> [!NOTE] Lemma > [[Real Matrix Scalar Multiplication|Scalar multiplication]] is [[Distributivity|distributive]] over [[Real Matrix Addition|addition of real matrices]]: that is, for all [[Real Matrices|real matrices]] $A,B\in \text{Mat}_{mn} (\mathbb{R})$ and $\lambda\in \mathbb{R},$ $\lambda(A+B)=\lambda A+\lambda B.$ **Proof**: Follows directly from definitions and [[Distributivity of Multiplication over Addition of Real Numbers]].