Find the change-of-coordinates matrix from B to the standard basis in R". 2 -2 3 8 -2 7 8. Pa
Find the change-of-coordinates matrix from B to the standard basis in R". 2 -2 3 8 -2 7 8. Pa
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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![**Linear Algebra: Change-of-Coordinates Matrix**
**Problem Statement:**
This exercise involves finding the change-of-coordinates matrix from the given basis \( B \) to the standard basis in \( \mathbb{R}^n \).
**Given Basis:**
\[ B = \left\{ \begin{pmatrix} 2 \\ 8 \\ 7 \end{pmatrix}, \begin{pmatrix} -2 \\ 0 \\ -5 \end{pmatrix}, \begin{pmatrix} 3 \\ -2 \\ 8 \end{pmatrix} \right\} \]
**Question:**
Find the change-of-coordinates matrix from \( B \) to the standard basis in \( \mathbb{R}^n \).
The change-of-coordinates matrix \( P_B \) will be displayed below the problem statement, typically as follows:
\[ P_B = \boxed{ \ \ \ } \]
**Explanation:**
The basis \( B \) is composed of three vectors in \( \mathbb{R}^3 \). The vectors are:
1. \( \begin{pmatrix} 2 \\ 8 \\ 7 \end{pmatrix} \)
2. \( \begin{pmatrix} -2 \\ 0 \\ -5 \end{pmatrix} \)
3. \( \begin{pmatrix} 3 \\ -2 \\ 8 \end{pmatrix} \)
To find the change-of-coordinates matrix \( P_B \), we need to form a matrix whose columns are these three vectors. This matrix, once inverted, will serve as the change-of-coordinates matrix from the given basis \( B \) to the standard basis in \( \mathbb{R}^n \).
**Detailed Steps (for educational clarity):**
1. **Form the Matrix \( B \)** with the vectors as columns.
\[ B = \begin{pmatrix} 2 & -2 & 3 \\ 8 & 0 & -2 \\ 7 & -5 & 8 \end{pmatrix} \]
2. **Compute the Inverse \( B^{-1} \)**. The result of this inversion will be the required change-of-coordinates matrix \( P_B \).
\[ P_B = B^{-1} \]
3. **Output the Matrix \( P_B \)**.
**Note:**
Finding the inverse of a matrix](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8700fa5c-0cb9-46f9-817d-473369b2beec%2F98d88b29-d4e5-4c16-9a1a-8cc0f97a756a%2F5wfh9h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra: Change-of-Coordinates Matrix**
**Problem Statement:**
This exercise involves finding the change-of-coordinates matrix from the given basis \( B \) to the standard basis in \( \mathbb{R}^n \).
**Given Basis:**
\[ B = \left\{ \begin{pmatrix} 2 \\ 8 \\ 7 \end{pmatrix}, \begin{pmatrix} -2 \\ 0 \\ -5 \end{pmatrix}, \begin{pmatrix} 3 \\ -2 \\ 8 \end{pmatrix} \right\} \]
**Question:**
Find the change-of-coordinates matrix from \( B \) to the standard basis in \( \mathbb{R}^n \).
The change-of-coordinates matrix \( P_B \) will be displayed below the problem statement, typically as follows:
\[ P_B = \boxed{ \ \ \ } \]
**Explanation:**
The basis \( B \) is composed of three vectors in \( \mathbb{R}^3 \). The vectors are:
1. \( \begin{pmatrix} 2 \\ 8 \\ 7 \end{pmatrix} \)
2. \( \begin{pmatrix} -2 \\ 0 \\ -5 \end{pmatrix} \)
3. \( \begin{pmatrix} 3 \\ -2 \\ 8 \end{pmatrix} \)
To find the change-of-coordinates matrix \( P_B \), we need to form a matrix whose columns are these three vectors. This matrix, once inverted, will serve as the change-of-coordinates matrix from the given basis \( B \) to the standard basis in \( \mathbb{R}^n \).
**Detailed Steps (for educational clarity):**
1. **Form the Matrix \( B \)** with the vectors as columns.
\[ B = \begin{pmatrix} 2 & -2 & 3 \\ 8 & 0 & -2 \\ 7 & -5 & 8 \end{pmatrix} \]
2. **Compute the Inverse \( B^{-1} \)**. The result of this inversion will be the required change-of-coordinates matrix \( P_B \).
\[ P_B = B^{-1} \]
3. **Output the Matrix \( P_B \)**.
**Note:**
Finding the inverse of a matrix
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