> [!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 $\varphi(0_{V})=0_{W}.$ **Proof**: By linearity of $\varphi,$ $\varphi(0_{V})=\varphi(0_{V}+0_{V})=\varphi(0_{V})+\varphi(0_{V}).$Adding $-\varphi(0_{V})$ to both sides gives $\varphi(0_{V})=0_{W}.$