Bayer is not one of a sour struggling clueless student. That being said, he is a brilliant mathematician who certainly knows the material inside and out and tries to help his students along the way to that kind of understanding as well. Here is a sampling of what I submitted for my courseworks evaluations for Professor Bayer: I think laying down a foundation and defining terms in very clear ways would be enormously helpful before going into different ways/tricks of solving problems. I felt like I didn't have any foundation whatsoever and that the basics were lost on me for the whole course. I also feel like the lectures were extremely all over the place and disorganized. I think going back to basics when introducing a topic would be extremely useful for the students and then the cool/special cases could be introduced so that we can actually understand them. Also, I feel like WEEKLY HOMEWORK IS EXTREMELY NECESSARY IN EACH AND EVERY MATH CLASS in order for the students to have weekly practice and know where they stand in their grasping of the concepts from getting feedback/grades on weekly homework. I did appreciate the previous exam material available online but because the answers were not posted and it wasn't clear where to look for questions pertaining to the particular unit that was being covered, I felt stuck and like I had no guidance. I ended up getting a tutor for this class, which is the only way I was able to do well on exams. I put a tremendous amount of effort into this course because I felt like I didn't learn hardly anything from lectures so I had to have my tutor reexplain everything for me. I usually do quite well in math classes so this is unusual for me  it is very unlikely that I just wasn't smart/quick enough to grasp the material the first time around.
Dec 2012 
Chen is smart, and is always willing to help. It's just that everything you can possibly get out of his lectures is all in your textbook, and the textbook does a better job in teaching. At least it doesn't murmur. He pretty much copies down the textbook on the blackboard. Even the examples are straight from the textbook. It makes things easier since you don't have to take notes (that are hard to understand if you haven't learnt the material beforehand) as long as you buy your textbook. I figured it out when I was studying for the final. It would've been much nicer if I knew that earlier. Chen uses his own notations of things like (*) and others. I still do not fully understand what some of them mean, and I didn't have to. The first half of the class is easy, since linear algebra is like that. However, you might have to actually study for the second half. The test problems are reasonable and wellwritten with okay curves. Make sure you understand the material enough so that you don't have problem with more computational (and thus easier) ones from the homework. You don't need to memorize the proofs at all; Chen might not be a good lecturer, but he is a nice guy. His demands are very reasonable and doable.
Dec 2012 
This is literally the worst class I have ever taken. Qile is completely incomprehensible as a teacher, making class time completely unproductive. If you want to succeed in this class, you will have to do so solely by reading the textbook. In addition, Qile is completely unhelpful outside of class, refusing to provide any review materials for the assessments or even introduce the class to our TA. Overall, this class is absolutely horrible. No student should have to suffer through this. This class has taken any interest I ever had in math and utterly obliterated it.
Jun 2012 
Totally brilliant. I'm not naturally passionate about math, but Linear Algebra instantly became my favorite class. I loved his teaching style. Despite a disorganized personal appearance, his lectures were crystal clear. He explained why he made certain choices as a professor. He liked to do examples in multiple ways so that all students could understand one method or another. Professor Bayer structures this class without graded assignments, so students have to be able to selfmotivate. I found myself falling behind in my own practice problems, then catching up before each test (there were three total). But as long as you are capable of learning things on your own without assignments, this is a great class. And I never used my textbook. He has lots of examples up online, so don't bother buying the book.
Apr 2012 
Bayer is very interesting as a person and very intelligent, but I think that doesn't necessarily equate to being able to teach. He's incredibly unorganized and assumes that everyone in the class knows what matrices are and how to work with them. One class, he breezed over a new topic but then spent about ten minutes talking to us about how to factor a second degree polynomial. He posts tons of practice exams on his site, which are the key to studying. However, I feel that, by doing his previous test problems, I'm learning a method for how to solve his specific questions, I'm not learning linear algebra. If I took another linear algebra course elsewhere I'd have to start essentially from scratch. So sure, if you're naturally really good at math and have never had to study it, you probably will get an A. For me, I had to work really hard to do decently, and I still don't feel as though I'm learning anything. I really don't recommend this class.
Jan 2012 
Let me start of by saying this: only around 50% of the students showed up to class. Maybe it was the time that this class was offered (6:107:25 pm), or his monotonous voice that seemed to put half of what was left in class to sleep, but it made it very appealing to most to just simply skip class and read out of the textbook (which he followed very closely). If you do show up to class though and listen closely, you can get a lot out of the class. Although many students got by skipping classes and handing in tedious problem sets, Professor Stein is very knowledgeable about both Linear Algebra and what exists beyond it. He goes into many proofs in class (albeit terse) that are not necessary for exams but may be of interest to some math majors. The last class he gave us a very brief preview of analysis and modern algebra, which was very interesting to say the least. Professor Stein is extremely helpful when it comes to office hours. He is exceptional proficient at fielding questions both from the homework and those outside of class and is always available for assistance. Professor Stein is unique in that he insists on solving problems his own way, and learning about the logic behind his methods is very helpful in understanding the underlying concepts behind the course. Although theory is not emphasized nearly as much as problem solving especially in the textbook, going to office hours is a huge help in this regard. Lastly, the exams. Around 6 problems each of the two midterms, and around 1011 for the final. Both midterms have around 5 problems modeled after homework problems, while the last problem is usually a more tricky question that involves some of theory and understanding the concepts behind the problem solving. The final is also straightforward except for some tricky questions. Going through past problem sets over again is easily the best preparation for all of his exams. Be sure to be extra careful with arithmetic calculations; you can lose a lot of points from simple miscalculations. Many that take this class find repetitive weekly problem sets and exams. But for those willing to show up to class and willing to go to office hours because they want to learn more about the problem solving and understand why linear algebra is important, there is a lot to be found with Professor Stein. The amount of interest you put into this class is directly proportional to how much you get out of it.
Jun 2011 
Catherine Williams is probably the best math professor I've had at Columbia. If you're in SEAS like myself and you have the choice of taking the math department linear algebra with her I would take that opportunity. I can't speak to which course is more difficult, but I've heard bad things about other linear algebra sections in both the math and applied math departments, which some of my classmates found to be painful, and certainly far from enjoyable experiences. Not so with Catherine Williams. I mean the fact that she speaks English clearly and fluently already makes her a cut above the rest for a science/math professor, but moreover the clarity in her proofs and explanations is second to none. Tests aren't easy, but generously curved, she's helpful in office hours and she'll bother to remember your name if you show up, which is a nice touch for a large lecture. I also agree with the other reviewer who mentioned that she's not exactly an eyesore either.
Jun 2011 
Bayer is not only a phenomenal teacher, he's also an all around incredible guy. Spend time talking to the man, he always has fascinating stories. He also is a great teacher, being in class is not necessary but very fun and helpful. You don't need the textbook since he'll make sure he teaches a ton of different approaches in class, including some really great short cuts. This class was easy, but only because he gave us the tools to make it easy. Take this class with Bayer, you'll get a good grade, learn something, and have fun while you're at it.
