Let A = = 6 -1 0-5 and define T: R³ → R² by T(x) = Ax. 3 (a) Find the images under T of u = and v= 8 (b) Determine if c = is in the range of the transformation T. 3 (c) Determine if the transformation is one-to-one.
Let A = = 6 -1 0-5 and define T: R³ → R² by T(x) = Ax. 3 (a) Find the images under T of u = and v= 8 (b) Determine if c = is in the range of the transformation T. 3 (c) Determine if the transformation is one-to-one.
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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(Only use elementary row operations and correct row operation notation.)
![### Linear Transformation Example
Given the matrix \( A \) and the transformation \( T \):
\[ A = \begin{bmatrix} -3 & 0 & 6 \\ -1 & 0 & -5 \end{bmatrix} \]
Define the transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) by \( T(x) = Ax \).
#### Problems:
(a) **Find the images under \( T \) of**
\[ u = \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix} \quad \text{and} \quad v = \begin{bmatrix} 3 \\ 1 \\ 8 \end{bmatrix}. \]
(b) **Determine if** \( \mathbf{c} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) **is in the range of the transformation** \( T \).
(c) **Determine if the transformation is** one-to-one.
### Solution Approach:
#### Part (a): Finding Images
To find the images of \( u \) and \( v \) under \( T \), compute \( T(u) = A \cdot u \) and \( T(v) = A \cdot v \).
\[ T(u) = A \cdot \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix} \]
\[ T(v) = A \cdot \begin{bmatrix} 3 \\ 1 \\ 8 \end{bmatrix} \]
#### Part (b): Determine if \( \mathbf{c} \) is in Range
Check if there exists \( x \in \mathbb{R}^3 \) such that \( T(x) = Ax = \mathbf{c} \).
#### Part (c): One-to-One Transformation
The transformation \( T \) is one-to-one if and only if the kernel of \( A \) (i.e., the solution to \( A \cdot x = 0 \)) contains only the zero vector.
### Detailed Steps to Solve:
1. **Matrix Multiplication for (a):**
\[ T(u) = \begin{bmatrix} -3 & 0 & 6 \\ -1 & 0 & -5 \end{bmatrix} \cdot](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8d07e113-70c7-44ea-a54a-4dad627b44c8%2Ff16e0b24-7583-4584-8921-7d79b481494f%2F1849l9p_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Example
Given the matrix \( A \) and the transformation \( T \):
\[ A = \begin{bmatrix} -3 & 0 & 6 \\ -1 & 0 & -5 \end{bmatrix} \]
Define the transformation \( T : \mathbb{R}^3 \rightarrow \mathbb{R}^2 \) by \( T(x) = Ax \).
#### Problems:
(a) **Find the images under \( T \) of**
\[ u = \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix} \quad \text{and} \quad v = \begin{bmatrix} 3 \\ 1 \\ 8 \end{bmatrix}. \]
(b) **Determine if** \( \mathbf{c} = \begin{bmatrix} 2 \\ 3 \end{bmatrix} \) **is in the range of the transformation** \( T \).
(c) **Determine if the transformation is** one-to-one.
### Solution Approach:
#### Part (a): Finding Images
To find the images of \( u \) and \( v \) under \( T \), compute \( T(u) = A \cdot u \) and \( T(v) = A \cdot v \).
\[ T(u) = A \cdot \begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix} \]
\[ T(v) = A \cdot \begin{bmatrix} 3 \\ 1 \\ 8 \end{bmatrix} \]
#### Part (b): Determine if \( \mathbf{c} \) is in Range
Check if there exists \( x \in \mathbb{R}^3 \) such that \( T(x) = Ax = \mathbf{c} \).
#### Part (c): One-to-One Transformation
The transformation \( T \) is one-to-one if and only if the kernel of \( A \) (i.e., the solution to \( A \cdot x = 0 \)) contains only the zero vector.
### Detailed Steps to Solve:
1. **Matrix Multiplication for (a):**
\[ T(u) = \begin{bmatrix} -3 & 0 & 6 \\ -1 & 0 & -5 \end{bmatrix} \cdot
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