{0} (a) Find the transition matrix from B to the standard ordered basis E= Consider the ordered bases B = T= (b) Find the transition matrix from C to B. S = (c) Find the coordinates of u = [B] [u]B = 18 Note: You can earn partial credit on this problem. and C= {}]} {D.Q} in the ordered basis B. for the vector space R².
{0} (a) Find the transition matrix from B to the standard ordered basis E= Consider the ordered bases B = T= (b) Find the transition matrix from C to B. S = (c) Find the coordinates of u = [B] [u]B = 18 Note: You can earn partial credit on this problem. and C= {}]} {D.Q} in the ordered basis B. for the vector space R².
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Linear Algebra Practice Problem: Bases and Coordinates**
Consider the ordered bases \( B = \left\{ \begin{bmatrix} 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 9 \\ 4 \end{bmatrix} \right\} \) and \( C = \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} \) for the vector space \( \mathbb{R}^2 \).
1. **Transition Matrix from \( B \) to the Standard Ordered Basis \( E \)**
Find the transition matrix from \( B \) to the standard ordered basis \( E = \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \).
\[
T = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]
2. **Transition Matrix from \( C \) to \( B \)**
Find the transition matrix from \( C \) to \( B \).
\[
S = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]
3. **Coordinates of \( \mathbf{u} \) in the Ordered Basis \( B \)**
Find the coordinates of \( \mathbf{u} = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \) in the ordered basis \( B \).
\[
[\mathbf{u}]_B = \begin{bmatrix}
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
**Note:** You can earn partial credit on this problem.
---
### Explanation of Diagrams:
This text includes three main parts. Each part requires finding matrices or coordinates based on provided vector bases and the vector space \( \mathbb{R}^2 \).
1. **Transition Matrix Diagram (Part a):**
The matrix \( T \) is a 2x2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F8d3e3c9d-58d7-433d-b77e-2a116894411f%2Fa1basdm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Linear Algebra Practice Problem: Bases and Coordinates**
Consider the ordered bases \( B = \left\{ \begin{bmatrix} 2 \\ -1 \end{bmatrix}, \begin{bmatrix} 9 \\ 4 \end{bmatrix} \right\} \) and \( C = \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix}, \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} \) for the vector space \( \mathbb{R}^2 \).
1. **Transition Matrix from \( B \) to the Standard Ordered Basis \( E \)**
Find the transition matrix from \( B \) to the standard ordered basis \( E = \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \).
\[
T = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]
2. **Transition Matrix from \( C \) to \( B \)**
Find the transition matrix from \( C \) to \( B \).
\[
S = \begin{bmatrix}
\boxed{} & \boxed{} \\
\boxed{} & \boxed{}
\end{bmatrix}
\]
3. **Coordinates of \( \mathbf{u} \) in the Ordered Basis \( B \)**
Find the coordinates of \( \mathbf{u} = \begin{bmatrix} 3 \\ -1 \end{bmatrix} \) in the ordered basis \( B \).
\[
[\mathbf{u}]_B = \begin{bmatrix}
\boxed{} \\
\boxed{}
\end{bmatrix}
\]
**Note:** You can earn partial credit on this problem.
---
### Explanation of Diagrams:
This text includes three main parts. Each part requires finding matrices or coordinates based on provided vector bases and the vector space \( \mathbb{R}^2 \).
1. **Transition Matrix Diagram (Part a):**
The matrix \( T \) is a 2x2
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