I am looking for a quick introduction to linear algebra that is
slick (i.e., proofs are easy to follow and don't require handwaving);
general and comprehensive enough to serve as foundation for finite field extensions (i.e., it works over any field, and does arbitrary vector spaces and not just $F^n$);
reasonably short;
and yet reasonably elementary that students who have mostly seen matrix computations and $\mathbb{R}^n$ will recognize ideas.
Background: I am teaching an abstract algebra
class to an audience of rather diverse skillsets. A significant number of the students will be bored by a lengthy from-scratch review of linear algebra, while another will be lost without at least some kind of remediation. The material should include whatever is necessary for the theory of finite field extensions (up to basic Galois theory) -- so, quotient spaces, direct sums, kernels at least.
I don't need determinants (I will do them anyway, following Strickland, so they can be assumed), infinite-dimensional vector spaces (except for their existence and the fact that $F\left[x\right]$ is an $F$-vector space with basis $\left(1,x,x^2,\ldots\right)$), diagonalization, nilpotency, orthogonality, bilinear forms, numerics, symmetric and alternating matrices, inequalities.
I don't mind if the proofs are terse, assuming that they can be expanded without too much trouble. What I care about is that the results are arranged in a way to make short proofs possible.
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