> [!NOTE] Lemma > Let $A,B$ be [[Real Matrices|real matrices]] of order $m\times n$ and $l\times m$ respectively. Let $BA$ denote their [[Real Matrix Product|product]]. Then the [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] of $BA$ is given by $L_{BA}=L_{B}\circ L_{A}$where $\circ$ denotes [[Function Composition|function composition]]. **Proof**: Let $\underline{v}\in \mathbb{R}^{n}.$ Then by [[Associativity of Multiplication of Real Matrices]], $L_{B}(L_{A}(\underline{v}))=B(A\underline{v})=(BA)\underline{v}=L_{BA}(\underline{v}).$