How can, for example, [itex]M_2(R)[/itex] have four dimensions? What I mean is how can a 2x2 matrix be considered a vector? Also how can the set of solutions to a linear differential equation be a set of vectors? Or are these examples supposed to be the idea of vector spaces applied outside the realm of actual vectors? Because vector spaces are introduced in the book as being abstract but up until now, I've thought they were pretty concrete. Is this where the idea of vector spaces becomes abstract?