. Consider the transformation LA R³ R³, and a basis 3 = (v1, V2, V3) where 2 4 -2 --( :-)--) -- 0 -- 0 A = 1 2 -1 = 1 = 2 -3 -6 3 Find the matrix A' = [LA]. Also find an invertible matrix Q such that A' = Q-¹AQ. 2 -3
. Consider the transformation LA R³ R³, and a basis 3 = (v1, V2, V3) where 2 4 -2 --( :-)--) -- 0 -- 0 A = 1 2 -1 = 1 = 2 -3 -6 3 Find the matrix A' = [LA]. Also find an invertible matrix Q such that A' = Q-¹AQ. 2 -3
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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Question
![Consider the transformation \( L_A : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), and a basis \( \beta = (v_1, v_2, v_3) \) where
\[
A = \begin{pmatrix} 2 & 4 & -2 \\ 1 & 2 & -1 \\ -3 & -6 & 3 \end{pmatrix}, \quad v_1 = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}.
\]
Find the matrix \( A' = [L_A]^\beta_\beta \). Also, find an invertible matrix \( Q \) such that \( A' = Q^{-1} A Q \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F14ceeca3-f7a4-4197-b06e-f279d8417957%2Fz0dkk38_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the transformation \( L_A : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), and a basis \( \beta = (v_1, v_2, v_3) \) where
\[
A = \begin{pmatrix} 2 & 4 & -2 \\ 1 & 2 & -1 \\ -3 & -6 & 3 \end{pmatrix}, \quad v_1 = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}, \quad v_3 = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}.
\]
Find the matrix \( A' = [L_A]^\beta_\beta \). Also, find an invertible matrix \( Q \) such that \( A' = Q^{-1} A Q \).
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