Exercises 29–34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1.3.
(a) Prove that if W1 and W2 are finite-dimensional subspaces of a vector space V, then the subspace W1 + W2 is finite-dimensional, and dim(W1 + W2) = dim(W1) + dim(W2) − dim(W1 ∩ W2). Hint: Start with a basis {u1, u2, . . . , uk} for W1 ∩ W2 and extend this set to a basis {u1, u2, . . . , uk, v1, v2, . . . vm} for W1 and to a basis {u1, u2, . . . , uk, w1, w2, . . . wp} for W2.
(b) Let W1 and W2 be finite-dimensional subspaces of a vector space V, and let V = W1 + W2. Deduce that V is the direct sum of W1 and W2 if and only if dim(V) = dim(W1) + dim(W2).
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