Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

A 2020 Vision of Linear Algebra

ocw.mit.edu

Post reply on HN
Print view View original

31–40 of 205 posts

Re: A 2020 Vision of Linear Algebra

#31

Post by bluquark »

CameronNemo wrote: post #12
willbw wrote: post #5

Earlier quoted context omitted.

Of course it is up to you what you do with your money but, they have a $17.5B endowment so there may be more needy causes if you were so inclined.

Is there a better way to incentivize educational institutions to offer free content?

A modest proposal: ban donating to them so they need to radically increase their student body (online or offline) to earn their keep with tuition, instead of relying on the donations of the extremely rich parents of legacy admissions students.

Re: A 2020 Vision of Linear Algebra

#33

Post by abdullahkhalids »

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…

Re: your last point, the determinant is the matrix function, which is fully anti-symmetric under permutation of rows and columns i.e. swapping a pair of rows or a pair of cols pulls out a minus sign. The definition of the determinant is related to the alternating representation of the symmetric group.

The permanant [1] is the matrix function which is fully symmetric, so permuting any rows or cols leaves it invariant. It emerges from the identity representation.

Finally, partially symmetric matrix functions are known as immanants [2], defined using the other irreps of the symmetric group.

[1] https://en.wikipedia.org/wiki/Permanent_%28mathematics%29

[2] https://en.wikipedia.org/wiki/Immanant

Re: A 2020 Vision of Linear Algebra

#36

Post by brmgb »

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ?

My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

Re: A 2020 Vision of Linear Algebra

#37

Post by mjburgess »

gfxgirl wrote: post #25
roenxi wrote: post #11

It is interesting to compare this with 3Blue1Brown's linear algebra introduction on YouTube. He seems to have been the only mathematician who has actually mastered the medium; linear algebra lends itself very well to animations. The mathematicians don't understand how badly they need to animate some of these concepts.

3Blue1Brown's linear algebra animations were fun to watch but they did almost nothing for me except the basic fact that the "linear" part means lines. The rest was effectively preaching to the choir so those that already know linear algebra nodded their heads and idiots like me were still flummoxed

That's interesting. I do think 3B1B's goal is probably to build better intuitions in people who already know it.

Re: A 2020 Vision of Linear Algebra

#38

Post by chubot »

brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

Yeah I would call this the engineering approach (matrices) vs the mathematical approach (algebra).

I took like 3-4 courses in the US involving the engineering approach, starting in high school and continuing through the college as a CS major. That was all that was required.

But I also like algebra, so I happened to take a 400-level course that only math majors take my senior of college. And then I got the group theory / vector space view on it. I don't think 95% of CS majors got that.

I don't think one is better than the other, but they should have tried to balance it out more. It helps to understand both viewpoints. (If you haven't seen the latter, then picture a 300-page text on linear algebra that doesn't mention matrices at all. It's all linear transformations and spaces.)

What country were you taught in? Wild guess: France?

Re: A 2020 Vision of Linear Algebra

#39

Post by ojnabieoot »

brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

I would not describe his approach as “the US approach” but it is a pretty standard approach to introducing linear algebra to engineers, which is the theme of the course.

I was also taught linear algebra this way, by an applied mathematician with a background in chemical engineering:

- start by solving Ax=b with row reduction

- develop theorems about linear independence and spanning sets of vectors based on these exercises

- introduce the determinant from the perspective of linear systems (rather than eg geometry or group theory)

- eigenvectors and eigenvalues

Later I switched from physics to math and TAed a more “algebraic” approach involving groups/rings/fields. But the matrix-first approach was more helpful for both my physics coursework and later courses in numerical linear algebra.

Re: A 2020 Vision of Linear Algebra

#40

Post by tadhgds »

brmgb wrote: post #36

After watching this and having read the comments, I am quite puzzled by the approach American seem to take to linear algebra. Are matrices viewed as the core of the subject in the USA ? My country curriculum introduces linear algebra through group theory and vector spaces. Matrices come later.

I can't say whether or not it is the standard approach but I do know that it is very common in many countries to teach a linear algebra course that is heavy on matrix operations, that you can come away believing that linear algebra is somehow _about_ matrices and their operations. I know many in my university class seemed to believe that.

A book I enjoyed is Axler's Linear Algebra Done Right[0], in which, if I remember correctly, doesn't contain a single matrix.

[0]https://zhangyk8.github.io/teaching/file_spring2018/linear_a...

Post reply on HN