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Definition

Let \(T:\mathbb{R}^2\rightarrow \mathbb{R}^2\) be given by \(T\left(\begin{bmatrix} x_1\\x_2\end{bmatrix}\right)= \begin{bmatrix} 2x_1-x_2 \\ x_1 - x_2\end{bmatrix}.\)

Let \(\Gamma = \left( \begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 1 \\ 1\end{bmatrix}\right)\) and \(\Omega = \left( \begin{bmatrix} 2 \\ 1\end{bmatrix}, \begin{bmatrix} 1 \\ 0\end{bmatrix}\right)\) be ordered bases for \(\mathbb{R}^2\).

Then \([T]_\Gamma^\Omega = \begin{bmatrix} 1 & 0 \\ 0 & a\end{bmatrix}\). What is \(a\)?