Background: Due to some unfortunate sequencing, I have developed my abstract algebra skills before most of my linear algebra skills. I've worked through Topics in Algebra by Herstein and generally liked his approach to vector spaces and modules. Besides a very elementary course in linear algebra (where most of the time went towards matrix multiplication), I have not developed any other linear algebra skills.
But it seems that it is now necessary for me to do so. Most of the topics that I am looking at now require some background of linear algebra and I still lack understanding of ideas like: bilinear forms, invariant subspaces, eigenvalues, requirements for diagonalization of a matrix and so forth.
This brings me to my question (provided there are any)
Question: What are some good texts that develop the theory of linear algebra from a perspective of general algebra?
Are there any texts that develop the key (elementary) ideas of linear algebra in an abstract setting? Thank you for the help!