Let B= (b,,b2} and C = {c1,c2) be bases for R. Find the change-of-coordinates matrix from B to C and the change-of-coordinates matrix from C to B. - 1 b = 1 b, = 3. C2 2 Find the change-of-coordinates matrix from B to C P = (Simplify your answers.) C-B
Let B= (b,,b2} and C = {c1,c2) be bases for R. Find the change-of-coordinates matrix from B to C and the change-of-coordinates matrix from C to B. - 1 b = 1 b, = 3. C2 2 Find the change-of-coordinates matrix from B to C P = (Simplify your answers.) C-B
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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![**Problem Statement:**
Let \( B = \{ \mathbf{b}_1, \mathbf{b}_2 \} \) and \( C = \{ \mathbf{c}_1, \mathbf{c}_2 \} \) be bases for \( \mathbb{R}^2 \).
Given the basis vectors:
\[
\mathbf{b}_1 = \begin{bmatrix} -1 \\ 3 \end{bmatrix},
\mathbf{b}_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix},
\mathbf{c}_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix},
\mathbf{c}_2 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}
\]
**Tasks:**
1. Find the change-of-coordinates matrix from \( B \) to \( C \).
2. Find the change-of-coordinates matrix from \( C \) to \( B \).
**Solution Approach:**
To solve this, we need to express each basis vector of \( B \) in terms of the basis vectors of \( C \), and vice versa. This is done by finding the coordinates of each basis vector of one basis with respect to the other basis.
**Change-of-Coordinates Matrix from \( B \) to \( C \):**
Represent \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \) as linear combinations of \( \mathbf{c}_1 \) and \( \mathbf{c}_2 \).
**Change-of-Coordinates Matrix from \( C \) to \( B \):**
This is basically the inverse of the change-of-coordinates matrix from \( B \) to \( C \).
**Matrix Representation:**
\[
P = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}
\]
\[
C \leftarrow B = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}
\]
(Simplify your answers.)
By solving the above transformations and simplifying the results, you will obtain the required change-of-coordinates matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9225fa68-a539-4743-ab4b-10a3968e5a82%2Fcd7310a8-db2c-4002-a2e9-3ca7d2a45004%2F8s6ltds_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( B = \{ \mathbf{b}_1, \mathbf{b}_2 \} \) and \( C = \{ \mathbf{c}_1, \mathbf{c}_2 \} \) be bases for \( \mathbb{R}^2 \).
Given the basis vectors:
\[
\mathbf{b}_1 = \begin{bmatrix} -1 \\ 3 \end{bmatrix},
\mathbf{b}_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix},
\mathbf{c}_1 = \begin{bmatrix} 1 \\ 3 \end{bmatrix},
\mathbf{c}_2 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}
\]
**Tasks:**
1. Find the change-of-coordinates matrix from \( B \) to \( C \).
2. Find the change-of-coordinates matrix from \( C \) to \( B \).
**Solution Approach:**
To solve this, we need to express each basis vector of \( B \) in terms of the basis vectors of \( C \), and vice versa. This is done by finding the coordinates of each basis vector of one basis with respect to the other basis.
**Change-of-Coordinates Matrix from \( B \) to \( C \):**
Represent \( \mathbf{b}_1 \) and \( \mathbf{b}_2 \) as linear combinations of \( \mathbf{c}_1 \) and \( \mathbf{c}_2 \).
**Change-of-Coordinates Matrix from \( C \) to \( B \):**
This is basically the inverse of the change-of-coordinates matrix from \( B \) to \( C \).
**Matrix Representation:**
\[
P = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}
\]
\[
C \leftarrow B = \begin{bmatrix} ? & ? \\ ? & ? \end{bmatrix}
\]
(Simplify your answers.)
By solving the above transformations and simplifying the results, you will obtain the required change-of-coordinates matrices.
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