> [!NOTE] Definition (Linear Isomorphism) > Let $V,W$ be [[Vector spaces|vector spaces]] over $\mathbb{F}$. A *linear isomorphism* is a [[Linear maps|linear map]] $\varphi : V \to W$ that is [[Bijection|bijective]]. > [!Example] > [[Complex numbers is isomorphic to R^2]]. # Properties > [!NOTE] Corollary (Isomorphic FDVSs have the same dimension) > If $V \cong W$ (i.e. there is an isomorphism between them) and $V$ is [[Vector spaces|finite dimensional]], then $W$ is also *finite dimensional* with $\dim V = \dim W$. > > In particular $\mathbb{R}^{n} \cong \mathbb{R}^{m}$ iff $n=m$. > > [!NOTE] Lemma (Isomorphisms preserve dimension of subspaces) > Suppose $\varphi:V \to W$ is an isomorphism and $U \subset V$ is a [[Vector subspace|subspace]] of $V$. Then $\varphi$ gives an isomorphism between $U$ and the *subspace* $\varphi(U)\subset W$. *In particular, if $U$ is [[Vector spaces|finite dimensional]]* then $\dim U = \dim \varphi (U)$. ^aa66b3 >*Proof.* Regarding $\varphi$ as a linear map $U \to W$, it is still injective, and of it is of course surjective onto its image $\varphi(U)$, therefore it is a bijection $U\to \varphi(U)$, and so is an isomorphism as claimed. > >Note that $U$ [[Basis of Vector Space#^61f832|must]] also be finite dimensional so the last line follows since *isomorphic FDVS have the same dimension* (above corollary). # Applications - [[Vector spaces|FDVS coordinate map]].