Jun 2011 
I loved Catherine's class! I've been into math through high school and college, but this was my first time taking linear algebra. Catherine was by far the best teacher I've had at Columbia so far. Her lessons were clear, and she was very helpful during office hours. Compared to other sections, I felt like her class was indeed harder, but put in the effort, and it'll be worth it. The exams were challenging, but she grades generously at the end.
May 2011 
Warning! This class is extremely easy. You have to be really dumb to not do well in this class. To elucidate my point, I'd like to point out that more than half the class got 100% on the first midterm, and the trend was more or less maintained through the second midterm and the final. I went to the class only for midterms and the final, and I got an A+. The secret to doing that is to solve all the previous exam questions he has on his website before the test. And you'll learn it by doing it. The questions are always asked in the same fashion as he has been for the last 10 years, so no surprises there. For all the people heading to the bookstore to buy/rent the textbook for this course, I'd say please don't. It will be completely useless with Bayer because you don't have any homework.
Apr 2011 
Linear Algebra is a requirement for many majors, so there were people from many different backgrounds, which was interesting. It is supposed to be very different from Calculus III  and it is. Though matrices might seem confusing at first, by the end of the course I realize how useful they can be in so many problems, especially if you are interested in economics. Bayer is an amazing instructor and focuses on intuitive approaches to math, rather than old textbook formulalearning. He writes on the black board extensively, so it is one of those classes where keeping up with lectures is a must. He has a sense of humor and also recalls many funny stories. He is also eager to push students into more advanced math classes and to explain complex topics which are not really in the scope of Linear Algebra, but fascinating nonetheless. I really enjoyed his approach and would recommend Bayer to anyone actually interested in Math.
Jan 2011 
I thought Catherine was an awesome prof! After being turned off math from a bad experience first semester of Freshmen year, I now wish I had majored in it. Partly I think this is because the material was pretty cool, and Catherine will never fail to point out how neat some of the theorems and proofs are (I agree!). But it sure helped that she was genuinely enthusiastic about linear algebra and tried to get us excited as well. She always stayed behind to entertain whatever questions, scheduling problems or vague mathematical ruminations we had. I thought it was incredibly nice of her to try and remember all our names. Catherine is really good at explaining abstract concepts and won't shy away from attempting an intuitive explanation or nonintuitive ideas. I'm a senior this year, and this is my first CULPA review. I just felt compelled to write one because I really think she's a star instructor.
Dec 2010 
Catherine was by far the best math teacher I've had at Columbia. Her lectures were extremely easy to follow, she made a fairly drab class pretty lively with her ebullient personality, and since I'm a nonmath major guy, most importantly, she's quite good looking. Catherine really cared about her students and made herself available often for office hours, and one thing that really stuck in my mind was that she made an effort to memorize all of the students' names, even though it was a 100 person lecture. I don't know if she's tenured and will be teaching again, but she should be her linear algebra class was my favorite class of the semester, mostly because of her teaching style.
Dec 2010 
ZOMG Catherine is amazing! Her lectures are really well structured and easy to follow. She does cover a lot of ground in a short time though, so its easy to miss major things. She also brings a level of enthusiasm to her lectures that is rare for a math class, and it helped keep my attention from wandering during class. I got the feeling Prof. Williams wanted us to have a really solid conceptual appreciation of the math. Homeworks and exam problems tended to be 1. tough 2. reliant on knowledge of the theory. You won't get very far by just knowing how to do the cookbook algorithms for problems. She's also very helpful discussing concepts or tough hw problems (lots of these) in office hours, which meant they tended to be overrun, but that's not her fault. She's also really nice, will remember you by name for the entire semester, and when her phone accidentally went off on the first day of class, the ringtone was the Star Wars cantina song! *heart*
Oct 2010 
Odd but brilliant and a fairly good teacher. This was a tough subject matter to teach in 6weeks, but Irena manged to get most points across. Proof intensive, so make sure you have time to absorb such ideas (unless such things are not an issue for you). Irena was always there after class to go over the points of the class. However sometimes her points were not so helpful as "look at it harder" doesn't always help. Nevertheless I found her willingness to go over the problems and each step of a proof to be helpful.
Aug 2010 
To Monty Python fans: Lectures play out as the lost Linear Algebra segment of Life or Death Struggles. http://www.youtube.com/watch?v=0pcY1XWN_Q It does become a life or death struggle to stay awake during lectures, but the achievement is worth more than the accompanying bragging rights: Ovi knows his stuff and consistently explains things clearly, not to mention the inclass examples are very useful when it comes time to do the problem sets, as the textbook is scrimps on the details in places. The midterms are fairly straightforward  similar to homework problem sets and most certainly the practice exams in difficulty and material. But the final was another beast entirely, as it was much trickier, more timeconsuming, and generally harder than the midterms (no helpful multiparted questions to walk you through to the major point  he goes straight for the jugular on the final). The general consensus was that the class average was too high going into the final, and Ovi decided this had to change. Read: a very manageable class up until the final 2 weeks or so, when you tie your brain in knots trying to remember every minute detail about linear in/dependence etc. and then try to think critically. It's not necessarily catastrophic, but there will be an explosion of some degree. Ovi is just as awkward oneonone as he is in class, but he is quite reasonable, even nice (apologetic may be pushing it) if you've been unfairly graded. Bonding with the TA's isn't necessary, but it wouldn't hurt your cause if you were unjustly docked points, which is a more common occurance in this class than others because of the extensive room for error in dealing with highvolume matrices. Overall, it's not a difficult class, but do expect to put in some effort. Actually sleeping at night is essential, because with all the numbers running around in the matrices, it is very, very easy to lose a lot of points on careless errors (speaking from a friend's experience). Not an earthshatteringly enlightening class, but recommended considering it opens the door to the fancier cryptography class on making and breaking codes (MATH V3025). And hey, Ovi's goofy mannerisms are kind of endearing (be on the lookout for his understated but abrupt expressions of personal opinion  even if you don't appreciate the perhaps unintentional comedy, it'll at least make you pay attention).
Mar 2010 
I had Linear Algebra with Dvora during the summer semester of 2009. I found her explanations to be very clear, in spite of the difficulty of the subject matter. I truly believe that any mature student who has a solid background in Calculus and Differential Equations will be ok in this course. Dvora seemed very approachable and had no problems answering questions. I don't know how accessible she was because I work full time but she did offer opportunities to meet before the class to go over the homework. I found this to be very helpful. I would take her again.
Dec 2009 
Elliott Stein does not deserve a silver nugget. True he may be fluent in English, but he teaches with so little enthusiasm, I (an applied math major) could not pay attention for any length of time. Also, he teaches straight out of the textbook, which in most cases is not a bad thing, except the textbook chosen for Linear Algebra was deplorable. It only teaches equations/applications, so there are no underlying concepts motivating the subject. Again, this seems like a minor issue, but when you're doing hours of repetitive homework every week, it begins to weigh heavily on the class. Now, about the homework. From the beginning, it took hours to get through. He assigns two of every type of problem, which is unnecessary for introductory ideas and timeconsuming for later ones. In short, the work was tedious. Perhaps the saving grace of the class was its exams. They were very straightforward: out of six questions on each of the two midterms, five followed straight from the homework and the last one was only slightly different. But beware, the grading is tough. My friend made one addition error and got less than half credit for the question. The final was basically two midterms. Honestly, I took this class because it is required and I wanted a decent grade. That is all I got out of it, too, since I learned next to nothing in terms of linear algebra. As to whether I would recommend it, well, if you are simply looking for a good grade and don't mind setting aside a few painful hours a week, this is the class for you. But still, keep your eyes peeled for a professor that can make the class less frustrating.
