A 2020 Vision of Linear Algebra

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." Like, this is math, the best answer should not be the only one, but it should be ideally the most popular. We would consider it a failure in my mind if the most popular explanation of the fundamental theorem of calculus was not some variation of "integrals undo derivatives and vice versa". I don't see this approach solving that. Furthermore there is a lot of focus from day one on this CR decomposition which serves to say that a linear transform from R^m to R^n might map to a subspace of R^n with smaller dimension r < min(m,n) and while in some sense this is true it is itself quite "unphysical"—if a matrix contains noisy entries then it will generally only be degenerate in this way with probability zero. (You need perfect noise cancelation to get degeneracies, which amounts to a sort of neglected underlying conserved quantity which is pushing back on you and demanding to be conserved.) In that sense the CR decomposition is kind of pointless and is just working around some "perfect little counterexamples". So it seems weird to see someone say "hold this up as the most important thing!!"

Have not watched the videos yet, but that seems to me more like an 1820 vision of linear algebra :-)

If you look at the order of topics in his book "An Introduction to Linear Algebra", you will find the topic "Linear Transformation" way back in chapter 8! Even after the chapters eigenvalue decomposition and singular value decomposition. But understanding that a matrix is just the representation of a linear transformation in a particular basis is probably the most important and first thing you should learn about matrices ...

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

> 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 numbers), while the determinant itself lives in the base field (rationals, reals).

How about:

- [Geometric Determinant] The determinant is the volume of the polytope (parallel-epiped) spanned by the column vectors of the matrix.

- [Coordinate Free Determinant] The determinant is the map induced between the highest exterior powers of the source and target vector spaces (https://en.wikipedia.org/wiki/Exterior_algebra)

- I think there is also a representation theoretic version, that characterizes the determinant as invariant under the Symmetric group acting by permutation on the columns/rows of the matrix.

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