Let's solve the problem step by step: ### **Step 1: Understand the Linear Transformation** Given: \[ T\left(\begin{bmatrix}1\\0\\0\end{bmatrix}\right) = \begin{bmatrix}5\\4\end{bmatrix}, \quad T\left(\begin{bmatrix}0\\1\\0\end{bmatrix}\right) = \begin{bmatrix}7\\8\end{bmatrix}, \quad T\left(\begin{bmatrix}0\\0\\1\end{bmatrix}\right) = \begin{bmatrix}0\\2\end{bmatrix} \] The action of \( T \) on the standard basis vectors gives us the columns of the matrix \( A \) for the transformation. **Explanation:** For a linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \), if \( T(\vec{e}_1) = \vec{a}_1 \), \( T(\vec{e}_2) = \vec{a}_2 \), \( T(\vec{e}_3) = \vec{a}_3 \), then the matrix representation is \( A = [\vec{a}_1 \ \vec{a}_2 \ \vec{a}_3] \). --- ### **Step 2: Write the Matrix** Each vector output becomes a column in \( A \): \[ A = \left[ \begin{array}{ccc} 5 & 7 & 0 \\ 4 & 8 & 2 \end{array} \right] \] **Explanation:** - The first column is \( \begin{bmatrix}5\\4\end{bmatrix} \) (from \( T(\vec{e}_1) \)), - The second column is \( \begin{bmatrix}7\\8\end{bmatrix} \) (from \( T(\vec{e}_2) \)), - The third column is \( \begin{bmatrix}0\\2\end{bmatrix} \) (from \( T(\vec{e}_3) \)). --- ### **Final Answer:** \[ A = \begin{bmatrix} 5 & 7 & 0 \\ 4 & 8 & 2 \end{bmatrix} \]