Let T : V → W be a linear transformation between finite-dimensional vector spaces V and W. Let B and C be bases for V and W, respectively, and let A = [T]_C←B.

If dim V = n and dim W = m, prove that L (V, W ) ≅ M_mn. (See the exercises for Section 6.4.) [Let B and C be bases for V and W, respectively. Show that the mapping φ(T) = [T]_C←B, for T in L (V, W ), defines a linear transformation φ : L (V, W ) → M_mn that is an isomorphism.]