Vector Spaces

The first four chapters all worked in a concrete three-dimensional space. This chapter abstracts “space” itself for the first time: a vector space is a set satisfying the addition and scalar multiplication axioms, and its elements need not be geometric vectors.

The abstraction immediately pays off — matrices, functions and polynomials can all be vectors in a vector space, and so each of them can have a basis and a dimension. A basis and a dimension define the “coordinate system” of the space, while a coordinate transformation describes how the representation of the same vector changes under a different basis — which corresponds directly to the similarity relation between matrices.

The section on subspaces explains that any subset of a vector space is itself a vector space, and that the null space and the image space are precisely subspaces built out of these concrete objects. The null space and the column space of the earlier chapters receive a unified explanation here.