Question: 4. Let V be a finite-dimensional vector space. For any subspace W of V , we define W = {T V': x W, T(x) =

4. Let V be a finite-dimensional vector space. For any subspace W of V , we define W = {T V': x W, T(x) = 0}. You may assume, without proving it, that if W is a subspace of V then W is a subspace of V0. (a) Suppose that W1 and W2 are subspaces of V . Prove that W1 = W2 if and only if W1 = W2. (b) Suppose that W1 and W2 are subspaces of V . Prove that (W1 + W2) = W1 W2 and (W1 W2) = W1 + W2. (c) Suppose that W is a subspace of V . Prove that dim(V ) = dim(W) + dim(W).

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