Theorem Any linear transformation T: Rn → Rm is also a matrix transformation, with the matrix given by
A = [T(e1) T(e2) ... T(en)].
Proof By an earlier exercise, any x ∈ Rn can be expressed as
x = x1e1 + x2e2 + ... + xnen.
Then by the linearity of T, we have
T(x) = T(x1e1 + x2e2 + ... + xnen) = x1T(e1) + x2T(e2) + ... + xnT(en).
In another earlier exercise, we already noticed the following equality
x1a1 + x2a2 + ... + xnan = Ax,
where a1 a2 ... an are the columns of A and x1, x2, ..., xn are the coordinates of x. (You are recommended to try to verify the equality for a general 3 by 3 matrix). Thus we conclude that
T(x) = Ax for A = [T(e1) T(e2) ... T(en)].