Nov 2009 
I had the unfortunate experience of having her for Linear Algebra over the summer. Dvora is someone you want to avoid at all costs, even if it means taking the class a different semester. She has no problem ridiculing a student in front of the entire class, and is unavailable to help when asked. She will ensure that all students in the class know which students are doing poorly. Her explanations are unclear and disorganized, and if you ask for clarification she will tell you how simple it is and that you should have understood it the first time. I cannot stress enough how much you want to avoid her.
Aug 2009 
As the first reviewer said, Thibaut is hands down one of the best instructors you could ever hope to have. He's a great communicator, and a great lecturer who really does care about whether or not his students learn. Outside of call, he's incredibly generous with his time (e.g. holding office hours every day for an hour during the summer term, when most summer instructors will only give an hour a week in the summer). He'll ask a lot of you, and work you incredibly hard, but you'll be rewarded with a very solid grasp of the underlying mathematical theory that should prove immensely helpful in future courses that use linear algebra. Highly recommended.
Aug 2009 
Professor Pugin was handsdown the best professor I've had for any math course. He's extremely demanding, but he's great at explaining and he works very hard to help the students, so you walk away learning an incredible amount. He's also French, which means that you learn a lot of theory and proofs in his class, and less applications. For me, that was great, I'm going on to pure math. For people who want to do applied stuff, well, it's still great. Because you'll understand the math you'll apply later. One problem was that it was a summer class and he didn't like the textbook. I agree the textbook sucked, but in a summer class you really have to do a lot of work at home, so you need to have a text that you can refer to. Not having that made the class very difficult. Also, his tests are rather difficult even if you understand the class material, and I've heard that this is true for other classes he has taught.
Aug 2009 
There is no doubt that Professor Zheng is a talented mathematician, but he clearly lacks basic teaching skills. You can barely hear him in class, and he faces the board most of the time, even when he is explaining things, which made it almost impossible to know what he is talking about. Also, his class was incredibly boring, so most students simply skipped his lectures and only attended class to take the exams. I went to class for the first month or so, but then I realized that I actually learned more from reading the textbook by myself and stopping by the Math Help Room when needed. The material from this class was not particularly difficult, so if you are willing to spend time reading the textbook and understanding the assigned homework problems, you should do fine. Now, that is not to say the class was easy. So do consider taking it with another professor if you can.
May 2009 
Professor Hou is a great math teacher. After having taken classes with him over the course of 3 years, he is one teacher that I can rely on. His lecturing style may be quick but each lecture is very clear. He breaks things down and explains concepts thoroughly, plus he draws really awesome pictures/graphs to go with almost everything. He seems very conscious of his English skills, but he speaks clearly albeit with a slight accent. On the board he is not only legible (which, in the math department is hard to find) but he writes nice and big and always steps away for you to copy down the notes. Although he may make a few mistakes during class, he doesn't make many (always double check with the textbook). His courses are the few math classes where I studied more often from my notes than from the textbook. For all of the courses, he has a different strategy for grade breakdown, but it never strays too far from a standard h/wmidtermmassive final mold. His course website always is uptodate with assignments, assignment solutions, and dates of tests/exams. He usually posts practice midterm(s) and final with solutions and gives a little review of topics before each. He may seem kinda awkward, that is because he is. Thus, he may seem unapproachable, but do not fear, you can usually find him napping/wearing awesome slippers in his office during office hours because nobody goes. He has office hours twice a week (for A+O an additional problem session on fridays) and he is always happy to explain things if you just ask! The bottom line: he covers material in a timely fashion, and he is organized. If you want to actually learn some math and not just stumble through a course, Hou is the man. It is totally possible to get a good grade in the course if you put the necessary effort in, and I would say the grading in the courses is consistently fair (curved to a B/B+ avg). If you have the opportunity, take Hou. (his vast collection of sweatervests is also a plus)
Feb 2009 
I highly disagree with the previous comment. Matt was a dedicated teacher, put into a difficult situation because of the range of studentability in the class and the fast paced summer schedule. Nevertheless, he had office hours every day and was always willing to meet at other times and discuss problems. The tests were fair in the sense that many of the problems were similar to the homeworks assigned with one or so problem more challenging. All in all I felt I walked away appreciating linear algebra and would recommend taking a course from Matt.
Jan 2009 
Bythebook (literally), fair, and utterly unremarkable professor. Mind you I'm writing this review more than a year after the course, but it was really quite ordinary, about what you'd expect from a large lecture where you do the problem sets, take the tests, and you're done. I got the tools I needed for more advanced classes and I have no complaints. Problem sets were weekly with one dropped, entirely fair. Lectures were boring when I went and followed the textbook almost to the letter. Professor Hou's English was fine, and he's nice enough. As I recall, the midterms were much easier than the final, which incorporated some notinthetextbook material introduced in the last lecture, which I didn't attend. Oops.
Dec 2008 
Worse teacher in the Math Department by far. I am a math major and I've seen a bunch of professors in the Math Department. This lady is the worse by far. Use itunes U; look up the MIT lectures if you really want to learn the material (at least that's what my friends and I did) since her lectures are a waste of time. There were 42 kids in the class. Less than 10 showed up to the lectures. Mirela will waste all of class on a single proof that she will not even finish. She does not draw any maps of subspaces nor any proper graphs in 3D space of the vectors she discusses. It is crucial to have an answer manual to the textbook. This will teach you how to do the homework problems. There was never an occasion when she did any problems remotely similar to the homework in class.
Dec 2008 
Mirela is neither a good teacher nor a particularly nice teacher. She refuses to move a midterm if you have a family emergency instead, she is "willing" to shift your grade to another midterm. That doesnt reall help. Teaches too long on the easy stuff and is forced to go really fast through all the easy things, often not teaching the stuff in class that is on the homework. The book is absolutely terrible so good luck learning anything from that either. While the first midterm may be easy, dont be fooled. The class gets hard and she just gets worse and worse. She is very hard grader and refuses to curve until the end of the class. I would suggest finding a different professor if you can.
Aug 2008 
Dvora is amazing! She is really clear and genuinely cares about students learning the material. She goes over the proofs and examples in such a way that teaches the students the methods rather than simply encouraging memorization. Her enthusiasm for teaching the material is contagiousI looked forward to going to class! The only thing is that she definitely likes a bellcurve distribution set around B, so she does not usually curve since this is generally where the exam grades fall. But go to class and do the work and you are fine!
Apr 2008 
First the good part: it's an easy A/A. Now the bad part: There is nothing you can learn from this class. The prof is condescending, and he never teaches the basics: he simply assumes you know them and then goes on to illustrate extensively his own methods (which by the way are useless) of solving his own problems. This class itself is completely useless: if you are a math major, it will leave you unprepared for higher level classes, and if you are taking it for personal enrichment, it will be a total waste of your time. The only use you can draw from this class is getting an A, but it will cost you 3 hours of your life, every week, for 45 months, for which you will get nothing else but the grade in return.
