> [!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 its [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]]. Then its [[Kernel of Linear Map|kernel]] equals the set of solutions to the [[Linear System of Equations|linear system]] $A\underline{v}=\underline{0}.$ **Proof**: By definition, $\underline{v}\in \ker L_{A} \iff A \underline{v} = \underline{0}$.