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 \({latex.inline[u_{1}, ..., u_{n}](u_{1}, ..., u_{n})} be a basis of U. Thus n = dim U, and by hypothesis we have that n = dim V. Thus \){latex.inlineu{1}, ..., u{n}} is a linearly independent list of vectors in V of length dim V. By 1753403620 - Axler 2.38 Linearly independent list of the right length is a basis|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.