Chapter 4.5, Problem 31E

Exercises 31 and 32 concern finite-dimensional vector spaces V and W and a linear transformation T : V → W.

31. 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 35 in Section 4.2. Prove that dim T(H) ≤ dim H.

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35. 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.