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A 2020 Vision of Linear Algebra

ocw.mit.edu

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Re: A 2020 Vision of Linear Algebra

#131

Post by Bootvis »

heinrichhartman wrote: post #27
crdrost wrote: post #21

So I like this outline. It is very MIT-ish where there is a sense of teaching someone to solve practical engineering problems with matrices. But, I do foresee some difficulties. One thing that I find really difficult, for example, is that I take undergrads who have had linear algebra and ask "what is the determinant?" and seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues…

> seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues." I found that the "best conceptual" answer depends a lot on taste, and what concepts you are familiar with. In this case: - Calculating exact eigenvalues of matrices larger than 4x4 is impractical, since it requires you to solve a polynomial of degree >4. - The EV exist only in algebraically closed fields (complex numb…

I don’t have an intuition for these concepts I’m afraid (I probably should watch the videos). What I don’t see for instance is how this relates to the fact that a matrix A with det(A) = 0 is not invertible.

Re: A 2020 Vision of Linear Algebra

#132

Post by kragen »

praptak wrote: post #29

I'm currently trying to grok the finite element method. Gilbert Strang's explanation of the transition from the Galerkin method to FEM did more for me in terms of connecting the dots than anything else I could find on the web. And it wasn't even a lecture, just a kind of an interview. I think it's this one: youtube.com/watch?v=WwgrAH-IMOk

https://www.youtube.com/watch?v=WwgrAH-IMOk

I feel like I don't really understand his explanation, because it's kind of vague. But I think that might be because you've seen the equations dozens of times, and I haven't seen them at all, so you were prepared to understand the video.

Re: A 2020 Vision of Linear Algebra

#133

Post by qorrect »

cat199 wrote: post #125
rcthompson wrote: post #120

Earlier quoted context omitted.

I had a very similar situation in my linear algebra course: in hindsight, I would literally have been better off teaching myself the material than listening to the professor. To this day it's still the main weak spot in my math/stats knowledge base. I'm really interested to check out these lectures.

haven't watched but based on summary these seem more about pedagogy than the subject - that said there is a full course worth of videos taught by same professor that are pretty good

Oh yeah Gilbert Strang's original course is amazing , https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...

Re: A 2020 Vision of Linear Algebra

#134

Post by qorrect »

raz32dust wrote: post #4
knzhou wrote: post #2

Gilbert Strang's linear algebra course blew my mind back in high school, and I still use insights from it every day. Strang has a particular lecturing style where he approaches every topic several times, often beginning many lectures before the main treatment. At first I thought it was a bit confusing, but later I realized it helped build fluency, just like a language class. I'm really thankful to MIT OCW for putting…

+1, I was very grateful to MIT OCW because when I learned Linear algebra, I could not have afforded it. Later when I got a job, I donated to OCW and I bought his book full price from his own site [1] just as a tribute to the guy. [1] https://math.mit.edu/~gs/linearalgebra/

Hey me too! ( All of it )

Re: A 2020 Vision of Linear Algebra

#135

Post by metreo »

rcthompson wrote: post #120
katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

I had a very similar situation in my linear algebra course: in hindsight, I would literally have been better off teaching myself the material than listening to the professor. To this day it's still the main weak spot in my math/stats knowledge base. I'm really interested to check out these lectures.

Intermediate Stats writing out Chi-Squares by hand on exams literally ended my academic inclinations. I had been using software to do this for a while and something about the process of being forced spend hours memorizing how to by hand just to "earn" a letter rubbed me the wrong way. I absolutely know much more about Chi-squares then I'd ever need to, possibly an imprinting of the bad experience.

Re: A 2020 Vision of Linear Algebra

#136

Post by mturmon »

srean wrote: post #24

I am familiar with the material of linear algebra but haven't read his books. Could someone who has absorbed linear algebra from different sources and familiar with Strang's books comment on what's good and bad and unique about them. In my time I had picked LA from Ben Noble, Halmos and Axler and the computation side of things from Golub & van Loan.

I like your bibliography!

Ben Noble's book was my entry to LA. I was an undergraduate and involved in a research activity that demanded a lot of knowledge of the eigenvalue problem. The concrete approach in that book helped a lot.

It was only later on that I took a class based on G&vL (implementing a bunch of basic LA factorizations in Matlab), and in my spare time read Halmos's book. I understand the coordinate-free algebraic approach, but I work on applications and that viewpoint has not stuck with me. The stuff on numerical accuracy in GvL really did stick, OTOH.

From the comments here, and Strang's book's table of contents, I gather that his book (which has a lot of fans) has a concrete geometric approach.

