Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?
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We had to learn a bit of Matrices in High School. And actually in my opinion a high school student should be able to tackle most parts of matrix algebra and some part of solving system of linear equations. Although I would advice against delving deeply into the the theory, vector spaces and linear transformations etc. Although there are not many books that restrict study to matrix algebra and avoid the discussion of Linear Algebraic theories. I can think of two books. But none of them are particularly great:
If you are a purist you will not find just a superficial course on Matrix theory amusing. You'll find it annoying in fact. But if you can get through it you'll find there are fantastic applications. We just had a couple of lectures by a teacher from North Carolina on the applications of Linear Algebra and its very motivating stuff. I believe this is the related text.
You could probably discuss the row reduction algorithm for solving systems of linear equations. It works quite demonstrably. The deep concept underlying this process is that every matrix has a unique reduced echelon form. You could motivate this through examples. Another related result is that the elementary $n\times n$ matrices giving the row operations actually generate the general linear group of $n\times n$ matrices.
A few ideas:
(1) Numerical Stuff: Look at various methods of solving linear systems or inverting matrices. Study performance (the number of operations involved), and what sorts of things can go wrong numerically. Show that the naive textbook methods don't work very well in practice. See the linear system example in "Why A Math Book Is Not Enough" (Forsythe).
(2) Relations to 3D Geometry: How different configurations of planes correspond with solutions of linear systems. Rank, determinants, etc. If you get through all of that, move on to eigenvectors and how they're related to "morphing" of shapes. Classification of conics (or even quadrics).
(3) Linear Programming: How inequalities describe polyhedra. Optimal sets contain vertices. Graphical solutions in 2D and 3D. Convexity. The simplex method. I guess this isn't very "deep" but it is most certainly "significant" in the real world.
From one high-schooler to another, I would say there are several areas of Linear algebra that can easily be explored. From a purely mathematical stand point here is what I would suggest (in increasing order of difficulty):
From an applications side, the edX course I linked to covers Linear Algebra in the context of programming. Many of the courses talk about applications of the algebra, which is probably the most motivating part of any given course.
Overall there are many individual topics that you can cover as a high school student, and the very basics of some of these topics are actually touched upon by your high school classes (very superficially, though). There are many courses in Linear algebra available online (such as the ones linked above, and this, et cetera), and I think that you might be interested in the beginning of these as the are (for the most part) not to complicated, and the courses available progress into more interesting and 'deep' concepts.
: I use matrix and vector interchangeably here, for simplicity.