What changes for linear algebra over a finite field?

This is a rather approximative overview of what generalizations can be explored in an early course of linear algebra.

The short answer is that all that does not use the fact that $\Bbb R$ is ordered, $\Bbb C$ has a norm, or that $\Bbb C=\Bbb R[i]=\overline{\Bbb R}$ carries on identically to all fields and it can, in principle and in point of fact, be taught directly as "linear algebra", instead of "$\Bbb R$-or-$\Bbb C$ linear algebra". More specifically

1. All the things that are genuinely linear, like basis, matricial representations for finite dimensional spaces, dual and bi-dual, Gaussian elimination, determinants, Rouché-Capelli theorem carry on verbatim or with very obvious adjustments.

2. The results around Jordan normal form stay unchanged for algebraically closed fields. Phenomena like "real Jordan normal form", though, use heavily the fact that $\dim_{\Bbb R}\Bbb C=2$, and need to be heavily amended to be generalized to other extensions (which are almost always of infinite degree) in an interesting way.

3. The "theory of real inner products on finitely dimensional spaces" is generalized by the theory of quadratic forms, and it is interesting even as part of an early course. It studies the symmetric and bilinear maps $\phi:k^n\times k^n\to k$. There is a generalized notion of orthogonality, of adjoint, of degenerate quadratic forms, of orthogonal maps (sometimes called isometries). The main differences revolve around the fact that:

   • $\Bbb R$ is ordered, and so there is a notion of sign and positive definiteness that can be used to control/distinguish a lot of things. For a general field, the only thing that can be controlled is the presence of vectors $v$ such that $\phi(v,v)=0$ (isotropic vectors) and/or such that $\phi(v,w)=0$ for all $w$ (orthogonal to the whole space). This reflects on terminology and choice of "canonical forms". If you want to quickly gage the flavour of it, have a look at these results by Witt.

   • Fields of characteristic $2$, and $\Bbb F_2$ especially, need (if any) a separate treatment. The issue is that, in fields where $1+1\ne0$, there is a bijective correspondence between symmetric bilinear maps and homogeneous polynomial functions of degree $2$ - i.e., maps $q:k^n\to k$ that can be written as $q(v)=\sum_{i,j}q_{ij}v_iv_j$ for some constants $q_{ij}$. This correspondence is established by calling $Q_{\phi}(v)=\phi(v,v)$, and $\Phi_q(v,w)=\frac{q(v+w)-q(v)-q(w)}{2}$. It's straight-forward to verify that $\Phi_{q_\phi}=\phi$ and $Q_{\Phi_q}=q$. You can't divide by $2$ when $1+1=0$, and it turns out that the map $\phi\mapsto Q_\phi$ is not injective in characteristic $2$.

   However, you may want to look into an actual textbook for further detail on (3); "Introduction To Quadratic Forms Over Fields" by Lam (or its earlier, more famous version "Algebraic Theory of Quadratic Forms") is something that you may find in your local library. It doesn't quite cover what happens in characteristic $2$, though.