a. Find the transition matrix from C to B. 188 b. Find the coordinates of u = [u] B [V] B 18 = - [³3] c. Find the coordinates of v in the ordered basis B if the coordinate vector of v in C is [v]c = H] 19 L-4] [4] In the ordered basis B.
a. Find the transition matrix from C to B. 188 b. Find the coordinates of u = [u] B [V] B 18 = - [³3] c. Find the coordinates of v in the ordered basis B if the coordinate vector of v in C is [v]c = H] 19 L-4] [4] In the ordered basis 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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![Certainly! Here is the transcribed text:
---
### Vector Spaces and Basis Transformations
Consider the ordered bases \( B = \left\{ \begin{bmatrix} 9 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \end{bmatrix} \right\} \) and \( C = \left\{ \begin{bmatrix} -3 \\ -4 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \end{bmatrix} \right\} \) for the vector space \( \mathbb{R}^2 \).
**a. Find the transition matrix from \( C \) to \( B \).**
\[
\left[ \begin{matrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{matrix} \right]
\]
**b. Find the coordinates of \( \mathbf{u} = \begin{bmatrix} 3 \\ -3 \end{bmatrix} \) in the ordered basis \( B \).**
\[
[\mathbf{u}]_B = \left[ \begin{matrix} \boxed{} \\ \boxed{} \end{matrix} \right]
\]
**c. Find the coordinates of \( \mathbf{v} \) in the ordered basis \( B \) if the coordinate vector of \( \mathbf{v} \) in \( C \) is \( [\mathbf{v}]_C = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \).**
\[
[\mathbf{v}]_B = \left[ \begin{matrix} \boxed{} \\ \boxed{} \end{matrix} \right]
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F1cc168c0-2e23-4dca-a5d0-9d67df98506f%2Fkgkv827_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Here is the transcribed text:
---
### Vector Spaces and Basis Transformations
Consider the ordered bases \( B = \left\{ \begin{bmatrix} 9 \\ 1 \end{bmatrix}, \begin{bmatrix} -1 \\ 0 \end{bmatrix} \right\} \) and \( C = \left\{ \begin{bmatrix} -3 \\ -4 \end{bmatrix}, \begin{bmatrix} 2 \\ 4 \end{bmatrix} \right\} \) for the vector space \( \mathbb{R}^2 \).
**a. Find the transition matrix from \( C \) to \( B \).**
\[
\left[ \begin{matrix} \boxed{} & \boxed{} \\ \boxed{} & \boxed{} \end{matrix} \right]
\]
**b. Find the coordinates of \( \mathbf{u} = \begin{bmatrix} 3 \\ -3 \end{bmatrix} \) in the ordered basis \( B \).**
\[
[\mathbf{u}]_B = \left[ \begin{matrix} \boxed{} \\ \boxed{} \end{matrix} \right]
\]
**c. Find the coordinates of \( \mathbf{v} \) in the ordered basis \( B \) if the coordinate vector of \( \mathbf{v} \) in \( C \) is \( [\mathbf{v}]_C = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \).**
\[
[\mathbf{v}]_B = \left[ \begin{matrix} \boxed{} \\ \boxed{} \end{matrix} \right]
\]
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