Let's solve the problem step by step: ### **Step 1: Understand the Linear Transformation and Standard Basis Vectors** A linear transformation \( F: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) can be represented by a matrix \( A \) such that \( F(\vec{x}) = A\vec{x} \). - The **standard basis vectors** for \( \mathbb{R}^2 \) are \( \vec{e}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \) and \( \vec{e}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix} \). - The columns of matrix \( A \) are \( F(\vec{e}_1) \) and \( F(\vec{e}_2) \). Given: \[ F\left(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) = \begin{pmatrix} 9 \\ 7 \end{pmatrix} \] \[ F\left(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\right) = \begin{pmatrix} 3 \\ 7 \end{pmatrix} \] \[ F\left(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\right) = \begin{pmatrix} 7 \\ 6 \end{pmatrix} \] --- ### **Step 2: Construct the Matrix for \( F \) with Respect to the Standard Basis** **Matrix Construction:** - The **first column** of the matrix is \( F(\vec{e}_1) = F\left(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\right) = \begin{pmatrix} 3 \\ 7 \end{pmatrix} \). - The **second column** is \( F(\vec{e}_2) = F\left(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\right) = \begin{pmatrix} 9 \\ 7 \end{pmatrix} \). So, the matrix is: \[ \boxed{ \begin{pmatrix} 3 & 9 \\ 7 & 7 \end{pmatrix} } \] --- ### **Final Answer:** \[ \boxed{ \begin{matrix} 3 & 9 \\ 7 & 7 \end{matrix} } \] --- #### **Explanation:** - We used the given outputs of the transformation on the standard basis vectors to directly form the columns of the matrix. No further calculation is needed because linear transformations are completely determined by their action on the basis vectors. The information about \( F\left(\begin{pmatrix} 1 \\ 1 \end{pmatrix}\right) \) was extra and is already consistent with the other values.