Concern finite-dimensional vector spaces V and W and a linear transformation T: V → W.

Let H be a nonzero subspace of V, and let T(H) be the set of images of vectors in H. Then T(H) is a subspace of W, by Exercise 47 in Section 4.2. Prove that dim T (H) ≤ dim H.

Data from in Exercise 47

Let V and W be vector spaces, and let T : V → W be a linear transformation. Given a subspace U of V, let T(U) denote the set of all images of the form T(x), where x is in U. Show that T(U) is a subspace of W.