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The Essence of Linear Algebra [video]

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Re: The Essence of Linear Algebra [video]

#21
post #11

The introductory video is absolutely right. I studied linear algebra at university but it's been only recently when I decided to re-learn it that I have finally got it, and for that understanding the geometry behind linear algebra is crucial. Now I find myself thinking about problems in terms of matrices and vectors and making all sorts of deductions, even writing mathematical proofs, which I never thought I would.

Sounds like you'd enjoy playing around with array programming languages like APL (dyalog, microapl), J, or Q w/ kdb+(kx systems). As one of the oldest paradigms, there are a few open source and proprietary-commercial offerings.

Re: The Essence of Linear Algebra [video]

#23
These videos are a treasure and I watch every new video that 3Blue1Brown puts out. They saved me last year when I had a graduate-level numerical linear algebra class and was struggling to grok the true meaning of all this linear algebra stuff (an embarrassing situation for a computer-graphics guy to be in!). Things that had always been "insert formula X to get result Y" started making a lot more sense.

The videos also make me angry because it frustrates me that such explanations were not available to me earlier in my life. What is it about the state of math education that this kind of explanation is not there in every class?

Re: The Essence of Linear Algebra [video]

#24
I took a Linear Algebra course last year: the relationship between linear transformations, matrices and basis was carefully explained and also given a clear geometrical interpretation but the determinant was introduced as some magical matrix function with all the property we needed (multilinear, alternating, ...) and the just proceeded to prove its existance and uniqueness. I have never even thought it was related to area before. Also the justification (in chapter 8 part 2) for the formal determinant calculation to obtain the cross product is amazing.

I'm still not sure what the essence of the cross product is: how it is related to divisions algebra (quaternions and octonions) and how bivectors fit in the context of general vector spaces. This was not in the scope of the course however.

Re: The Essence of Linear Algebra [video]

#25
I have to echo what the posters above have already said: I learned linear algebra way back in the day, and it's only now, watching the videos, that I suddenly _got_ matrix multiplication etc. I'm not joking when I say I almost cried with joy. These videos are incredible, and I'm so sad I didn't see them earlier in my life.

Re: The Essence of Linear Algebra [video]

#26

These videos are a treasure and I watch every new video that 3Blue1Brown puts out. They saved me last year when I had a graduate-level numerical linear algebra class and was struggling to grok the true meaning of all this linear algebra stuff (an embarrassing situation for a computer-graphics guy to be in!). Things that had always been "insert formula X to get result Y" started making a lot more sense. The videos als…

I think the problem is that until recently, the costs of producing and distributing these videos was much higher. With the rise of YouTube, it has become possible for a Salman Khan or John Green to become a celebrity eduvideographer without a lot of capital investment.

One of the flaws of common core is that it seems that the proponents did a poor job of marketing it to the broader adult community. Doing so would have:

1) Given parents the answer to "why is this change happening at all? I learned math just fine as a kid!"

2) Engaged some people in trying to think about the best ways to explain that material and accelerated the formation of a community that gains status with each other by coming up with better and better explanations of the common core curriculum.

Re: The Essence of Linear Algebra [video]

#29
There are great, like everyone 3b1b makes. I'm comfortable with lin alg but I'm skimming them to see what he's chosen to emphasize. I love - for example - that he points out how "span" becomes more interesting in 3d, obv with an example, and the underemphasis on the mechanical operation of matrix multiplication relative to the geometric one.

I do wish the formulas for the dot product and determinant were derived from the geometric explanation, rather than justified with it afterwards. I always appreciated that in classes.

There are some more advanced topics in lin alg that I would have loved to see get the full visual intuitive treatment when I was learning these things.

- SVD, because it's more general and less pathological than eigenvalue decomposition, and often more useful. - A linear transformation as consisting of (I think, it's been a little while) a choice of eigenvectors and eigenvalues, "divided" by the extra degrees of freedom from duplicate eigenvalues. - The "taxonomy" of normal matrices and the polar decomposition (obviously comes after complex matrices)

And there's a nice visualization of the mechanical algorithm of matrix multiplication that looks way more "plausible" than the normal one: draw your two input matrices and your output matrix on grids on 3 sides of a rectangular prism around a corner. Then each value in the output matrix is the dot product of the vectors that intersect at that coordinate, and the whole thing only looks right if all the dimensions match up correctly.

Re: The Essence of Linear Algebra [video]

#30
post #3

Wow thanks for posting this link. I was wondering does anyone know more channels (or any other resource) just like this one, for Statistics and Calculus?

These are fun too, not directly calculus but some infinities and series videos https://www.youtube.com/channel/UCOGeU-1Fig3rrDjhm9Zs_wg

Wow, my Saturday afternoon just disappeared.
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