N M Define the linear transformati T:R³ R3 So that somation. XI m -n₂+ 242 X2 → -2₁ - 2n2+ 3 X3 2n1 + 12 + 23 @ Find the standard patrix of I One to one ? Explain. is T is T Onto? Explain. if there is any find a Vector V such that T (V) - V Who V = 2 -1 3
N M Define the linear transformati T:R³ R3 So that somation. XI m -n₂+ 242 X2 → -2₁ - 2n2+ 3 X3 2n1 + 12 + 23 @ Find the standard patrix of I One to one ? Explain. is T is T Onto? Explain. if there is any find a Vector V such that T (V) - V Who V = 2 -1 3
Algebra and Trigonometry (6th Edition)
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Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Linear Transformations in R³
**Problem 1: Define the Linear Transformation**
Given linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is defined such that:
\[ T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - x_2 + x_3 \\ 2x_1 + 2x_2 + x_3 \\ x_1 + 2x_2 + 3x_3 \end{bmatrix} \]
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**Problem 2: Standard Matrix of T**
(a) **Find the Standard Matrix of T**
To define the standard matrix \( A \) for the transformation \( T \), we express the transformation in terms of a matrix multiplication \( T(\mathbf{x}) = A \mathbf{x} \).
The transformation is defined as:
\[ T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - x_2 + x_3 \\ 2x_1 + 2x_2 + x_3 \\ x_1 + 2x_2 + 3x_3 \end{bmatrix} \]
This can be written as:
\[ T(\mathbf{x}) = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \]
Therefore, the standard matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \]
(b) **Is T one-to-one? Explain.**
A linear transformation \( T \) is one-to-one if and only if the matrix \( A \) has full column rank (which means that the columns of \( A \) are linearly independent).
To check if \( A \) is one-to-one, we need to determine if \( \text{det](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F158c51ef-5dd2-4ce5-bed6-950ee0630429%2F7ee823c4-b7c5-4d48-826a-bcba978fc8ef%2Fdkoqz9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Linear Transformations in R³
**Problem 1: Define the Linear Transformation**
Given linear transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) is defined such that:
\[ T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - x_2 + x_3 \\ 2x_1 + 2x_2 + x_3 \\ x_1 + 2x_2 + 3x_3 \end{bmatrix} \]
---
**Problem 2: Standard Matrix of T**
(a) **Find the Standard Matrix of T**
To define the standard matrix \( A \) for the transformation \( T \), we express the transformation in terms of a matrix multiplication \( T(\mathbf{x}) = A \mathbf{x} \).
The transformation is defined as:
\[ T \left( \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \right) = \begin{bmatrix} x_1 - x_2 + x_3 \\ 2x_1 + 2x_2 + x_3 \\ x_1 + 2x_2 + 3x_3 \end{bmatrix} \]
This can be written as:
\[ T(\mathbf{x}) = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \]
Therefore, the standard matrix \( A \) is:
\[ A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix} \]
(b) **Is T one-to-one? Explain.**
A linear transformation \( T \) is one-to-one if and only if the matrix \( A \) has full column rank (which means that the columns of \( A \) are linearly independent).
To check if \( A \) is one-to-one, we need to determine if \( \text{det
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