> [!NOTE] Lemma (Inverse of isomorphism is also linear) > Let $V,W$ be [[Vector spaces|vector spaces]] over a [[Field (Algebra)|field]] $\mathbb{F}.$ Let $\varphi:V\to W$ be a [[Linear Isomorphism|linear isomorphism]]. Then the [[Function Inverse|inverse]] of $\varphi$ is also a linear map $\varphi^{-1}: W\to V.$ **Proof**: Let $w_{1},w_{2}\in W$ and $\mu_{1},\mu_{2}\in \mathbb{F}$. Then since $\varphi$ is a bijection, there are $v_{1},v_{2}\in V$ so that $w_{1}=\varphi(v_{1})$ or equivalently $v_{1}=\varphi^{-1} (w_{1})$ and similarly $w_{2}$. By linearity of $\varphi$, we have $\begin{align} \varphi(\mu_{1}v_{1}+\mu_{2}v_{2}) &= \mu_{1} \varphi(v_{1}) + \mu_{2} \varphi(v_{2}) \\ & = \mu_{1}w_{1} + \mu_{2} w_{2} \end{align}$and now applying $\varphi^{-1}$ to both sides gives the desired result. $\square$