matrix of a map relative to bases

A linear map T : V -> W is an abstract, coordinate-free object. To compute with it you choose a basis B = (b_1, ..., b_n) of the domain and a basis C of the codomain, and then T turns into a matrix. The recipe: apply T to each domain basis vector, express the result T(b_j) in coordinates relative to C, and make that coordinate column the j-th column of the matrix. The columns of the matrix ARE the images of the basis vectors, written in the target basis.

Once the matrix [T]_{C<-B} is built, computation reduces to matrix-vector multiplication: if x is the coordinate column of a vector v relative to B, then [T]_{C<-B} x is the coordinate column of T(v) relative to C. The abstract map and its matrix do the same thing; the matrix is just the map seen through two chosen coordinate windows. Composition of maps corresponds to multiplication of their matrices, in the matching order.

The crucial caveat is basis-dependence. Change either basis and the matrix changes, even though the map is untouched. A new domain basis multiplies on the right by a change-of-basis matrix, a new codomain basis multiplies on the left by another. So the same map has infinitely many matrices, and any property you want to attribute to the map itself must be invariant under these changes — that is what rank, and for operators the determinant, trace, and eigenvalues, actually are.

This is the bridge between the two halves of the subject: abstract reasoning about maps, and concrete computation with matrices. Theorems are cleanest stated coordinate-free; arithmetic is done in a basis. Knowing how to pass between them — and which facts survive the passage — is the working mathematician's fluency.