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14 1. Preliminaries Exercises 1. Show that if U and W are subspaces of V, then the intersection U ∩ W is a subspace. 2. Give an example of subspaces U and W of R2 for which U ∪W is not a subspace. 3. Show that for subspaces U and W, the set U ∪ W is a subspace if and only if either U ⊆ W or W ⊆ U. 4. Let U and W be subspaces of V. Show that the following conditions are equiv- alent: (a) For each element z of U + W, there is exactly one choice of u ∈ U and w ∈ W such that z = u + w. (b) U ∩ W = {0}. 5. Let U and W be finite-dimensional subspaces of V. Prove the formula dim(U + W) = dim(U) + dim(W) − dim(U ∩ W). Illustrate with the example of two planes in R3. 6. Let T : V → W be a linear transformation from V to W. (a) Let R = {w1,..., wk} be a linearly independent set in T (W). For each i = 1,...,k, choose vi in the inverse image of wi, that is, T (vi) = wi. Show that the set P = {v1,..., vk} is linearly independent. (b) Show by example that if S is a linearly independent set of vectors in V, then T (S) can be linearly dependent. (c) Show that if T is injective and S is a linearly independent set of vectors in V, then T (S) is linearly independent. (d) Show by example that if S spans V, then T (S) need not span W. (e) Show that if T is surjective and S spans V, then T (S) spans W. 7. Show that an m × n matrix A has rank one if and only if there are nonzero column vectors x ∈ Fm and y ∈ Fn such that A = xyT . 8. Let A and B be matrices such that the product AB is defined. Show that rank(AB) ≤ rank(A) and rank(AB) ≤ rank(B). 9. (a) Show that an m × n matrix A has rank k if and only if there is an m × k matrix B and a k × n matrix C such that rank(B) = rank(C) = k and A = BC. Note that Exercise 7 is the special case k = 1. (b) Referring to part 9(a), let b1,..., bk be the columns of B, and let cT 1 , . . . , cT k be the rows of C. (So ci is a column vector and thus ci T is a row.) Let Ci denote the k × n matrix with ci T in the ith row and zeroes in all the other rows. Using C = ∑k i=1 Ci, show that A = BC gives (1.15) A = k i=1 bici.t Equation (1.15) decomposes the rank k matrix A into a sum of k matrices of rank one.