Scalar Multiplication of Matrix

The product of a scalar and a matrix is the matrix obtained by multiplying each entry in the matrix by the scalar:

$$\lambda \left(\begin{array}{ccc}{A}_{1,1}& \dots & A_{1,n}\\ ⋮& & ⋮\\ A_{m,1}& \dots & A_{m,n}\end{array}\right)=\left(\begin{array}{ccc}\lambda A_{1,1}& \dots & \lambda A_{1,n}\\ ⋮& & ⋮\\ \lambda A_{m,1}& \dots & \lambda A_{m,n}\end{array}\right)$$

The matrix of a scalar times a linear map is equal to the scalar times the matrix of the linear map.

Suppose $\lambda \in \mathbb{F}$ and $T\in \mathcal{L}(V,W)$. Then $\mathcal{M}(\lambda T)=\lambda \mathcal{M}(T)$.