Problem 2 Let V be a subspace of Rn and W a subspace of V So

Problem 2. Let V be a subspace of Rⁿ and W a subspace of V .

Solution

The Dimension theorem states, inter-alia, that if V is a linear subspace of Rⁿ which has a basis of k vectors, then any set of more than k vectors in V is linearly dependent

Let W, V be linear subspaces of Rⁿsuch that W V. Let dim(V ) = m and dim(W) = n . Also, let B₁, .., B_nform a basis of W. Since the B’s are also independent vectors in V, n m by the Dimension Theorem.