題組內容

4. (10%) Let $L$ be a linear operator on $\mathbb{R}^2$, and let $v_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$, $v_2 = \begin{bmatrix} -3 \\ 5 \end{bmatrix}$, $v_3 = \begin{bmatrix} -9 \\ 1 \end{bmatrix}$.

(b) If \( L(\pmb{v}_1) = \begin{bmatrix} 2 \\ 4 \end{bmatrix} \) and \( L(\pmb{v}_2) = \begin{bmatrix} 3 \\ 5 \end{bmatrix} \), find the value of \( L(\pmb{v}_3) \).