> [!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 $\ker L_{A}$ its [[Kernel of Linear Map|kernel]]. Then the [[Dimension of Finite Dimensional Real Vector Space|dimension]] of $L_{EA}$ satisfies $\dim \ker L_{EA}=\dim \ker L_{A}$ **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. Let $U=\ker L_{A} \subset \mathbb{R}^{n}$, we have $\ker L_{AF} = L_{F}^{-1}(U)$ and these must have equal dimension by [[Linear Isomorphism gives Isomorphism Between Subspace of Domain and Image of Subspace]].