Where can I find linear algebra described in a pointfree manner? Clearly, some of linear algebra can be described in a pointfree fashion. For example, if $X$ is an $R$-module and $A : X \leftarrow X$ is an endomorphism of $X$, then we can define that the "eigenspace function of $A$" is the map $\mathrm{Eig}_A : \mathrm{Sub}(X) \leftarrow R$ described by the following equalizer.
$$\mathrm{Eig}_A(\lambda) = \mathrm{Eq}(A,\lambda \cdot \mathrm{id}_X)$$
In fact, this makes sense in any $R$-$\mathbf{Mod}$ enriched category with equalizers.
Anyway, I did some googling for "pointfree linear algebra" and "pointless linear algebra" etc., and nothing really came up, except for an article called "Point-Free, Set-Free Concrete Linear Algebra" which really isn't what I'm looking for. So anyway...

Question. Where can I find linear algebra described in a pointfree manner?


A commutative algebra textbook. 
Here is a better description of how eigenspaces work from this perspective: a pair consisting of an $R$-module and an endomorphism of it is the same thing as an $R[X]$-module. If $R$ is a field, then finitely generated $R[X]$-modules are classified by the structure theorem for finitely generated modules over a PID. They consist of a torsion-free part and a torsion part which is a direct sum of modules of the form $R[X]/f(x)^d$ where $f(X) \in R[X]$ is irreducible. 
If $R$ is an algebraically closed field, the only $f(X)$ that occur are the linear polynomials $f(X) = X - \lambda$, and the $(X - \lambda)$(-power) torsion submodule is precisely the (generalized) $\lambda$-eigenspace. But if $R$ is not algebraically closed or, worse, not a field, then more complicated things can happen, and accordingly the idea of eigenspaces is less useful. 
