Let \(V\) and \(W\) be finite-dimensional vector spaces over the field \(F\). Prove that \(V\) and \(W\) are isomorphic if and only if \(\dim V = \dim W\).
Solution
Suppose first that \(V\) and \(W\) are isomorphic, and let \(T:V\to W\) be an isomorphism. If \(v_1,\ldots,v_n\) is a basis of \(V\), then \(Tv_1,\ldots,Tv_n\) is linearly independent because \(T\) is injective, and it spans \(W\) because \(T\) is surjective. Hence it is a basis of \(W\), so \[ \dim V=\dim W. \]
Conversely, suppose \(\dim V=\dim W=n\). Choose bases \(v_1,\ldots,v_n\) of \(V\) and \(w_1,\ldots,w_n\) of \(W\). There is a unique linear map \(T:V\to W\) with \[ T(v_i)=w_i\qquad(1\le i\le n). \] It sends a basis to a basis, hence is both injective and surjective. Therefore \(T\) is an isomorphism.