(Worth 10 points total) Let the linear transformation \(L:\m…
(Worth 10 points total) Let the linear transformation \(L:\mathbb{R}^2\rightarrow \mathbb{R}^2\) be defined by \(L\left(\begin{bmatrix} x_1\\x_2 \end{bmatrix}\right) = \begin{bmatrix}x_2 \\ 2x_2 + x_1\end{bmatrix}\) Part A) Find the matrix representing \(L\) with respect to the standard basis \(\left\{\vec{e}_1 = \begin{bmatrix} 1\\0 \end{bmatrix},\vec{e}_2=\begin{bmatrix} 0\\1 \end{bmatrix} \right\}\) for \(\mathbb{R}^2\). Part B) Find the matrix representing \(L\) with respect to the following basis for \(\mathbb{R}^2\) (for both the input and output vectors): \(\vec{v}_1 = \begin{bmatrix} 1\\-1 \end{bmatrix}\quad \quad \vec{v}_2 = \begin{bmatrix} 0\\2 \end{bmatrix}\) (If your final answer involves a matrix product, you may leave your answer as a product without actually multiplying them together.) Write your final answers in the text box below. Your full work will be submitted to Gradescope as soon as you submit on Canvas.
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