Just a quick point - I think there's some confusion here.
Linear algebra is not high school stuff.
To control engineers it is a means of solving simultaneous differential equations,
generally by using parameters specified in (real,s) where s is the differential operator with respect to time.
so s is d/dt, s^2 is d2/dt^2 etc
the point is you can manipulate these algebraically as polynomials, which is a lot easier than solving ordinary differential equations .
The simultaneous equations are arranged as matrices and vectors
and mathematical methods are used to manipulate these.... check out Laplace transforms, eigenvectors , eigenvalues, hamiltonians, jacobi transforms, determinants
etc etc.
regards
Allan
ps it's been years since I studied this stuff........