0.2.2 Linear transformations.
Let U be an n-dimensional vector space and let V be an m-dimensional vector space, both over the same field F; let BU be a basis of U and let BV be a basis of V. We may use the isomorphisms x → [x]BU and y → [y]BV to represent vectors in U and V as n-vectors and m-vectors over F, respectively. A linear transformation is a function T : U → V such that T (a1x1 + a2x2) = a1T (x1) + a2T (x2) for any scalars a1, a2 and vectors x1, x2. A matrix A ∈ Mm,n(F) corresponds to a linear transformation T : U → V in the following way: y = T (x) if and only if [y]BV = A[x]BU . The matrix A is said to represent the linear transformation T (relative to the bases BU and BV ); the representing matrix A depends on the bases chosen. When we study a matrix A, we realize that we are studying a linear transformation relative to a particular choice of bases, but explicit appeal to the bases is usually not necessary.
page5 Matrix Analysis Second Edition