Theorem

Suppose that V is finite-dimensional and U is a subspace of V such that dim U = dim V. Then U = V.

Proof

Let ${latex.inlineu_{1}, ..., u_{n}} be a basis of U. Thus n = dim U, and by hypothesis we have that n = dim V. Thus ${latex.inlineu_{1}, ..., u_{n}} is a linearly independent list of vectors in V of length dim V. By 2.38, we get that the list of u’s is a basis of V. In particular, every vector in V is a linear combination of the list of u’s, which means U = V.