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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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A lot depends on what you mean by "not overly complicated". – Lepidopterist Mar 9 '14 at 0:21
@Lepi... -- Presumably he means "understandable by a high school student". – bubba Mar 9 '14 at 0:26
Are you the high schooler or the teacher? In my answer below, I assumed that you are the teacher, which is why my comments were rather terse. – bubba Mar 9 '14 at 6:16
@bubba I am a highschooler. – user112829 Mar 9 '14 at 14:43

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:

  • Richard Bronson Schaums Outline of Theory and Problems of Matrix Operations (1988)
  • Krishnan Namboodiri - Matrix Algebra, An Introduction
  • Matrices - Shanti Narayan, P.K. Mittal (This book is a rather good one though. But don't think anything more than maybe the first couple of chapters are accessible to a high school student)

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.

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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.

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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.

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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):

  • You can explore the basic matrix [1] operations. These are probably covered in Algebra II or Pre-calculus in your high school. Khan Academy shows how to do these operation in the linked playlist. This edX course covers this in their second module (you must register for edX, for free, to access the course).

  • You can see the applications of matrices in solving linear equations of $n$ dimensions. This is probably also covered in your algebra II or pre-calculus class in high school. This module from MIT 18.06 (Linear Algebra) talks about this, and covers the intuitive basis for this application of matrices. The course progresses from there, and gets into advanced topics which might be harder to grasp, but are interesting and worthwhile.

  • If you have taken AP calculus or an equivalent then you can see the application of matrices in calculus. Derivative of Vectors are part of the AP Calc. BC curriculum, and College Board provides this set of practice problems on the subject. Furthermore, you can learn this from this modul from MIT 18.02 (multivariate calculus).

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.

[1]: I use matrix and vector interchangeably here, for simplicity.

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