Assume that T is a linear transformation. Find the standard matrix of T. T: R2→R“, T(e,) = (8, 1, 8, 1), and T (e2) = (-6, 9, 0, 0), where e, = (1,0) and e2 = (0,1).
Assume that T is a linear transformation. Find the standard matrix of T. T: R2→R“, T(e,) = (8, 1, 8, 1), and T (e2) = (-6, 9, 0, 0), where e, = (1,0) and e2 = (0,1).
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
![**Problem Statement:**
Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).
**Given:**
\[ T: \mathbb{R}^2 \to \mathbb{R}^4 \]
\[ T(\mathbf{e}_1) = (8, 1, 8, 1) \]
\[ T(\mathbf{e}_2) = (-6, 9, 0, 0) \]
where \( \mathbf{e}_1 = (1, 0) \) and \( \mathbf{e}_2 = (0, 1) \).
**Objective:**
Determine the standard matrix representation of the linear transformation \( T \).
**Explanation:**
To find the standard matrix \( A \) of the transformation \( T \), we use the images of the standard basis vectors in the domain:
\[
A = \begin{bmatrix}
| & | \\
T(\mathbf{e}_1) & T(\mathbf{e}_2) \\
| & |
\end{bmatrix}
=
\begin{bmatrix}
8 & -6 \\
1 & 9 \\
8 & 0 \\
1 & 0
\end{bmatrix}
\]
Thus, the standard matrix \( A \) for the transformation \( T \) is a \( 4 \times 2 \) matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e6b254a-d25d-479c-b42b-752b8b5bbcbb%2Fafcab268-98e2-4e33-bea5-d1216fdd2698%2F6ws354_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Assume that \( T \) is a linear transformation. Find the standard matrix of \( T \).
**Given:**
\[ T: \mathbb{R}^2 \to \mathbb{R}^4 \]
\[ T(\mathbf{e}_1) = (8, 1, 8, 1) \]
\[ T(\mathbf{e}_2) = (-6, 9, 0, 0) \]
where \( \mathbf{e}_1 = (1, 0) \) and \( \mathbf{e}_2 = (0, 1) \).
**Objective:**
Determine the standard matrix representation of the linear transformation \( T \).
**Explanation:**
To find the standard matrix \( A \) of the transformation \( T \), we use the images of the standard basis vectors in the domain:
\[
A = \begin{bmatrix}
| & | \\
T(\mathbf{e}_1) & T(\mathbf{e}_2) \\
| & |
\end{bmatrix}
=
\begin{bmatrix}
8 & -6 \\
1 & 9 \\
8 & 0 \\
1 & 0
\end{bmatrix}
\]
Thus, the standard matrix \( A \) for the transformation \( T \) is a \( 4 \times 2 \) matrix.
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