\(S\) has matrix \(A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\) and \(T\) has matrix \(B=\begin{pmatrix}2&0\\0&3\end{pmatrix}\). What is the matrix of \(T\circ S\)?
\(S\) acts first and then \(T\), so the matrix is \(BA=\begin{pmatrix}2&2\\0&3\end{pmatrix}\). The product \(AB\), in the opposite order, gives a different matrix.
The matrix \([T]\) of a linear transformation \(T\), with respect to given bases, has as columns the images of the domain basis vectors, written in coordinates of the codomain basis.
\[[T(v)]=[T]\,[v]\]
More practice problems: matrix of a linear transformation