Definition:Matrix Scalar Product

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring whose zero is $$0_R$$ and whose unity is $$1_R$$.

Let $$\mathbf{A} = \left[{a}\right]_{m n}$$ be an $m \times n$ matrix over $$\left({R, +, \circ}\right)$$.

Let $$\lambda \in R$$ be any element of $$R$$.

The scalar product of $$\lambda$$ and $$\mathbf{A}$$ is defined as follows.

Let $$\lambda \circ \mathbf{A} = \mathbf{C}$$.

Then: $$\forall i \in \left[{1 \,. \, . \, m}\right], j \in \left[{1 \,. \, . \, n}\right]: c_{i j} = \lambda \circ a_{i j}$$

Thus $$\left[{c}\right]_{m n}$$ is the $$m \times n$$ matrix composed of the scalar product of $$\lambda$$ and the corresponding elements of $$\mathbf{A}$$.

The matrix space $\mathcal {M}_{R} \left({m, n}\right)$ of all $m \times n$ matrices over $$R$$ is a module.

Proof
This follows as $$\mathcal {M}_{R} \left({m, n}\right)$$ is a direct instance of the module given in the Module of All Mappings, where $$\mathcal {M}_{R} \left({m, n}\right)$$ is the $$R$$-module $$R^{\left[{1 \,. \, . \, m}\right] \times \left[{1 \,. \, . \, n}\right]}$$.

The $$S$$ of that example is the set $$\left[{1 \,. \, . \, m}\right] \times \left[{1 \,. \, . \, n}\right]$$, while the $$G$$ of that example is the $$R$$-module $$R$$.