Use the above alogirithm to find P for the bases B = {1, 62, 63} and C = {₁, C2, C3} C-B for R³ where ~0~B~0~0~0~8 - [8]. b3 C₁ = 2 b₁ = b₂ = = = 2 3
Use the above alogirithm to find P for the bases B = {1, 62, 63} and C = {₁, C2, C3} C-B for R³ where ~0~B~0~0~0~8 - [8]. b3 C₁ = 2 b₁ = b₂ = = = 2 3
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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![**Algorithm for Finding \( P_{C \leftarrow B} \)**
Given two bases \( \mathcal{B} = \{\vec{b}_1, \ldots, \vec{b}_n\} \) and \( \mathcal{C} = \{\vec{c}_1, \ldots, \vec{c}_n\} \) for \( \mathbb{R}^n \), putting the augmented matrix
\[
[P_C \,|\, P_B]
\]
in reduced row echelon form produces the augmented matrix
\[
[I \,|\, P_{C \leftarrow B}]
\]
**Example Application:**
Use the above algorithm to find \( P_{C \leftarrow B} \) for the bases \( \mathcal{B} = \{\vec{b}_1, \vec{b}_2, \vec{b}_3\} \) and \( \mathcal{C} = \{\vec{c}_1, \vec{c}_2, \vec{c}_3\} \) for \( \mathbb{R}^3 \) where
\[
\vec{b}_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \vec{b}_2 = \begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix}, \quad \vec{b}_3 = \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}
\]
\[
\vec{c}_1 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \vec{c}_2 = \begin{bmatrix} 1 \\ 4 \\ 1 \end{bmatrix}, \quad \vec{c}_3 = \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}
\]
This algorithm provides a structured approach to finding the change of basis matrix from one basis to another in vector space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F545e4110-447f-42e0-86d8-f710095b2322%2F7747f994-7ee8-4353-8dfc-43ef3888fac6%2Fdjt9ave_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Algorithm for Finding \( P_{C \leftarrow B} \)**
Given two bases \( \mathcal{B} = \{\vec{b}_1, \ldots, \vec{b}_n\} \) and \( \mathcal{C} = \{\vec{c}_1, \ldots, \vec{c}_n\} \) for \( \mathbb{R}^n \), putting the augmented matrix
\[
[P_C \,|\, P_B]
\]
in reduced row echelon form produces the augmented matrix
\[
[I \,|\, P_{C \leftarrow B}]
\]
**Example Application:**
Use the above algorithm to find \( P_{C \leftarrow B} \) for the bases \( \mathcal{B} = \{\vec{b}_1, \vec{b}_2, \vec{b}_3\} \) and \( \mathcal{C} = \{\vec{c}_1, \vec{c}_2, \vec{c}_3\} \) for \( \mathbb{R}^3 \) where
\[
\vec{b}_1 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \quad \vec{b}_2 = \begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix}, \quad \vec{b}_3 = \begin{bmatrix} 0 \\ 2 \\ 2 \end{bmatrix}
\]
\[
\vec{c}_1 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \vec{c}_2 = \begin{bmatrix} 1 \\ 4 \\ 1 \end{bmatrix}, \quad \vec{c}_3 = \begin{bmatrix} 2 \\ 0 \\ 3 \end{bmatrix}
\]
This algorithm provides a structured approach to finding the change of basis matrix from one basis to another in vector space.
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