### Linear Transformation and Matrix Representation#### Problem StatementShow that $ T $ is a linear transformation by finding a matrix that implements the mapping. Note that $ x_1, x_2, x_3, \ldots $ are not vectors but are entries in vectors.#### Given Transformation\[ T(x_1, x_2, x_3) = (x_1 - 9x_2 + 6x_3, x_2 - 4x_3) \]#### RequiredFind the matrix $ A $ such that:\[ T(\mathbf{x}) = A\mathbf{x} \]where $ \mathbf{x} $ is the vector $(x_1, x_2, x_3)$.#### Solution StepsTo find the matrix $ A $, we need to express the transformation $ T $ as a product of a matrix $ A $ and the vector $\mathbf{x}$.1. **Express the given transformation in matrix form:** Rewrite the transformation in the form of a matrix multiplication: \[ T \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} x_1 - 9x_2 + 6x_3 \\ x_2 - 4x_3 \end{pmatrix} \]2. **Determine the structure of the transformation matrix $ A $:** The transformation matrix $ A $ should satisfy: \[ \begin{pmatrix} x_1 - 9x_2 + 6x_3 \\ x_2 - 4x_3 \end{pmatrix} = \begin{pmatrix} 1 & -9 & 6 \\ 0 & 1 & -4 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} \]Therefore, the matrix $ A $ that implements the given linear transformation $ T $ is:\[ A = \begin{pmatrix}1 & -9 & 6 \\0 & 1 & -4\end{pmatrix}\

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Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,, X2, ... are not vectors but are entries in vectors. T(x1,X2.x3) = (X1 - 9x2 + 6x3, X2 = 4x3) A = (Type an integer or decimal for each matrix element.)

Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,, X2, ... are not vectors but are entries in vectors. T(x1,X2.x3) = (X1 - 9x2 + 6x3, X2 = 4x3) A = (Type an integer or decimal for each matrix element.)

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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Question

### Linear Transformation and Matrix Representation

#### Problem Statement
Show that $ T $ is a linear transformation by finding a matrix that implements the mapping. Note that $ x_1, x_2, x_3, \ldots $ are not vectors but are entries in vectors.

#### Given Transformation
\[ T(x_1, x_2, x_3) = (x_1 - 9x_2 + 6x_3, x_2 - 4x_3) \]

#### Required
Find the matrix $ A $ such that:
\[ T(\mathbf{x}) = A\mathbf{x} \]
where $ \mathbf{x} $ is the vector $(x_1, x_2, x_3)$.

#### Solution Steps
To find the matrix $ A $, we need to express the transformation $ T $ as a product of a matrix $ A $ and the vector $\mathbf{x}$.

1. **Express the given transformation in matrix form:**

Rewrite the transformation in the form of a matrix multiplication:
\[
T \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} = \begin{pmatrix}
x_1 - 9x_2 + 6x_3 \\
x_2 - 4x_3
\end{pmatrix}
\]

2. **Determine the structure of the transformation matrix $ A $:**

The transformation matrix $ A $ should satisfy:
\[
\begin{pmatrix}
x_1 - 9x_2 + 6x_3 \\
x_2 - 4x_3
\end{pmatrix} = \begin{pmatrix}
1 & -9 & 6 \\
0 & 1 & -4
\end{pmatrix} \begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
\]

Therefore, the matrix $ A $ that implements the given linear transformation $ T $ is:
\[
A = \begin{pmatrix}
1 & -9 & 6 \\
0 & 1 & -4
\end{pmatrix}
\

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