Re: A 2020 Vision of Linear Algebra

#137

Post by crispyambulance »

ansible wrote: post #111
katzgrau wrote: post #72

I had very intelligent linear algebra professor in college but he was, in my opinion, a very poor communicator. I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. Unfortunately I'm one of those people who tends to reject the…

> I paid attention to lectures and stared at the text, but couldn't really understand the material. For the first part of a linear algebra course, students who don't mind blindly following mechanical processes for solving problems can do very well. I had a similar, though sort of opposite experience. In high school, I breezed through the material, and started teaching myself calculus during the summer to prepare for…

I think that whether or not students do well, there's a common theme in university math curricula for non-math majors. Basically, math gets taught as a kind of "toolbox" of techniques. Unless there's a strong follow-up in subject matter courses (for example in engineering coursework), those math skills effectively evaporate.

Some places use a rigorous "proof-theoretic" approach in math curricula. It's much harder and takes more time, but it's better than merely grinding on hundreds of easy calc-101/diff-eq problems, because students gain an understanding that doesn't erode as easily once they forget "the tricks".

More CS, engineering and science students, IMHO, should dabble in math department courses beyond the the usual "required" sequence for their majors. It can be eye-opening and provide long lasting benefit to take a hardcore real-analysis course, abstract algebra or a number of other courses in math.

Re: A 2020 Vision of Linear Algebra

#138

Post by vlasev »

dmvaldman wrote: post #103
nothis wrote: post #94

Earlier quoted context omitted.

I guess a mathematician might look down upon sticking with 2d and 3d stuff because it leaves out all the interesting things that happen at 92382 or negative infinity. But yea, matrices are basically just a convenient way to write rows and rows of "ax + by + cz...". In linear algebra, you just do it so often, people made up their own syntax. And nothing can visualize it like transforming graphics, IMO. You don't even…

I think these are good examples, but to me "linear algebra thinking" lies in it's generality. For example, the derivative is a linear operator, so how do you write it down as a matrix? Google's PageRank is a solution of a matrix equation, what does that matrix represent? Etc.

> For example, the derivative is a linear operator, so how do you write it down as a matrix?

Consider polynomials in X of degree up, but not including N. The powers 1,X,...,X^(n-1) form a basis. Then the coefficients of the polynomial can be put in a column vector. If D is the derivative operator, DX^n = nX^(n-1), so the derivative matrix can be expressed as a sparse matrix with D_(n,n+1) = n. Visually, it's a matrix with the integers 1,2,...,n-1 on the super-diagonal.

You can also see that this is a nilpotent matrix for finite N, since repeated multiplication sends the entries further up into the upper right corner.

You can extend this to the infinite case for formal power series in X, too, where you don't worry about convergence.

> Google's PageRank is a solution of a matrix equation, what does that matrix represent?

Isn't it just the adjacency matrix of a big graph?

Anyway, I agree with you. Matrices and linear algebra is a really good inspiration for higher level concepts like vector spaces and Hilbert spaces and so on. That's where the real power lies. But even in such general domains, matrices are often used to do concrete computations on them, because we have a lot of tools for matrices.

Re: A 2020 Vision of Linear Algebra

#139

Post by heinrichhartman »

Bootvis wrote: post #131
heinrichhartman wrote: post #27

Earlier quoted context omitted.

> seldom get back the "best" conceptual answer, "the determinant is the product of the eigenvalues." I found that the "best conceptual" answer depends a lot on taste, and what concepts you are familiar with. In this case: - Calculating exact eigenvalues of matrices larger than 4x4 is impractical, since it requires you to solve a polynomial of degree >4. - The EV exist only in algebraically closed fields (complex numb…

I don’t have an intuition for these concepts I’m afraid (I probably should watch the videos). What I don’t see for instance is how this relates to the fact that a matrix A with det(A) = 0 is not invertible.

The geometric version is the most intuitive for me:

If the volume of the prallel-epiped is zero, then there will be directions in the target space, that you did not hit. Hence he matrix can not be invertible.

Re: A 2020 Vision of Linear Algebra

#140

Post by Koshkin »

swiley wrote: post #76
pengaru wrote: post #66

It's ridiculous how much random college-level linear algebra textbook material I stared at before things clicked in the course of just jumping in and exploring 3D graphics and writing my own 3D vector, matrix multiplication and 3D transform headers and using them in making some games in plain C. At some point it's like "Wait, is linear algebra really just about heaps of multiplication and addition? Like every dimensi…

I think the best thing anyone told me about linear algebra was that “matrices are just the coordinate form of a linear map.” So applying the map is equivalent to multiplying it’s matrix etc.

Reality, as always, is not that simple. Matrix analysis is a huge area in itself; and matrices can also be used to represent tensors (which generally are not seen as linear maps) and some other things.
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