> [!NOTE] Lemma > Let $V,W$ be [[Vector spaces|vector spaces]] over a [[Field (Algebra)|field]] $\mathbb{F}$ with zeros $0_{V},0_{W}$ respectively. Let $\varphi:V\to W$ be a [[Linear maps|linear map]]. Then its [[Image of Linear Map|image]] is a [[Vector subspace|subspace]] of $W.$ **Proof**: First note that $\text{Im }\varphi$ is non empty since by [[Linear Map Fixes Zero]], $\varphi(0_{V})=0_{W}$ since $0_{W}\in \text{Im }\varphi.$ Let $w_{1},w_{2}\in \text{Im} \, \varphi$ and $\mu_{1},\mu_{2}\in \mathbb{F}$. Then by the definition of the image, there exists $v_{1},v_{2}\in V$ with $w_{1}=\varphi(v_{1})$ and $w_{2}=\varphi(v_{2})$. By linearity of $\varphi,$ $\mu_{1}w_{1}+\mu_{2}w_{2}=\mu_{1}\varphi(v_{1})+\mu_{2}\varphi(v_{2}) = \varphi (\mu_{1}v_{1}+\mu_{2}v_{2})$ also lies in $\text{Im} \, \varphi$, as required.