I am looking for books/papers/chapters that deal with space $\mathcal{S}$ containing MATRICES. $\mathcal{S}$ satisfies
- Addition: If $\mathbf{A},\mathbf{B} \in \mathcal{S}$ then $\mathbf{A}+\mathbf{B} \in \mathcal{S}$
- Scalar Multiplication: If $\mathbf{A}\in \mathcal{S}$ and $k\in \mathbb{R}$ then $k\mathbf{A} \in \mathcal{S}$
I found questions like Determining Bases of Space Spanned by (perhaps) Infinitely Many Matrix. and Space spanned by matrices helpful in getting an intuition but I am looking for more formal/rigorous treatment of this topic.
Unless you are just looking for a number of diverse examples of vector spaces, you are probably trying to learn about spaces (and algebras) of linear operators on vector spaces. (The concept of a linear operator has an invariant--i.e., coordinate-independent definition; that of a matrix, does not.)
Spaces of linear operators on finite-dimensional spaces are finite-dimensional themselves, so their properties are the properties shared by all finite-dimensional vector spaces (whatever plays the role of the individual vector). If you are looking to learn about these, I recommend Halmos's Finite-dimensional vector spaces and Linear algebra problem book.
It gets a lot more interesting and challenging when you get into linear operators on (infinite-dimensional) Banach spaces: here, the algebra is intertwined with the topology. If you are new to this, I recommend, first, becoming acquainted with representation theory (the first 60 pages of Serre's Linear Representations of Finite Groups will do), and then operator algebras (to get a flavor, look through http://projecteuclid.org/euclid.bams/1183510397, and then visit https://en.wikipedia.org/wiki/C*-algebra)