Given the coordinate matrix of x relative to a (nonstandard) basis 8 for R, find the coordinate matrix of x relative to the standard basis B-((3,-1), (0, 1)}, [x]s-
Given the coordinate matrix of x relative to a (nonstandard) basis 8 for R, find the coordinate matrix of x relative to the standard basis B-((3,-1), (0, 1)}, [x]s-
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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![Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis.
\[ B = \{(3, -1), (0, 1)\} \]
\[
[x]_B =
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]
\[ [x]_S =
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]
Explanation:
- The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis.
- The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \).
- The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\).
- The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0fc3766d-bc8c-44dc-b95b-e319f0e06bc0%2Faa7a6ec2-3906-4785-b124-94d755aa2a77%2Fldvquw8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Given the coordinate matrix of \( x \) relative to a (nonstandard) basis \( B \) for \(\mathbb{R}^n\), find the coordinate matrix of \( x \) relative to the standard basis.
\[ B = \{(3, -1), (0, 1)\} \]
\[
[x]_B =
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]
\[ [x]_S =
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]
Explanation:
- The problem involves converting the coordinate matrix of a vector \( x \) from a given nonstandard basis \( B \) to the standard basis.
- The basis \( B \) consists of the vectors \( (3, -1) \) and \( (0, 1) \).
- The coordinate matrix of \( x \) relative to \( B \) is \(\begin{bmatrix} 3 \\ 5 \end{bmatrix}\).
- The expression \([x]_S\) represents the coordinate matrix of \( x \) relative to the standard basis, which needs to be determined.
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