**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

Answered: Find the change-of-coordinates matrix…

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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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**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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