1 -6 - 22 1 If T is defined by T(x)= Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = -2 and b = 12 4 - 6 Find a single vector x whose image under T is b. X =
1 -6 - 22 1 If T is defined by T(x)= Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A = -2 and b = 12 4 - 6 Find a single vector x whose image under T is b. X =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Linear Transformations: Finding a Vector
Let's explore how to find a vector \( \mathbf{x} \) whose image under a given linear transformation \( T \) is \( \mathbf{b} \). Assume \( T \) is defined by the matrix multiplication \( T(\mathbf{x}) = A\mathbf{x} \).
Given:
\[ A = \begin{pmatrix} 1 & -6 & -22 \\ -2 & 4 & 12 \end{pmatrix} \]
\[ \mathbf{b} = \begin{pmatrix} -1 \\ -6 \end{pmatrix} \]
**Problem:**
Find a single vector \( \mathbf{x} \) whose image under \( T \) is \( \mathbf{b} \) and determine whether such a vector \( \mathbf{x} \) is unique.
**Solution:**
To find \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \) that satisfies \( A\mathbf{x} = \mathbf{b} \), we set up the following system of linear equations based on the matrix multiplication \( A\mathbf{x} = \mathbf{b} \):
\[ \begin{pmatrix} 1 & -6 & -22 \\ -2 & 4 & 12 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} -1 \\ -6 \end{pmatrix} \]
Next, solve the system of equations to determine \( \mathbf{x} \).
### Interactive Section
**Find a single vector \( \mathbf{x} \) whose image under \( T \) is \( \mathbf{b} \).**
\[ \mathbf{x} = \boxed{ \begin{pmatrix} \, \, \end{pmatrix} } \]
In this space, students might be encouraged to use methods such as Gaussian elimination or matrix inversion to solve the system.
---
To see the complete steps for solving the system, additional resources are available in the section on Solving Systems of Linear Equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F965f2633-6838-4c2d-b94e-32b85de15479%2Fbd759fba-cb1d-4256-bf77-8bcf9ce0825f%2Fjwdvaib_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformations: Finding a Vector
Let's explore how to find a vector \( \mathbf{x} \) whose image under a given linear transformation \( T \) is \( \mathbf{b} \). Assume \( T \) is defined by the matrix multiplication \( T(\mathbf{x}) = A\mathbf{x} \).
Given:
\[ A = \begin{pmatrix} 1 & -6 & -22 \\ -2 & 4 & 12 \end{pmatrix} \]
\[ \mathbf{b} = \begin{pmatrix} -1 \\ -6 \end{pmatrix} \]
**Problem:**
Find a single vector \( \mathbf{x} \) whose image under \( T \) is \( \mathbf{b} \) and determine whether such a vector \( \mathbf{x} \) is unique.
**Solution:**
To find \( \mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \) that satisfies \( A\mathbf{x} = \mathbf{b} \), we set up the following system of linear equations based on the matrix multiplication \( A\mathbf{x} = \mathbf{b} \):
\[ \begin{pmatrix} 1 & -6 & -22 \\ -2 & 4 & 12 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} -1 \\ -6 \end{pmatrix} \]
Next, solve the system of equations to determine \( \mathbf{x} \).
### Interactive Section
**Find a single vector \( \mathbf{x} \) whose image under \( T \) is \( \mathbf{b} \).**
\[ \mathbf{x} = \boxed{ \begin{pmatrix} \, \, \end{pmatrix} } \]
In this space, students might be encouraged to use methods such as Gaussian elimination or matrix inversion to solve the system.
---
To see the complete steps for solving the system, additional resources are available in the section on Solving Systems of Linear Equations.
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