Suppose A is the matrix for T: R3 - R3 relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. 0 -2 -1 1 A = 0 0 -1 B' = {(-1, 1, 0), (2, 1, 0), (0, 0, 1)}
Suppose A is the matrix for T: R3 - R3 relative to the standard basis. Find the diagonal matrix A' for T relative to the basis B'. 0 -2 -1 1 A = 0 0 -1 B' = {(-1, 1, 0), (2, 1, 0), (0, 0, 1)}
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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![Suppose \( A \) is the matrix for \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) relative to the standard basis. Find the diagonal matrix \( A' \) for \( T \) relative to the basis \( B' \).
\[
A = \begin{bmatrix}
0 & -2 & 0 \\
-1 & 1 & 0 \\
0 & 0 & -1
\end{bmatrix}
\]
\[
B' = \{ (-1, 1, 0), (2, 1, 0), (0, 0, 1) \}
\]
The diagram below shows a \( 3 \times 3 \) matrix representation for \( A' \):
\[
A' =
\begin{bmatrix}
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}} \\
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}} \\
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}}
\end{bmatrix}
\]
Arrows indicate transformations needed to convert \( A \) to \( A' \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8bcdc0e5-d655-4cad-a91d-e7e9f676adb0%2Fde720c0a-e2ae-4547-9e95-0ac945b98933%2Fjwvic1f_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose \( A \) is the matrix for \( T: \mathbb{R}^3 \to \mathbb{R}^3 \) relative to the standard basis. Find the diagonal matrix \( A' \) for \( T \) relative to the basis \( B' \).
\[
A = \begin{bmatrix}
0 & -2 & 0 \\
-1 & 1 & 0 \\
0 & 0 & -1
\end{bmatrix}
\]
\[
B' = \{ (-1, 1, 0), (2, 1, 0), (0, 0, 1) \}
\]
The diagram below shows a \( 3 \times 3 \) matrix representation for \( A' \):
\[
A' =
\begin{bmatrix}
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}} \\
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}} \\
\boxed{\hphantom{1}} & \boxed{\hphantom{1}} & \boxed{\hphantom{1}}
\end{bmatrix}
\]
Arrows indicate transformations needed to convert \( A \) to \( A' \).
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