Number 10 and 12 please show all work ### 1.8 Exercises1. **Let \( A = \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix} \) and define \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \).** Find the images under \( T \) of \( \mathbf{u} = \begin{bmatrix}1 \\ -3\end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix}a \\ b\end{bmatrix} \).2. **Let \( A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{bmatrix} \), \( \mathbf{u} = \begin{bmatrix}1 \\ 1 \\ 1\end{bmatrix} \), and \( \mathbf{v} = \begin{bmatrix}a \\ b \\ c\end{bmatrix} \). Define \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) by \( T(\mathbf{x}) = A\mathbf{x} \).** Find \( T(\mathbf{u}) \) and \( T(\mathbf{v}) \).In Exercises 3-6, with \( T \) defined by \( T(\mathbf{x}) = A\mathbf{x} \), find a vector \( \mathbf{x} \) whose image under \( T \) is \( \mathbf{b} \), and determine whether \( \mathbf{x} \) is unique.3. \( A = \begin{bmatrix}2 & 1 \\ -2 & -1\end{bmatrix} \), \( \mathbf{b} = \begin{bmatrix}-2 \\ 2\end{bmatrix} \)4. \( A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \), \( \mathbf{b} = \begin{bmatrix}-6 \\ -12\end{bmatrix} \)5. \( A = \begin{bmatrix}1 & -5 \\ 3 & 7\end{

Name: Number 10 and 12 please show all work ### 1.8 Exercises1. **Let \( A =
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: Number 10 and 12 please show all work ### 1.8 Exercises1. **Let \( A = \begin{bmatrix}2 & 0 \\ 0 & 2\end{bmatrix} \) and define \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) by \( T(\mathbf{x}) = A\mathbf{x} \).** Find the images under \( T \) of \( \math