Apr 2008 
So, initially the class looked really good because Spiegelman is a nice guy (VERY BUSY though outside of class) and easy to talk to and ask questions. However, I ended up not going to class for pretty much half of the semester mostly because I got a lot more out of Strang's book than I did from lecture. It was also the added complication of the Smart Board and premade slides which made it very difficult to take notes because he moved so fast. Also, watching some of Strang's lectures was pretty useful. All in all, a relatively straightforward math class. Read the book and PRACTICE problems and you'll do fine.
Feb 2008 
This man speaks English fluently and without an accent. Hallelujah. Professor Stein may not be a dazzling speaker, but he presents the material clearly and concisely. Occasionally he works through short proofs, but this is always enlightening and rarely confusing. The book was a nice complement to his lectures. The class is based less around proofs than the other LA sections, which is a relief. Stein's exams were fair and usually left a lot of time to go back and check answers (not just for me, for most people in the class). Bottom line: good professor, certainly a good catch for a math class.
Dec 2007 
I've taken two other semesters with Prof. Hou (Calc III and IV) and I expected no less from him this semester. I've reviewed him before and again I must say he is as straightforward as you could possibly want from the Math department. He follows the text, but will extend to more challenging material even though it won't be on the exams. He emphasizes what will be on the exams, which are primarily drawn from the homework. He is very nice and extremely approachable, and is also very knowledgeable. Even though many people say to take Linear Algebra through the Applied Math dept. for various reasons, if you still want the theory and a good instructor, go with Hou.
Aug 2007 
Truly horrendous experience. He is not interested at all in teaching. He copies his own notes on the board quickly, mostly in theoremprooftheoremprooftheoremproof mode (obscurely, and he often screws up on the board too), and then gives you quizzes and exams that have nothing to do with what he talked about, or even with the contents of the textbook!!!. He marks no transitions, and he makes no emphasis on important points or essential contents. And he doesn't allow formula sheets on exams either. As a result, the grading of homework, quizzes and exams is absolutely unfair, so not only will you learn nothing from his class, but your GPA will suffer unduly. Any resemblance between teaching (even bad teaching) and what he does in a classroom is entirely coincidental. I am truly amazed that he is allowed to teach independently at Columbia. The worst is I actually was very interested and excited to learn linear algebra. I guess I'll have to buy the Schaum's Outline...
Jul 2007 
Dave is probably the most brilliant math teacher that you will come across. You finally get to see the practical applications of math rather than running around in endless proofs which seem of no consequence. The curve is mighty mighty generous. If you studied only the night before the exam, but studied well, you can easily get an A. Yeah, you probably didn't understand linear algebra really well, cause the way he teaches it is unconventional without the proofs. But you can see the way he thinks through analogies, and you start to think like that, and then math surprisingly becomes a hell lot of fun. Oh, and don't touch the text book. Its useless. Do only the practice exams.
Jun 2007 
Prof. Ciperiani doesn't know how to teach. People won't show up to class just because it's so useless. If you spend 3 hours a week studying on your own and teaching yourself, it would be better than attending a class that would put you to sleep. The tests are easy. Just do the Homework problems and you will be fine. As for the problem sets, take them serious and attend the TA sessions, you won't be able to do well otherwise.
Jun 2007 
Thaddeus is a nice guy who genuinely loves the subject and shows it in class. His lectures, while difficult are actually quite clear so as long as you copy down the board and pay somewhat attention as he explains it you should understand the material. The only problem is his grading setup: no homework or midterm which is really cool but since there is no graded homework it's hard to motivate yourself to do it. But you definitely should do the homework because they are a big help for his 8 30minute pop quizzes given during class. These are really annoying because they can be unexpected and detailed. He is very prooforiented rather than focusing on just problem solving which I found annoying. All that said I struggled with around a 50% quiz average but I studied for the final and after the curve I ended with a B in the class. Good professor if he just changes his pop quiz system.
May 2007 
Mirela is obsessed with cheating. She thought people were stealing homeworks from her box, so she made them due a day earlier. People got points taken off their homeworks for plagiarizing. She made up two different versions of the tests and handed a different one to every other person. And she even warned people during the final that she saw someone looking at another person's paper. The class itself was good at the beginning and bad by the end. I never really got a good complete picture of what we were trying to do. I still don't understand what an orthogonal projection is and what inner products have to do with linear algebra. And I went to class and did well in the course. Note: this course becomes proof based after the first midterm. Between one third and a half of the second midterm and final consisted of proofs. So if you don't know how to write a proof, you are screwed. But you won't know it until it's too late to drop the class.
May 2007 
When I was first trying to decided if I wanted to take LA with Bayer, I was thoroughly confused by the conflicting reviews on CULPA. Having just finished his class, I decided to write this definitive review, the only one YOU need to read to know all about Bayer. Let none question, or argue afterwards. First of all, my final score of 85, was in the 33rd percentile, and I got an A. I'm pretty happy with that. The text book is indeed useless. Even though he posts chapters that he supposedly is covering, reading those chapters won't help you at all for the tests. The best way to get high grades is to have one friend go to all the classes, take all the notes (especially on the day before the test when he tells you EXACTLY what's going to be on the test). Then have said friend, teach everyone else taking the class how to do the problems the night before the test. Presto, instant high scores. Although you do have to be careful, he isn't so clear in his explainations of how to do the problems sometimes and your friend can be wrong some times. Also, he will throw curve balls when everyone's getting high scores. To be on the safe side, you'd better better go to office hours and ask him for help so that you are sure you know how to do the problems that he told you will be on the test. Yes, as a lecturer, Bayer goes on a lot of tangents. They are mildly entertaining, but nothing really endearing. These tangents really aren't so bad considerring your mind can take a break and day dream a little. And yes, when you go to office hours, he can be a little condescending, but unless you're the type who expects profs to kiss ass all the time, it's really not that bad. He's perfectly willing to answer your questions and help you. And yeah, if you're really into math, you'll have fun with him during office hours. Once you're armed with knowing exactly what's on the test, with Bayer's own solutions to the problems, and with your friend's notes, you're pretty much set. I mean, 30% of the class gets A's. Com'on.
May 2007 
Honestly, if you spend the time in class to study on your own, you will be better off. You could do very well in this class since the midterms are easy and straight forward. The material is interesting. But I bet that you would NEVER use your class notes, so just study on your own and you'll be more than fine!
Apr 2007 
Prof. Cepiriani is one of the most boring Professors EVA! You would DEFINITELY learn more by studying on your own rather than going to her class. Use the TA office hours, and try your best to do well. There is NO CURVE in this class.
Apr 2007 
So Bayer is very laid back and very approachable. He's friendly and class is relatively interesting, especially considering the pretty boring nature of most topics covered. He grades fairly and generously, his curve is usually centered almost around an A is the impression I got and he himself admits to grading too generously at times. I'd recommend him to anyone with one warning; he kind of likes to use methods that he "creates" for some simple problems (like finding the inverse of certain matrices) and expects to see you use his strategy on exams.
