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I believe I've had the "clicked" moment. It's all about basis vectors, right?

I.e., What would these given vectors mean if their basis vectors were these; what are their vectors in terms of these basic vectors. Matrices can be seen as simply groups of basis vectors. That is why multiplying two matrices together corresponds to composition of their respective transformations; because, when doing AxB, you are rewriting the B's basis vectors in terms of A, and the and thus this new matrix has the basis vectors that are exactly what you would get if you had done Av Bv (v is a vector) separately. And matrix by vector multiplication is rewriting with the Matrix' basis vectors.

E.g., Iterating with Markov matrices is simply generating new points in space as if their components were weighted by the probabilities in the markov matrix. So the markov matrix is also a list of basis vectors. Also the iterative power method for finding eigvenvectors converges because we keep generating vectors that approach the analogues [1, 1, 1] vector.

Also gaussian elim./equation stuff is simply a special case of all this? It was first motivated by those problems but then we figured out it was all about linear mappings with these basis vectors, right?

Moderators, please don't close this post if it doesn't conform to the math exchange decorum/etc. I need to know if my insight is right.

And vector spaces are simply generalizations of the spaces of number vectors. Anything linear with an analogous plus/scale is amenable to this theory - all the stuff proved about plain old number vector spaces.

EDIT: my god, it all makes sense now. the stretching, rotating, shrinking, composition, relationship to systems of linear equations.

EDIT 2: Without linear independence in vectors, we don't gain extra dimensions in our transformation.

And is the eigenvalue basically a descriptor of what happens to area before/after transformation? Is it the analogous 1x1 square in the new space? I know this is a ramble, but can anybody understand anything I'm trying to convey ?

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  • $\begingroup$ Basis vectors are important, but they are not the end-all-be-all of linear algebra. Several things you write don't make sense or don't sound correct. You write for example "AvBv", which does not make sense for most examples. "Matrix by vector multiplication is rewriting with the Matrix' basis vectors" I disagree, its more than that. $\endgroup$
    – JMoravitz
    Jul 4, 2018 at 19:18
  • $\begingroup$ I should have set A(B(b)). But it can always be seen that way? The definition is so natural when I see it as all about the basis vectors. Anyways, apologies, I just graduated from undergrad so my mathematical background is very limited. $\endgroup$
    – Alex
    Jul 4, 2018 at 19:22
  • $\begingroup$ The idea that a matrix is a row vector whose entries are column vectors is a good one; similarly that it is a column of row vectors. More generally, there is a notion of partitioning a matrix into blocks; i.e. a block matrix. $\endgroup$
    – user14972
    Jul 4, 2018 at 19:47
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    $\begingroup$ I'm voting to close this question because it's not really a question. That said, congratulations on your aha! moment about bases. There's more to linear algebra than that, but it's a good place to start. $\endgroup$
    – Ethan Bolker
    Jul 4, 2018 at 22:41

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Congratulations on an epiphany, but I think you've only just figured out the utility bases. Having a feeling for that makes it easier to learn linear algebra, but that is certainly not the whole of linear algebra. And a lot of undergradutes never really figure out the utility of bases either, and appear to be convinced that they're another torture device invented by mathematics teachers to make things hard.

Choosing a set of basis elements amounts to picking coordinates with which to express the space and its transformations. On one hand this makes things "more concrete" and lets you reason elementarily about the objects, but on the other hand it can be "too zoomed in" and can distract you from more core concepts.

You can also learn about linear algebra from the geometric viewpoint and downplay the use of bases and coordinates. One such book is Kaplansky's Linear algebra and geometry: a second course. It's important to know about this half of the topic, too, or else all the bookkeeping with bases only amounts to a lot of busy-work.

Maybe another epiphany you can immediately have is "oh, linear algebra isn't just about bases and matrices?" I have a feeling lots of students could use THAT epiphany...

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