basis of R² is given as B₁ = { x₁ = (1,1), x₁ = (1,0) }; Find the change of basis matrix P, from b whore R ·5-11 2) - (221)
basis of R² is given as B₁ = { x₁ = (1,1), x₁ = (1,0) }; Find the change of basis matrix P, from b whore R ·5-11 2) - (221)
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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### Change of Basis in \(\mathbb{R}^2\)
**Problem Statement:**
Given two bases \( B_1 \) and \( B_2 \) in \(\mathbb{R}^2\):
- Basis \( B_1 \) is defined by vectors:
\[
x_1 = (1,1), \quad x_2 = (1,0)
\]
- Basis \( B_2 \) is defined by vectors:
\[
y_1 = (4,3), \quad y_2 = (3,2)
\]
Find the change of basis matrix \( P \) from basis \( B_1 \) to basis \( B_2 \).
**Solution Approach:**
1. **Construct the Matrix Representation of Basis Vectors:**
Represent the vectors of \( B_1 \) and \( B_2 \) as columns in matrices \( A \) and \( B \), respectively.
\[
A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}
\]
2. **Determine the Change of Basis Matrix:**
The change of basis matrix \( P \) from \( B_1 \) to \( B_2 \) is found by:
\[
P = B A^{-1}
\]
3. **Calculate the Inverse of \( A \):**
Compute the inverse of matrix \( A \):
\[
A^{-1} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{-1}
= \begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}
\]
4. **Compute the Change of Basis Matrix \( P \):**
Multiply \( B \) by \( A^{-1} \):
\[
P = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}
= \begin{pmatrix} 3 & 1 \\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2c8c9d8-cb7d-4bb3-b241-ba372e4ba26e%2F78fb4762-d74e-4ac4-880b-0aaa1e626296%2Fi5jutg_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Change of Basis in \(\mathbb{R}^2\)
**Problem Statement:**
Given two bases \( B_1 \) and \( B_2 \) in \(\mathbb{R}^2\):
- Basis \( B_1 \) is defined by vectors:
\[
x_1 = (1,1), \quad x_2 = (1,0)
\]
- Basis \( B_2 \) is defined by vectors:
\[
y_1 = (4,3), \quad y_2 = (3,2)
\]
Find the change of basis matrix \( P \) from basis \( B_1 \) to basis \( B_2 \).
**Solution Approach:**
1. **Construct the Matrix Representation of Basis Vectors:**
Represent the vectors of \( B_1 \) and \( B_2 \) as columns in matrices \( A \) and \( B \), respectively.
\[
A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix}
\]
2. **Determine the Change of Basis Matrix:**
The change of basis matrix \( P \) from \( B_1 \) to \( B_2 \) is found by:
\[
P = B A^{-1}
\]
3. **Calculate the Inverse of \( A \):**
Compute the inverse of matrix \( A \):
\[
A^{-1} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^{-1}
= \begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}
\]
4. **Compute the Change of Basis Matrix \( P \):**
Multiply \( B \) by \( A^{-1} \):
\[
P = \begin{pmatrix} 4 & 3 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}
= \begin{pmatrix} 3 & 1 \\
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