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

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Description: Number 12 ### 1.8 Exercises1. **Let \( A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} $, and define $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ by $ T(x) = Ax $. Find the images under $ T $ of $ u = \begin{bmatrix} -1 \\ 3 \end{bmatrix} $

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Answered: -1 3 and let A be the matrix in…

Math

Advanced Math

-1 3 and let A be the matrix in Exercise 10. Is -1 Let b = 4 b in the range of the linear transformation x+ Ax? Why or why not?

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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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

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Question

Number 12

### 1.8 Exercises

1. **Let $ A = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} $, and define $ T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ by $ T(x) = Ax $. Find the images under $ T $ of $ u = \begin{bmatrix} -1 \\ 3 \end{bmatrix} $ and $ v = \begin{bmatrix} a \\ b \end{bmatrix} $.**

2. **Let $ A = \begin{bmatrix} 1 & 3 & 0 \\ 3 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $, $ u = \begin{bmatrix} 1 \\ 0 \\ -5 \end{bmatrix} $, and $ v = \begin{bmatrix} a \\ b \\ c \end{bmatrix} $. Define $ T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 $ by $ T(x) = Ax $. Find $ T(u) $ and $ T(v) $.**

In Exercises 3-6, with $ T $ defined by $ T(x) = Ax $, find a vector $ x $ whose image under $ T $ is $ b $, and determine whether $ x $ is unique.

3. **$ A = \begin{bmatrix} 1 & 1 \\ 2 & -1 \end{bmatrix} $, $ b = \begin{bmatrix} 5 \\ -3 \end{bmatrix} $**

4. **$ A = \begin{bmatrix} 1 & 2 \\ 2 & -5 \end{bmatrix} $, $ b = \begin{bmatrix} -6 \\ 4 \end{bmatrix} $**

5. **$ A = \begin{bmatrix} 1 & 3 \\ 1 & 5 \end{bmatrix} $, $ b = \begin{bmatrix} -5 \\ 7 \end{bmatrix} $**

6. **\( A = \begin{bmatrix} 1 & -3 \\

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