A philosopher's tools are paper and pencil. A mathematician's tools are paper, pencil, and wastebasket. There is no "philosophy" of linear algebra. A vector space is an abelian group with a scalar multiplication that satisfies distributive laws. Unlike most other mathematical objects, they can be classified by a single quantity: their dimension. Every vector space is isomorphic to the set of functions from S to F, F^S = {v:S -> F}, where S is a set and F is the underlying field. It's dimension is the cardinality of S. A linear transformation is a function preserving the linear structure. A matrix is a representation of a linear transformation using a particular basis. Matrix multiplication is composition of linear transformations. As Whistler remarked when an aficionado said "There are two great painters in the world, you and Velazquez." His reply, "Why drag Velazquez into this?" Why drag philosophy into this? Just buckle down and learn the math, as everyone before you did as a matter of course.