Linear algebra questions that a high-schooler could explore Are there any deep/significant concepts in linear algebra that are not overly complicated that a high schooler could explore in depth?

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.
