-1 2 The standard matrix for a linear transformation T : R³ → R' is 0 1 1 -1 the change-of-basis formula to find its matrix with respect to the basis Use B =
-1 2 The standard matrix for a linear transformation T : R³ → R' is 0 1 1 -1 the change-of-basis formula to find its matrix with respect to the basis Use B =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 7**: The standard matrix for a linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) is
\[
\begin{bmatrix}
-1 & 2 & 1 \\
0 & 1 & 3 \\
1 & -1 & 1
\end{bmatrix}
\]
Use the change-of-basis formula to find its matrix with respect to the basis
\[
\mathcal{B} = \left\{
\begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix},
\begin{bmatrix}
0 \\
2 \\
3
\end{bmatrix},
\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}
\right\}.
\]
**Explanation:**
You are given a standard matrix for a linear transformation in \(\mathbb{R}^3\) and asked to find its representation with respect to a new basis \(\mathcal{B}\). This involves utilizing the change-of-basis formula, which is a fundamental concept in linear algebra used to transform matrix representations from one basis to another.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9107650-c916-4694-865d-5231de163068%2F00066d34-e141-48bf-b59c-508155630a9b%2Ffmvhmmm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 7**: The standard matrix for a linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) is
\[
\begin{bmatrix}
-1 & 2 & 1 \\
0 & 1 & 3 \\
1 & -1 & 1
\end{bmatrix}
\]
Use the change-of-basis formula to find its matrix with respect to the basis
\[
\mathcal{B} = \left\{
\begin{bmatrix}
1 \\
0 \\
-1
\end{bmatrix},
\begin{bmatrix}
0 \\
2 \\
3
\end{bmatrix},
\begin{bmatrix}
1 \\
1 \\
1
\end{bmatrix}
\right\}.
\]
**Explanation:**
You are given a standard matrix for a linear transformation in \(\mathbb{R}^3\) and asked to find its representation with respect to a new basis \(\mathcal{B}\). This involves utilizing the change-of-basis formula, which is a fundamental concept in linear algebra used to transform matrix representations from one basis to another.
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