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: 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. 
A: You don't want "pointfree", you obviously want "categorical" or "category theoretic" or simply "generalized". Also notice that category theory offers the notion of a generalized element or point. For example, a generalized point of your eigenspace $\mathrm{Eig}_{\lambda}(A)$ (you called it $\mathrm{Eig}_A(\lambda)$, which is not common) is really a generalized point $x \in X$ with $A(x)=\lambda x$. (I have used generalized elements extensively in my work because that way I could find and do calculations with elements which otherwise would have become huge intractable diagrams.) Regarding the question, you would have to define "linear algebra in a point-free manner" first. Usually linear algebra is seen as the study of vector spaces and linear maps, and obviously many constructions can be cast in context of the category of vector spaces and linear maps, and then be generalized to many other categories. As you said, the eigenspace constructions works for all linear categories with equalizers. Ok, but what about characteristic polynomials, normal forms, etc.? Probably the categories have to be quite special in order to generalize linear algebra to them. Anders Kock has done something in that direction.
