Concern finite-dimensional vector spaces V and W and a linear transformation T: V → W.

Let H be a nonzero subspace of V, and suppose T is a one-to-one (linear) mapping of V into W. Prove that dim T(H) = dim H. If T happens to be a one-to-one mapping of V onto W, then dim V = dim W. Isomorphic finite-dimensional vector spaces have the same dimension.