Let B = {(1, 3), (-2, -2)} and B' = {(-12, 0), (-4, 4)} be bases for R2, and let %3D 4 3 A = 0 2 be the matrix for T: R? - R? relative to B. (a) Find the transition matrix P from B' to B. P= (b) Use the matrices P and A to find [v]g and [T(v)]g, where [v]g = [4 -3]7. %3D [v]s [T(v)]s (c) Find P-1 and A' (the matrix for T relative to B'). p-1= A'= (d) Find [T(v)]g' two ways. [T(v)]s = P-[T(v)l3 [T(v)]g = A'[v]g'
Let B = {(1, 3), (-2, -2)} and B' = {(-12, 0), (-4, 4)} be bases for R2, and let %3D 4 3 A = 0 2 be the matrix for T: R? - R? relative to B. (a) Find the transition matrix P from B' to B. P= (b) Use the matrices P and A to find [v]g and [T(v)]g, where [v]g = [4 -3]7. %3D [v]s [T(v)]s (c) Find P-1 and A' (the matrix for T relative to B'). p-1= A'= (d) Find [T(v)]g' two ways. [T(v)]s = P-[T(v)l3 [T(v)]g = A'[v]g'
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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Let B = {(1,3), (-2,-2)} and B' = {(-12,0), (-4,4)} be bases for R2, and let
A=
| 4 | 3 |
| 0 | 2 |
be the matrix for T: R2 -> R2 relative to B
![The image presents a problem involving linear algebra and matrix transformations. Here's a transcription suitable for an educational website:
---
**Linear Algebra: Transformations and Change of Basis**
Let's consider:
- **Bases for \( R^2 \):**
- \( B = \{(1, 3), (-2, -2)\} \)
- \( B' = \{(-12, 0), (-4, 4)\} \)
- **Matrix \( A \):**
\[
A = \begin{bmatrix} 4 & 3 \\ 0 & 2 \end{bmatrix}
\]
This is the matrix representation for the transformation \( T: R^2 \to R^2 \) relative to the basis \( B \).
### Tasks:
(a) **Find the Transition Matrix \( P \) from \( B' \) to \( B \).**
- Determine the matrix \( P \):
\[
P = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
(b) **Use matrices \( P \) and \( A \) to compute \([v]_B\) and \([T(v)]_B\), where \( [v]_{B'} = [4 \ -3]^T \).**
- First, express \( [v]_B \):
\[
[v]_B = \begin{bmatrix} \quad \\ \quad \end{bmatrix}
\]
- Then, find \([T(v)]_B\):
\[
[T(v)]_B = \begin{bmatrix} \quad \\ \quad \end{bmatrix}
\]
(c) **Find \( P^{-1} \) and \( A' \) (the matrix for \( T \) relative to \( B' \)).**
- Inverse of \( P \):
\[
P^{-1} = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
- Matrix \( A' \):
\[
A' = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
(d) **Calculate \([T(v)]_{B'}\) in two ways.**
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7d6fc442-7ad3-4518-9add-f310e8d32dde%2Fb8344f9c-6c18-4ea5-823f-b48cfea361d4%2Fzwxearf_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents a problem involving linear algebra and matrix transformations. Here's a transcription suitable for an educational website:
---
**Linear Algebra: Transformations and Change of Basis**
Let's consider:
- **Bases for \( R^2 \):**
- \( B = \{(1, 3), (-2, -2)\} \)
- \( B' = \{(-12, 0), (-4, 4)\} \)
- **Matrix \( A \):**
\[
A = \begin{bmatrix} 4 & 3 \\ 0 & 2 \end{bmatrix}
\]
This is the matrix representation for the transformation \( T: R^2 \to R^2 \) relative to the basis \( B \).
### Tasks:
(a) **Find the Transition Matrix \( P \) from \( B' \) to \( B \).**
- Determine the matrix \( P \):
\[
P = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
(b) **Use matrices \( P \) and \( A \) to compute \([v]_B\) and \([T(v)]_B\), where \( [v]_{B'} = [4 \ -3]^T \).**
- First, express \( [v]_B \):
\[
[v]_B = \begin{bmatrix} \quad \\ \quad \end{bmatrix}
\]
- Then, find \([T(v)]_B\):
\[
[T(v)]_B = \begin{bmatrix} \quad \\ \quad \end{bmatrix}
\]
(c) **Find \( P^{-1} \) and \( A' \) (the matrix for \( T \) relative to \( B' \)).**
- Inverse of \( P \):
\[
P^{-1} = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
- Matrix \( A' \):
\[
A' = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix}
\]
(d) **Calculate \([T(v)]_{B'}\) in two ways.**
1
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