Mar 2007 
Dave is the best math teacher ever. I never thought Linear Algebra could be presented as easily as he did. Have trouble with math? Dave breaks down Linear Algebra so a 3rd grader could understand it. Dave teaches you his own methods for doing complex tasks that other professors wouldn't be able to test on. The guy is a damn genius. He does all of the lecture with no script (like many professors) doesn't screw up, and everything always comes out as whole numbers. People can say his class is easy all they want, but there were a lot people with hurt feelings when they saw that they were at the bottom of the curve. If you're looking to have your hand held like a 16 year old, you're in the wrong place. You're whole grade is based off of 4 tests, so you have to have the dicipline to keep up on your own. This also rules out the possibility that you can just cram for 2 test and ace the class.
Jan 2007 
the worst teacher ever, in my opinion. he tries to make the class interesting but he always fails. midterms and final exam questions are very analogous to the exam questions from the last few years.
Jan 2007 
Mirela Ciperiani is not a good professor. She is generally a nice woman, except when you dare to breathe too loudly in her class (in which case you will get an evil glare and/or be asked if you have a question). Her biggest downside is that she focuses on easy topics for far too long and is then forced to breeze through the tougher/more useful topics. For example, she spent an entire class on matrix addition and simple multiplication... On the other hand, her tests are very, very easy. In fact, they are so easy that she has to curve the class down. BEWARE, Ciperiani considers a straight (no curve) 9095 an A.
Jan 2007 
Take this course if you don't want to work for a good grade and could care less if you've learned the material. Dave comes up with his own ways for solving problems and totally ignores the book, which I've heard is very good but is useless since he teaches different crap. He expects you to come up with generic formulas after doing a problem on an exam. A nearly impossible task. Good thing is, his curve is amazing so your likely to do fine if you don't get good grades on the exams.
Jan 2007 
She is unbelievably nice and sweet. She is also the easiest teacher in the math department. I suck at math, but got an A+ in her class. Barely any work, and her tests come straight out of the homework/book. You will not only learn Linear Algebra in her class, but will have an easy and very unstressful time.
Jan 2007 
Lipshitz LOVES linear algebra. Like the poster next to me, I found Lipshitz to be the best Math teacher I took at Columbia. It was his first semester, straight from Stanford. He's a man of small proportions but his passion for linear algebra is anything but small. Now LinAl as I like to call it isn't really a class anyone looks forward to taking. But Lipshitz does the best job to make this sometimes boring material interesting. He does encourage his students to participate and does answer ALL questions from the suck up questions regarding future material and to the slacker questions that you should have known 2 classes ago. He just wants you to learn the material. Saying he's organized is an understatement. The guy loves using courseworks which is great because I'm one of those people who check courseworks every time I'm on a computer. He posts lecture notes (very helpful supplement to the book when doing the HW), grades a week after their due, and a simple syllabus. But the best thing is that whenever he updates CW he writes what he updated under the Intro section. How many times have your profs updated CW w/o telling exactly what changed, leaving you to stumble through the Lecture or Syllabus or Grades changing wondering what was changed? Now, as the other reviewer did, I'll say something about the negatives of this class. With Lipshitz's ability to make the class pretty interesting and his outstanding organizational skills comes hard HW. Notice the other reviewer liked to start the HW's early. I would've liked to start early, too.. but I can't ever do that. I'm one of those people. I started the Monday before it was due and it would take me from 7pm to 4am to finish them sometimes (working slower as I got more tired and sleepy). It could take anywhere from 5 to 12 hours and on average 8 or 9. Why was the HW so hard? Lipshitz loves proofs which aren't too easy. And the "simple" computation problems are cooked (he loves this word) so that they're not so easy after all. The book that we use (LEON) is terrible and I found myself using MIT's online lectures (recommended) and Google (good luck) often to find solutions to the problems. I'm not a math major and LinAl is never going to be more than a tool for me but I stuck through the class. Half the people from the first day dropped  I'm not kidding. So the class became selfselective and the curve put me in the middle or even slightly above. If you're tired of not learning anything and you want to remember something a week after the final, take this class. Linear Algebra has TONS of applications and we talk about some (Google Pagerank). Lipshitz is the best Math/Sci teacher I've had here. Start the HW early if you can (some people really can't). I'm sure you'll do fine . It's not as much of a terror as I made it out to be. It's just a class called Linear Algebra made me automatically think it was going to be one of the least priority classes that semester when I should have made it a higher priority.
Jan 2007 
I totally second the opinion of the first reviewer. Robert Lipshitz is a pleasure to learn from. It is clear that he really cares whether the students learn the material. He is extremely nice and very approachable. The students who transferred out of his class in the early going should regret having done so. His first couple of problem sets were way, way too long, prompting mass defection of students, but, after the remaining students complained, he paired the problem sets down to a reasonable length. He wasn't the easiest professor I've ever had, nor the hardest, but struck a good balance so that you were challenged but not overwhelmed and discouraged. If you did your homework and attended class you would be prepared for his exams, which were fair, and I would say of about average difficulty compared to other math professors' exams. If you have a chance, take this guy!
Jan 2007 
Despite being a grownup Jeff Spicoli, Professor Bayer's lectures made sense of the potential madness known as linear algebra. I sat in for two graduate studenttaught linear classes during the summer and they consisted either of intense, nonstop writing or class discussions that left the class with more questions than answers. Prof B understands the flaws behind these methods and has developed his own innovative style. He genuinely cares about his teaching and adapts to accomodate. Always entertaining, he often goes off on tangents in his own lectures but his classes never left me confused nor bored. He explains just about everything clearly and the material he discusses ambigulously can easily be researched on the web (i.e. casting out nines on wikipedia). Plus he's very receptive to questions about linear algebra, rock climbing and swimming. With the Fall 2006 class, he decided to have three exams and a final, with the final being the remaining material not covered. Prior to Fall 2006, he had only two exams and a final. The exams and final consist of five problems with the first four being straight forward while the fifth was a bit more tricky. As written by another reviewer, Prof B recommends solving problems from the text but doesn't collect homework so homework doesn't figure into the class grades. In fact, Prof B's lectures diverge so much from the required text that a student could earn a good grade without purchasing it. He suggested this in class and his class website recommends ways to save money for those interested in purchasing the textbook. Prof B grades on a fair curve. However, those who didn't attend class regularly tended to score poorly. I highly recommend Professor Bayer for students willing to attend every class and solve problems without the incentive to do so.
