> [!NOTE] Lemma > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Let $L_{A}:\mathbb{R}^{n}\to \mathbb{R}^{m}$ denote the [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] of $A,$ which takes $\underline{v}\in \mathbb{R}^{n}$ to $A\underline{v}\in \mathbb{R}^{m}.$ Then $L_{A}$ is indeed a [[Linear maps|linear map]]. **Proof**: By [[Distributivity of Multiplication over Addition of Real Matrices]], for all $v,w\in \mathbb{R}^{n},$ $L_{A}(v+w)=L_{A}(v)+L_{A}(w).$ Let $\lambda\in \mathbb{R}$ and $v\in \mathbb{R}^{n}.$ Clearly, $L_{A}(\lambda \underline{v})=\lambda L_{A}(\underline{v}).$