> [!NOTE] Lemma > Let $V,W$ be [[Vector spaces|vector spaces]] over a [[Field (Algebra)|field]] $\mathbb{F}$ such that $V$ is [[Vector spaces|finite dimensional]] and there exists a [[Linear Isomorphism|linear isomorphism]] $\varphi:V\to W$: that is, $V\cong W.$ Then $W$ is also finite dimensional and their [[Dimension of Vector Space|dimensions]] satisfy $\dim V = \dim W.$ **Proof**: Let $B\subset V$ be a basis of $B.$ Then $|B|<\infty$ and by [[Linear Isomorphism Preserves Basis]], $\varphi(B)$ is a basis of $W.$ Since $\varphi$ is a bijection, $|\varphi(B)|=|B|<\infty$ so $W$ is also finite dimensional with the same dimension.