Dec 2006 
Professor Lipshitz is outstanding. He is a phenomenal lecturer; if you go to class, you won't need to study to understand what he says. He is perfectly clear, answers all questions extremely thoroughly, is very accessible, and is glad to help. He is very bright; you can ask him anything and he'll give you a very tothepoint answer. Mathematics is difficult to teach, but Professor Lipshitz has a talent for it. I would not only recommend him, I would strongly suggest him. Lipshitz's problem sets are EXTREMELY difficult, but his tests are VERY easy. He also grades very generously. With these facts combined with his superb lecturing, you'll come away with both an indepth knowledge of the material and a good grade. If you listen to his lectures, you'll be able to do the problem sets, and if you do the problem sets, the tests should be simple. The problem sets will grill you on the mechanics of solving different problems and really make sure you know your linear, and his tests are very conceptual and make sure you "get it". Putting both together, you'll come away from Lipshitz's class on top of your game. Our class ended up finishing the textbook pretty quickly and then went more in depth going entirely off of his lectures (i know it might sound scary to not have a textual reference, but he posts his lectures online, too). However, though I think we tackled much more than other linear classes did, it was very manageable with the strength of Professor Lipshitz's explanations. If you take his class, you'll end up with a greater knowledge of linear than the syllabus suggests. On top of being a great lecturer, professor lipshitz is a downright good guy. You can tell that he's a true math dork, and its very endearing. He's kind, apologetic, friendly, and approachable. Much of what he does is comedic, though I don't know whether he intends it to be; his quirks and mannerisms add entertainment to his enlightening lectures. To him, certain theorems are "miracles", and certain proofs are "beautiful". Its tough not to love him. Take a class with lipshitz while you're at Columbia.
Nov 2006 
Wow. This guy is awesome  the best math professor I've had at Columbia by SO far. He's a young guy, not a full professor yet, but don't think that that detracts from his teaching. He explains everything beautifully, from involved examples to complex proofs. He's always prepared to answer questions, even if the question is only minimally related to the material we're learning. He simply wants to share his knowledge of math with anyone interested, and it seems that he is really rooting for everyone to understand the material. He genuinely loves math, and he clearly wants to pass his love down to his students. Also, he's VERY organized, and always has the problem sets posted online over a week before they are due (before he goes over any of the material they cover). I personally find this very helpful, since they take a while and it's nice to be able to get an early start. Finally, he's pretty much the nicest guy ever. He's extremely approachable, he jokes with the class often, and I get the feeling that everyone in the class really really appreciates him. That said, the class isn't easy. The computational stuff is fine, but Lipshitz goes beyond that and delves into the theory behind the computation. He does make a lot of the more proofbased problems on the homework optional (for up to 5% extra credit at the end of the semester), but you will be required to really think about why/how things work. If you're going to take linear algebra anyway, might as well come away really understanding it! In my opinion, he strikes the perfect balance between challenging theorybased problems and more computational, easier problems. If you've never taken a proofbased class before, don't let this scare you! It was my first time, and I certainly feel that I've come away with a greater appreciation for math.
Jul 2006 
This was a good course with a good instructor. Peters is a nice guy, is very receptive to students, and has a good command of the subject. He also has an engaging and interactive method of teaching, so you feel like you're discovering the material, rather than simply memorizing definitions and properties. He's only a first year grad student, so he sometimes struggled to give a feeling for why the material was important (in the larger, mathematical sense). The class was very computationally oriented, so we did come away with a good number of immediately useful mathematical tools (least squares, GrammSchmidt, singular value decomposition, etc.).
May 2006 
No sense of humor and acts like he's in the army. If you're not prepared to go to class, I wouldn't take him. Class is unnecessary, but he knew that I didn't go to class and when I needed to talk to him about a homework that he lost, he was unfriendly and not helpful. I hear he is helpful in office hours, but I was too scared to go. He'll know if you don't go to class. If you're the kind of student who goes to all the classes, this is a good class for you. Many problems from class on exams.
Mar 2006 
Really found the guy not to be a good teacher at all. I scraped through the class by memorising the practice exams. This wasn't all his fault, as I had been misadvised by my program director, who told me it was an ideal course to take alongside Calc 1 (Calc 3 is a prerequisite) I agree with the other poster regarding the obnoxious student, he was a class A prick
Mar 2006 
A nice guy, but pretty mediocre on the teaching. His accent and handwriting are very difficult to interpret which definitely impact my grades as I have trouble referring to what he is specifically talking about (n or m, v or u, capital or lowercase). His teaching is very heavy on proofs, which is pretty useless if you are not going into mathematics. Skip this class, if you can.
Jan 2006 
Professor Neel is by far the best mathematics professor I've experienced here at Columbia. His lectures are consise and straightforward, and he makes it clear what you need to know. He doesn't do anything that is above and beyond what is asked, but the routine he sets makes for a very well structured course. An occasional joke helps digest the math with a smile on your face. The problem sets have a range of difficulty, starting with easier problems, working up to what is conceptually difficult. If you fully understand the problem sets, the tests just help reinforce what you know. There are no surprises. He emphasizes the concepts and strays away from tedious algebra as much as he possibly can. He is always available in office hours and expresses a desire to help should you seek it out. Overall, this was just a very enjoyable course. If Neel is an option for you, I would seek him out at all costs.
Jan 2006 
Prof. Liu comes prepared to class with clear notes and only a very slight accent you get over by the end of the first lecture. He seems quite shy and many times fails to understand the simple questions from the students. In general, not a difficult class at all...although I'm not too sure how much I actually learned from it. He solely focuses on the methods and completely ignores the applications of Linear Algebra. Many times he fails to give clear instructions on his homework assignments whenever he decides to modify them resulting in a horrible homework average. For once I relied on the exams to get a decent grade. Basically, go to class the second half of the semester to get certain concepts clarified and make sure to memorize the practice midterms/final since they are pretty much exactly like the actual exams.
Dec 2005 
First off, the man can't speak english well at all, which is a big hinderance as a professor and can be extremely funny when annoying nerds try to ask unneccesarily complex questions. That said, I liked the man. He was funny and brutally honest with his students (which I appreciate). I went to class about half the time, and his lectures were only marginally more useful than the book. His midterms and finals were basically the exact same as the practice tests, which (obviously) was very helpful. Anyways, another math professor who can't speak english and a class where you don't have to go, but I thought he was a little funnier than most and I ended up liking the guy (even If I was completely unable to communicate with him).
Dec 2005 
Prof. Liu is new to the Columbia Math Department, and he did an excellent job in Linear Algebra for Fall 2005. He followed the textbook (Linear Algebra, by Steven Leon) closely, but this is good thing because the textbook is wellorganized and contains the right material. First, his lectures were great. Prof. Liu improved upon the textbook by clarifying a lot of the explanations, especially the proofs, where the textbook fell short. He is very wellspoken, and he explains complex concepts very clearly. Professor Liu was also a very friendly guy personally. He should be commended on the fact that he included in the class a detour into mathematical logic. It was very eyeopening, helped a lot with the proofs we had to do, and it will likely pay dividends in later math classes. I have two very minor complaints: First, the class was at 9AM, so everyone was exhausted, and I think that a large chunk of the class just skipped the lecture (probably at their own peril). Second, there was this obnoxious student (who said he already knew linear algebra) who would sit at the front of the room during each lecture and guffaw and sigh at Prof. Liu while he was lecturing. Sometimes, this guy would fall asleep and snore, and sometimes he would get up out of his seat and stamp loudly as he walked out of the room (he said that he was bored and he thought that the material being taught was too easy for him  which raises the question of why he's in this class in the first place). Prof. Liu was always very classy about it and ignored this student. But, honestly, no one should have to put up with this kind of regular disruption, and I wish that Prof. Liu had told this guy to shut up or not attend the lectures. There were weekly homeworks (15%), two midterms (25% each), and a final (35%). The homeworks were incredibly long, and, towards the end of the course, they took at least twelve hours each, due to both the number of questions assigned and the computational intensiveness of the material. Although I did not like it at the time, now I'm not sure if the length of the homeworks was such a bad thing, because they provided so much practice before the exams. The two midterms and final each contained one proof, and I think that everyone found this to be the hardest part. However, if you memorize the proofs in the book and lectures in addition to the proofs that you must do in the homeworks, you can always easily adapt them in order to get the proofs on the exams. The other questions on the exams were fair  not horribly difficult, not too easy. It is very difficult to teach Linear Algebra well  there is a severe tradeoff between understand and rigor. However, I think that this particular textbook (by Linear Algebra, by Steven Leon) and Prof. Liu together pulled it off without much loss of either one. I don't think that you will find a Linear Algebra class anywhere that is better than the one Professor Liu gave.
