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).