Theorem: let v(f) and w(f) be a n and m dimensional vector spaces respectively and t: v w be linear transformation and b, and b, are standard basis for v and w

Theorem: Let V(F) and W(F) be a n and m dimensional vector spaces respectively and T: V W be linear transformation and B, and B, are standard basis for V and W respectivey. Let A be the matrix of T relative to the standard basis for V and W, then ITlea, = Pa,AP Proof: Let v e V then IT(V)l, = [Tne, [Vle. [v=Pa,[T(v)le, [T(V)] = Ps, IT(V)l, Now, T(v)] = A[V] IT(V)la, = P,T(V) = PpA(v) So, PaAPB, [Vla, So, e,8,=P, AP,