Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Educational Content: Understanding Linear Transformations and Standard Matrices
#### Problem Statement:
Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).
#### Given:
\[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \]
- \( T(\mathbf{e_1}) = (1, 9) \)
- \( T(\mathbf{e_2}) = (-6, 2) \)
- \( T(\mathbf{e_3}) = (9, -5) \)
where \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) are the columns of the \( 3 \times 3 \) identity matrix.
#### Task:
Find the matrix \( A = \begin{bmatrix} \quad \end{bmatrix} \)
*(Type an integer or decimal for each matrix element.)*
#### Explanation:
The transformation \( T \) maps vectors from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \). The standard matrix \( A \) representing this transformation can be constructed using the images of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) under \( T \).
The columns of matrix \( A \) are the images of these standard basis vectors. Therefore, the standard matrix \( A \) is constructed as follows:
\[
A = \begin{bmatrix}
1 & -6 & 9 \\
9 & 2 & -5 \\
\end{bmatrix}
\]](https://content.bartleby.com/qna-images/question/704a64b5-5250-41d0-9c29-5aaf5a50e535/3a0ce225-9618-4fae-8e8f-429d91e8010d/wo17qdc_processed.jpeg)
Transcribed Image Text:### Educational Content: Understanding Linear Transformations and Standard Matrices
#### Problem Statement:
Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).
#### Given:
\[ T: \mathbb{R}^3 \rightarrow \mathbb{R}^2 \]
- \( T(\mathbf{e_1}) = (1, 9) \)
- \( T(\mathbf{e_2}) = (-6, 2) \)
- \( T(\mathbf{e_3}) = (9, -5) \)
where \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) are the columns of the \( 3 \times 3 \) identity matrix.
#### Task:
Find the matrix \( A = \begin{bmatrix} \quad \end{bmatrix} \)
*(Type an integer or decimal for each matrix element.)*
#### Explanation:
The transformation \( T \) maps vectors from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \). The standard matrix \( A \) representing this transformation can be constructed using the images of the standard basis vectors \( \mathbf{e_1}, \mathbf{e_2}, \mathbf{e_3} \) under \( T \).
The columns of matrix \( A \) are the images of these standard basis vectors. Therefore, the standard matrix \( A \) is constructed as follows:
\[
A = \begin{bmatrix}
1 & -6 & 9 \\
9 & 2 & -5 \\
\end{bmatrix}
\]
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