> [!NOTE] Lemma > Let $A$ be a [[Real Matrices|real matrix]] of order $m\times n.$ Let $F$ be a product of $n\times n$ [[Elementary Matrices|elementary matrices]]. Let $L_{A}$ denote the [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] of $A$ and $\text{Im } L_{A}$ its [[Image of Linear Map|image]]. Then $\text{Im } L_{EA}=\text{Im } L_{A}$ **Proof**: By [[Elementary Matrices are Invertible]], $F$ is invertible and so by [[Left Multiplication Linear Map of Invertible Square Matrix is Linear Isomorphism]], $L_{F}$ is an isomorphism. Since $L_{F}$ is surjective, $L_{A}(L_{F}(\mathbb{R}^{n}))=L_{A}(\mathbb{R}^{n})$.