Matrix Addition

Assume that $U,V$ and $W$ are finite-dimensional and that a basis has been chosen for each of these vector spaces. Thus for each linear map from $V$ to $W$, we can talk about its matrix with respect to the chosen bases.

The sum of two matrices of the same size is the matrix obtained by adding the corresponding entries in the matrices:

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

Something important to note is that the matrix of the sum of two linear maps is equal to the sum of the matrices of the two maps:

Matrix of the sum of linear maps

Suppose $S,T\in \mathcal{L}(V,W)$. Then $\mathcal{M}(S+T)=\mathcal{M}(S)+\mathcal{M}(T)$.