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many rows and columns does A have? and T (x) = Ax for some matrix 4 and for each x in R. How 0

many rows and columns does A have? and T (x) = Ax for some matrix 4 and for each x in R. How 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Number 10 and 12 please show all work
### 1.8 Exercises 1. **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{
Transcribed Image Text:### 1.8 Exercises 1. **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{
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