**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.

Answered: Image /qna-images/answer/cd7310a8-db2c-4002-a2e9-3ca7d2a45004.jpg

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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Question

**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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