-1 2 A = 2 1. pe the matrix for T: R3 - R3 relative to B. (a) Find the transition matrix P from B' to B. P= 2.
-1 2 A = 2 1. pe the matrix for T: R3 - R3 relative to B. (a) Find the transition matrix P from B' to B. P= 2.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.2: Exponents And Radicals
Problem 37E
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![Certainly! Below is the transcription of the image as it might appear on an educational website:
---
**Let** \( B = \{(0, 1, 1), (1, 1, 0), (1, 0, 1)\} \) **and** \( B' = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \) **be bases for** \( \mathbb{R}^3 \), **and let**
\[
A =
\begin{bmatrix}
3 & -2 & -1 \\
5 & 2 & 1 \\
2 & 1 & 2
\end{bmatrix}
\]
**be the matrix for** \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) **relative to** \( B \).
(a) **Find the transition matrix** \( P \) **from** \( B' \) **to** \( B \).
\[
P =
\begin{bmatrix}
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}}
\end{bmatrix}
\]
(b) **Use the matrices** \( P \) **and** \( A \) **to find** \([v]_B\) **and** \([T(v)]_B\), **where**
\[
[v]_{B'} =
\begin{bmatrix}
0 \\
1 \\
-1
\end{bmatrix}
\]
\[
[v]_B =
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}
\]
\[
[T(v)]_B =
\begin{bmatrix}
-2 \\
3 \\
0
\end{bmatrix}
\]
(c) **Find** \( P^{-1} \) **and** \( A' \) **(the matrix for** \( T \) **relative](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1f02723-d9e1-4704-83ab-f90dc08f102d%2Ff82d5833-999b-4384-b2b7-783d09457ecd%2F29owcsn_processed.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription of the image as it might appear on an educational website:
---
**Let** \( B = \{(0, 1, 1), (1, 1, 0), (1, 0, 1)\} \) **and** \( B' = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \) **be bases for** \( \mathbb{R}^3 \), **and let**
\[
A =
\begin{bmatrix}
3 & -2 & -1 \\
5 & 2 & 1 \\
2 & 1 & 2
\end{bmatrix}
\]
**be the matrix for** \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) **relative to** \( B \).
(a) **Find the transition matrix** \( P \) **from** \( B' \) **to** \( B \).
\[
P =
\begin{bmatrix}
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}} \\
\boxed{\phantom{0}} & \boxed{\phantom{0}} & \boxed{\phantom{0}}
\end{bmatrix}
\]
(b) **Use the matrices** \( P \) **and** \( A \) **to find** \([v]_B\) **and** \([T(v)]_B\), **where**
\[
[v]_{B'} =
\begin{bmatrix}
0 \\
1 \\
-1
\end{bmatrix}
\]
\[
[v]_B =
\begin{bmatrix}
-1 \\
0 \\
1
\end{bmatrix}
\]
\[
[T(v)]_B =
\begin{bmatrix}
-2 \\
3 \\
0
\end{bmatrix}
\]
(c) **Find** \( P^{-1} \) **and** \( A' \) **(the matrix for** \( T \) **relative
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