Matrix (Representation of a Linear Map)

definitionvector-spaces

Definition

Let $T\in \mathcal{L}(V,W)$, and $v_1,\dots ,v_n$ be a basis of $V$, and $w_1,\dots ,w_m$ by a basis of $W$. The matrix of $T$ with respect to these bases is the $m\times n$ matrix $M(T)=M(T,(v_1,\dots ,v_n),(w_1,\dots ,w_m))$ whose entries $A_{i,j}$ are given by $T{v}_k=A_{1,k}{w}_1+A_{2,k}{w}_2+\cdots +A_{m,k}{w}_m$.

$$\begin{array}{rl}M(T)=\begin{array}{c}{w}_1\\ ⋮\\ w_m\end{array}& \left(\begin{array}{c}{A}_{1,k}\\ ⋮\\ A_{m,k}\end{array}\right)\\ & \begin{array}{c}{v}_1\cdots v_k\cdots v_n\end{array}\end{array}$$

Examples