Dec 2005 
I wish I could say a lot about this professor, but, after the first two classes, I made the rational decision that any CU student should be capable of and stopped going to his lectures. I'm sure Liu might be a great person, but he is as good a professor as a random man picked off the street, given a textbook, and put to the blackboard. His lectures are straight out of the book, word for word, and so are the examples he gives in class. I remember on one occasion he could not even solve the example he gave in class (which, once again, was straight out of the book). So you will probably be more productive reading the material on your own. He has a mild language problem, not too bad though. TAKE LIU'S CLASS if you have insomnia problems. He talks to the board and is hard to hear this droning will help you fall asleep nicely and easily. You can be sure that the rest of the class will be joining you in your nap, too. And I mean, everybody stops going. Out of my 40ish people class, only about 10 actually came to the final review. So if you can manage to get your homework done by studying on your own (and attending lectures would be a big mistake and waste of time), you'll be okay. The course is relatively not too difficult, def. more fun than calc2 but can be demanding if you are weak on conceptual thinking and doing proofs. Liu BORING but innocuous.
Dec 2005 
Professor Neel is an excellent teacher of mathematics. His lectures are systematic: he defines terms, proves theorems, and then provides examples very similar to those on his problem sets. The problems are well chosen and instructive. The difficult problems are conceptually challenging, with very few tedious algebra or arithmetic problems (with the exception of matrix multiplication problems, which are unavoidable in linear algebra). His midterms are comprehensive and fair (I have not taken his final yet). His quirks and odd turns of phrase will make you smile, and add needed humor to the 9:00 am lecture slot. He's not a startlingly innovative teacher, but if you show up to class and do the problem sets you will learn the math, and you might even enjoy it. I have.
May 2005 
There were a lot of proofs in this class, in the lectures, homeworks, and exams. Overall he's decent, but the textbook is just as good as the lecture. Class being at 9 am, many people chose the textbook (including myself). Midterms were fair, final was a little on the challenging side, but I heard he curves generously.
Jan 2005 
He's a really nice guy and really cares. He sets a lot of extra time to help review for the final and is really helpful out of class as well. That being said, he can't teach. He gives plenty of examples and really tries, but they're just not explained well enough. Or, in some cases, they are explained well, but he uses different notation from the textbook and then you can't understand the textbook. However, the textbook is incredibly bad I swear the publishers tried to save paper by limiting the number of wellexplained examples in the textbook (you can even see, the textbook is way smaller and thinner than most even though it still costs you like 80 bucks). We didn't even finish what we were supposed to cover, and, without a good textbook to back up what we "learned" in class, this was nearly a complete waste of time.
Jan 2005 
By far, the BEST!!! Math has NEVER been easier, and the professor was really really good. (One good hint about this class: there are a couple lectures that I found a bit challenging: when he gets to talking about subspaces, and you don't really understand what he's talking about, don't worry...no one really understood the first time. That's when you start going to MIT's Strang website and start watching his online lectures which are just fantastic). I did the problem sets (~2 hours a week), and watched about 10 of Strang's (author of the book at MIT) online lectures, and easity aced the class.
Jan 2005 
Cao was decent. I'd agree with the previous reviewer that going to class is a question of personal preference. He mostly used examples that were explained in the textbook. The only things to be careful of are his personal notation preferences  they're not quite the same as the book's all the time. Also, the beginning of the class was incredibly easy so I didn't take it very seriously. I got a 90something on the first midterm without studying, but, while continuing to consider the class a joke, I bombed the second midterm. Oops. So it's important to be conscious of the fact that the later material is a good bit harder than the initial stuff. Don't blow off the work and you should be fine.
Jan 2005 
Well, Professor Dzhamay is definitley one of the nicest professors you will ever meet. He is dedicated to helping his students understand math, often giving extra help outside of class (even in his office at night if he is there  he has definitley helped us at 10PM just because he was already there). However, there is a definite downside to his class. His problem sets are quite involved and difficult, getting much harder as the semester progresses. He explains the material well and you think you understand it, until you try to do the homework. Often the TAs in the helproom cannot get the questions right. The midterms were extremely long, and downright insanely difficult usually with a low mean and a fairly crappy curve. The final was cumulative, and was so hard that it would make you think you hadnt learned anything. Linear Algebra itself is not a hard subject, but his course is very demanding and it makes it very difficult. As a person he is great, but the exams are just too ridiculous for my taste. Do not take a class with him if you are looking for an easy decent grade, because you will get violated. If you take his class, you will learn the material very comprehensively and will be able to use it in any other field that you would need it for, but your grade will not reflect how much you actually learn.
Jan 2005 
Linear Algebra is a hard class, and Prof. Cao doesn't make it easier. But at least he doesn't make it harder, which is better than any other math teacher I've had a Columbia. He is fairly clear as a lecturer and he often gave numerous "hints" about what would appear on the exams. However, other than the "hints", there is very little reason to attend class. If you are able to stay awake throughout his lecture, then it is worth attending class just for the hints. Otherwise, just stick to the book, and save yourself the trip to class. There's no reason to avoid him, although there is really very little reason to take him either.
Dec 2004 
This is the most caring professor I have encountered at Columbia. As previously stated, he starts each lecture asking if anyone has a question. He has office hours, but he will really help you anytime. I emailed him constantly and he responded at all sorts of weird hours. One thing is that he is a lot harder than the other linear algebra professors, definitely not an easy A at all. However, if you want to learn take this class. A lot of people were scared away. We started with 50+ students and ended with 27. The homeworks are graded relatively harshly and can take a long time (anywhere from 1.5 to 4 hours per weel...maybe not soooo long). I'd definitely say take a class with him, he is not one to avoid. His marking I do not feel was harsh at all. His tests are all impossible, but he definitely seems to take that into account when making up the marks.
Dec 2004 
A good professor, but not the best teacher. Prof. Dzhamay is approachable, nice, and genuinely excited about the material. However, he either covers the concepts through general theory or examples so specific that it is difficult to apply themnothing in between. The tests are ridiculously hard and long, but generously curved.
