space of linear maps

In a first course you study one linear map at a time. The shift in Volume II is to treat ALL the linear maps from a space V to a space W as a single collection and ask what structure that collection has. The answer is clean: you can add two linear maps, and you can scale a linear map by a number, and both results are again linear. So the set of all linear maps from V to W is itself a vector space.

We write it Hom(V,W), or sometimes L(V,W). Addition is pointwise, (S + T)(v) = S(v) + T(v), and scaling is (c T)(v) = c (T(v)). The zero map is the additive identity. Once you fix a basis for V and a basis for W, every map turns into a matrix, and adding maps adds matrices entry by entry. So Hom(V,W) is, in coordinates, just the space of all m by n matrices.

Counting dimensions makes this concrete. If dim V = n and dim W = m, then a map is pinned down by where it sends each of the n basis vectors of V, and each target is a free vector in the m-dimensional W. That is n independent choices, each living in m dimensions, so dim Hom(V,W) = m times n. The matrix picture confirms it: an m by n matrix has m times n free entries.

The payoff is conceptual leverage. Treating maps as points in a vector space lets you talk about families of maps, limits of maps, the dual space (the special case W = the scalars), and bilinear forms — all by reusing the linear-algebra you already trust, one level up.