I highly recommend http://www.axler.net/DwD.html for developing a good intuition about what eigenvalues and eigenvectors actually are.

Not sure if they’re useful (I haven’t watched them), but Axler made a series of videos about the core content of his book https://www.youtube.com/watch?v=lkx2BJcnyxk&list=PLGAnmvB9m7...

I don't think that's really true. The book itself puts it much more simply and directly, at the very beginning in 'Preface for the Instructor':

"You are about to teach a course that will probably give students their second exposure to linear algebra."

Axler tries to teach students how to understand linear algebra.

If all that you want to do is use it, the prospect of that understanding may not be very motivating.

My point is that the comment that you quoted from the preface in no way changes the fact that the book's point is to convey an understanding of linear algebra that is primarily of interest to people going on in math.

Now I happen to think it is the right way to understand linear algebra and is how people in other fields should think about it. Because it is easier to figure out again if you've not done it in a while. But this point of view is primarily going to motivate would be mathematicians.

Sure, but the topic is 'Given Strang, what's the deal with Axler'. It's a perfectly sensible question in its own right.

It’s certainly not aimed at numerical analysis students, or engineering students, or physics students. (Which isn’t to say that those students can’t take pure math courses if they want.)

A (very imperfect) analogy might be something like the GoF book vs Peter Norvig's essay on design patterns.

Another important point to appreciate is that eigenvalues and eigenvectors frequently require complex numbers. For example the eigenvalues of [[0 1] [-1 0]] are i and -i. This geometrically correlates with some kind of "rotation".

[edit: the rest of my original comment was not correct]

If an eigenvalue is unique, the corresponding eigenvectors are on a line. If it appears twice, the corresponding eigenvectors are [edit: could be] on a plane. In general, they span [edit: could span] a subspace of dimension equal to the multiplicity.

The generalized eigenvectors are on a plane.

The case where the eigenvectors do not span the space corresponds to matrices whose Jordan normal form has upper triangular bits.

I would consider rephrasing the article as you do in the above comment, "when you find an eigenvector for one eigenvalue, then you can construct infinite number of eigenvectors by scaling that one".