Following the hint, we simply need to find the result of the linear transformation on each basis vector:
and so the matrix that represents the linear transformation T {\displaystyle T} is
![{\displaystyle {\begin{aligned}T\left(\left[{\begin{array}{c}1\\0\\0\end{array}}\right]\right)&=\left[{\begin{array}{c}1\\0\\0\end{array}}\right]\times \left[{\begin{array}{c}0\\1\\0\end{array}}\right]=\left[{\begin{array}{c}0\\0\\1\end{array}}\right],\\T\left(\left[{\begin{array}{c}0\\1\\0\end{array}}\right]\right)&=\left[{\begin{array}{c}0\\1\\0\end{array}}\right]\times \left[{\begin{array}{c}0\\1\\0\end{array}}\right]=\left[{\begin{array}{c}0\\0\\0\end{array}}\right],\\T\left(\left[{\begin{array}{c}0\\0\\1\end{array}}\right]\right)&=\left[{\begin{array}{c}0\\0\\1\end{array}}\right]\times \left[{\begin{array}{c}0\\1\\0\end{array}}\right]=\left[{\begin{array}{c}-1\\0\\0\end{array}}\right],\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/03fea1178075254fb3b47fb127bbf384f2eef8d5)
is
![{\displaystyle \left[{\begin{array}{ccc}0&0&-1\\0&0&0\\1&0&0\end{array}}\right].}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6cf94310a306a3fb7cec9cad45bf2b52391db8d0)
