Define \(T: M_{22} \rightarrow \mathbb{C}^1\) by \(T \bigg(\begin{bmatrix}a&b\\c&d\end{bmatrix} \bigg) = a + b + c - d\). Find the pre-image \(T^{-1}(3)\).
A pre-image of an output in the codomain is a subset of all inputs in the domain which, when linearly transformed, generate the output. To find \(T^{-1}(3)\), we need to find all \(\begin{bmatrix}a&b\\c&d\end{bmatrix}\), such that \(T \bigg(\begin{bmatrix}a&b\\c&d\end{bmatrix} \bigg) = 3\).
\(a + b + c - d = 3\)
\(d = a + b + c - 3\)
If we plug in this expression for \(d\), then
\(T^{-1}(3) = \bigg\{\begin{bmatrix}a&b\\c&a+b+c-3\end{bmatrix} \bigg| a, b, c \in \mathbb{C} \bigg\}\)
All following examples belong to the subset of pre-image:
\(\begin{bmatrix}1&1\\1&0\end{bmatrix}, \begin{bmatrix}0&0\\0&-3\end{bmatrix}, \begin{bmatrix}1&2\\3&3\end{bmatrix}\)