### Linear Transformations: Finding a VectorLet'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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Answered: 1 -6 - 22 1 If T is defined by T(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

Problem 1RQ

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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.

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