standard matrix

The matrix $A$ with $T(\mathbf x) = A\mathbf x$ , columns $T(\mathbf e_i)$ .

For linear $T:\mathbb R^n\to\mathbb R^m$ , the standard matrix is the $m\times n$ matrix whose $j$ th column is $T(\mathbf e_j)$ , so that $T(\mathbf x)=A\mathbf x$ for every $\mathbf x$ . It follows from linearity applied to $\mathbf x=x_1\mathbf e_1+\cdots+x_n\mathbf e_n$ , and it is the unique matrix agreeing with $T$ on every input.