Yes, we usually use the term Abstract Algebra for the study of algebraic structures: groups, rings, fields, vector spaces etc. , so Linear Algebra can be considered as a type of Abstract Algebra.
When I took linear algebra, it started off with "concrete" examples (e.g. vectors) which later switched to more abstract concepts. I'm undecided whether linear algebra should be considered a type of abstract algebra.
modules (Lang, Dummit and Foote, Hungerford, etc.) or, if you'd counter this with 'these aren't vector spaces!' pick up the book A Second Course in Linear Algebra by William Brown which has chapters with names such as 'Functorial Properties of Tensor Products'. If this doesn't satisfy you, you can branch out into other abstract subjects such as functional analysis and representation theory where linear algebra takes up a large portion of the theory and see how abstract that gets. If none of the above strikes a chord and you insist that linear algebra is the study of matrices, there is a vast and deeply theoretical topic known as 'matrix analysis' which divorces, for the most part, matrices as being representations of linear transformations given a fixed basis but considers them as algebraic objects in their own right. For this subject you could look at Matrix Analysis-Horn and Johnson, Topics in Matrix Analysis-Horn and Johnson, or Matrix Analysis-Bhatia.