> [!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 [[Kernel of Linear Map|kernel]] is a [[Vector subspace|subspace]] of $V.$ **Proof**: First note $\ker \varphi$ is non-empty since by [[Linear Map Fixes Zero]], $0_{V}\in \ker \varphi.$ Let $v,w\in \ker \varphi.$ Then by linearity, $\varphi(v+w)=\varphi(v)+\varphi(w)=0_{W}+0_{W}=0_{W},$ thus $v+w\in \ker\varphi.$ Let $\lambda\in \mathbb{F}.$ By linearity, $\varphi(\lambda v)=\lambda \varphi(v)=\lambda 0_{V}=0_{V}$ by [[Scalar Multiple of Zero of Real Vector Space]]. Thus $\lambda v \in \ker \varphi.$