Dec 2004 
I took linear algebra with him that met once a week from 4:10 to 7:00. Three hours is pretty brutal, but the class was pretty good. Some of the material is difficult to grasp, but that's linear algebra. He does a very good job of keeping the class interesting, and I found him a good teacher. I learned a lot about linear algebra. He is not easy though, as I found the tests very difficult.
Sep 2004 
this guy doesn't no how to teach, he gives a lot of abstract explanation in linear algebra with lots of unfamiliar symbols in his explanation which makes his lecture almost impossible to understand, he definitely does not have experience in teaching as my class is made up of only 20 ppl and he still insisted a curve for my class, my summer class experience with him was simply horrible, i would say , try to avoid him at all means, he is the worst teacher i have ever come across in my life
Jul 2004 
I honestly really enjoyed Prof. Bayer's class. True, you can get through it with a good grade without understanding linear algebra if you just learn a few tricks and study past exams. But if you do want to learn the material, Bayer teaches it. And explains it in several clear, logical ways in addition to the standard book method. I'm not a math major, but I was interested in the subject and he treated me with nothing but respect. Then again, I went to class and didn't try to cheat. Overall, Prof. Bayer's a odd but pretty entertaining guy. I'd go to class wondering what today's very quotable random aside would be. If you need to take the class, you can get by without a huge effort. If you're interested in the subject, you can learn it in his class's much more relaxed atmosphere, provided you don't depend on either just the lectures or just the text.
Jun 2004 
David Bayer's class was a colossal waste of time, and he is an arrogant, pompous, condescending, disorganized mess as a professor. First of all, his presentation of the material is sloppy. He's so busy trying to sneak in stupid anecdotes from his time as a student or making political jokes (which are always innappropriate, but particularly irritating when you disagree with him), that he wastes a good portion of class time. He often makes mistakes on the board, and if you don't catch them then you're screwed. He also teaches the class as though he's talking to himself, as he scribbles away on the board babbling as if he's just working out a problem and we're not even there. He is so annoying, and the lecture is so useless that if you are going to class, you'd better double up on the Ritalin beforehand. Because of this, few people go to class, and fewer can sit through the whole lecture. Homework is assigned, but not collected or graded and solutions are not posted. Same goes for sample exams, for the first midterm he didnt even post any samples; later on, samples were available but solutions were not. Basically, you learn nothing in class, you have no indicator of your progress or problems (since there are no quizzes/homeworks or solutions) until the individual exams, at which point 3040% of your grade is at stake. While the material appears to be easy, Bayer is very sneaky and bitter about the lack of attendance in class, so he purposely creates traps for people to do poorly. Even if you walk out of an exam confident that you did well, don't be so sure. He also arranges the curve so that about 5% (maybe more) of the class will receive an F as a final grade (keep in mind, that Fs are irrevocable once assigned). David Bayer does not use his class to teach Linear Algebra (Lord knows you don't learn it!) but to make a point. He thinks that since no one comes to class (which wouldn't be the case if he wasn't too lazy to grade problem sets or give quizzes) he should impose some consequence after the fact. While he doesn't know who anyone is in the class, he deals out this punishment blindly just to make an example of his students. He even came up with a complex scheme to catch people cheating which he devoted an entire semester to; perhaps if he put that much effort into teaching the course none of these other schemes would be necessary. He also has no respect for anyone who is not a math major (or something close to it). Overall, a stupid class, a horrible teacher, and an overbearing hidden agenda...avoid it like the plague.
May 2004 
If someone says that Bayer is a good teacher then he/she does not know anything about linear algebra. Instead of teaching the concepts and the different approach of linear algebra Bayer teaches you faster ways to solve his easy problems. He posts previous exams which are almost identical to actual exams. I really did not learn anything about linear algebra. However, I got an A in this class. I never understood the material conceptually but I was able to solve his problems. More precisely, easy class, bad teacher, you'll learn nothing about linear algebra. If you want to learn something go with the other section. If you want an A take his class. There are no homeworks. So be careful with the exams because if you screw up one question in one of the exams that may cost you almost a grade.
Apr 2004 
Best professor I have encountered at Columbia thus far (and ironically enough, he is a Barnard professor). He knows linear algebra inside out, and for that reason, he is able to teach it so well. He knows exactly how to approach the material and make it interesting. I really wasn't expecting to enjoy this class, but it turned out to be GREAT! Without a doubt, if you are planning on taking this class, do what you can to take it with Prof. Bayer.
Nov 2003 
I want to agree with everything else that is said and add to it a little bit. Professor Wang is an excellent math teacher. The only bad part about this class is the homework. It takes about 45 hours to do a week (sometimes even longer) but the midterms seem to be a step below what is asked on the homework so if you do fine on the homework you will do well on the tests. I definately reccommend him as a teacher and would take any class in the future that he is teaching.
Nov 2003 
Absolutely the best math professor I've ever had. The thing that puts him above the rest is that he explains what you are doing when you are proving an equation, or anything like that. Other professors don't do that, they just do the proof. Wang makes math both understandable and interesting.
Sep 2003 
Imagine the best math teacher humanly possible. Okay, Wang is a notch or two BETTER! He explains everything from the bottom up (it's impossible to "get lost" along the way), and always stops for questions. Don't be afraid of his Asian appearance; Wang is quite articulate and he even speaks at an audible volume. I only wish I'd had him for every math class since 1st grade. I challenge anyone to take this class without enjoying it and its patient, gentle, thoughtful professor.
Jul 2003 
Bayer's class is pretty straightforward. The two midterms and final are pretty much the same as previous years, which are online as practice tests. Minimum stress class, with easy grades.
Jun 2003 
Professor Bayer was great! He has really clear and organized lectures that runs through the material he stresses. There are also plenty of sample/practice exams for midterms and the final, and the format is essentially the same for the real exams, just different numbers. He's pretty easy with grades, so if you have a choice, definitely take a class with him! He's clear and organized, and speaks in a language that students actually understand, is easy at grading...etc. I mean, what else can you ask for?
Mar 2002 
MuTao was definitely a pleasant surprise as math teachers go. Don't be scared off by the foreign name; he speaks English very well, and his handwriting makes up for the very occasional deficiency. His explanations were generally pretty helpful and clear, and he always tried to be receptive to questions in class. Tests were fair. It certainly wasn't exciting, but it had pretty much everything you look for in a good math class.
Nov 2001 
A rare Chinese Math professor who speaks good English. Eloquent and clear lectures. Thoroughly knowledge of material. Reasonably responsible and responsive. Not extremely nice but never mean. Harvard PhD but not arrogant. Overall a good professor and extraordinary compared to the others in the Math Department.
Nov 2001 
Mu Tao Wang is a very good professor. He speaks English just fine, with just the slightest accent. He knows everything backwards and forwards, and is responsive to questions. There is nothing extraordinarily good or bad about him, he is simply a good professor and there is eminently capable of teaching any class you are taking well.
Jan 2000 
Patrick Gallagher is not a young professor, but is far more down to earth and happier to work with students than most professors. He speaks clearly and not too quickly, responds to questions well, and lectures well. While he may get a bit off track with a not as relevent proof that takes up much of a lecture, he also displays suprising humor from time to time. This all makes him an excellent choice for lower level math courses.