> [!NOTE] Lemma > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Let $E$ be a product of $m\times m$ [[Elementary Matrices|elementary matrices]]. Let $L_{A}$ denote the [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] of $A$ and $\ker L_{A}$ its [[Kernel of Linear Map|kernel]]. Then $\ker L_{EA}=\ker L_{A}$ **Proof**: By [[Elementary Matrices are Invertible]], $E$ is invertible and so by [[Left Multiplication Linear Map of Invertible Square Matrix is Linear Isomorphism]], $L_{E}$ is an isomorphism. Since $L_{E}$ is injective and [[Linear Map Fixes Zero]], $L_{E}(L_{A}(\underline{v})) = \underline{0}\iff L_{A}(\underline{v})=\underline{0}.$ **Proof**: Follows directly from [[Reduced Row Echelon Form Preserves Solutions to Linear Homogenous System]]. **Proof**: Follows from [[Elementary Row Operations do not Alter Set of Solutions to Linear System of Equations]].