-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
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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