Suppose \(T:\mathbb {M}_{2,2}\rightarrow \RR \) is a linear transformation such that
\[T\left (\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}\right )=3\]
\[T\left (\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}\right )=-1\]
\[T\left (\begin{bmatrix}1 & 0\\1 & 0\end{bmatrix}\right )=0\]
\[T\left (\begin{bmatrix}0 & 0\\0 & 1\end{bmatrix}\right )=0\]
Find \(T\left (\begin{bmatrix}a & b\\c & d\end{bmatrix}\right )\)
\[T\left (\begin{bmatrix}a & b\\c & d\end{bmatrix}\right )=\answer {3a+2b-3c}\]

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 7.1.4 (d).