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Let A = 2 1 4 so that A4(t) = (t–1)³. Define T : R³ → R³ by T() = Ax. %3D 1 0 3 1 by first (a) Find a basis B for the generalized eigenspace K1 belonging to A finding a basis for E = N(A – 11), extending to a basis for N(A – 1)², and then extending to a basis for Kı = N(A – 11)³, if necessary. (b) Find the matrix [T]B ofT relative to the basis B and find a matrix P such that P-'AP = [T]B. ||

Let A = 2 1 4 so that A4(t) = (t–1)³. Define T : R³ → R³ by T() = Ax. %3D 1 0 3 1 by first (a) Find a basis B for the generalized eigenspace K1 belonging to A finding a basis for E = N(A – 11), extending to a basis for N(A – 1)², and then extending to a basis for Kı = N(A – 11)³, if necessary. (b) Find the matrix [T]B ofT relative to the basis B and find a matrix P such that P-'AP = [T]B. ||

Advanced Engineering Mathematics
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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-1 0 -4 2 1 1 0 . Let A so that Aa(t) = (t– 1)³. Define T : R³ → R³ by T(x) = Ax. 3 (a) Find a basis B for the generalized eigenspace K1 belonging to A finding a basis for E1 = N(A – 1), extending to a basis for N(A – 11)², and then extending to a basis for K1 = N(A – 1I)³, if necessary. 1 by first (b) Find the matrix [TB of T relative to the basis B and find a matrix P such that P-'AP = [T]B.
Transcribed Image Text:-1 0 -4 2 1 1 0 . Let A so that Aa(t) = (t– 1)³. Define T : R³ → R³ by T(x) = Ax. 3 (a) Find a basis B for the generalized eigenspace K1 belonging to A finding a basis for E1 = N(A – 1), extending to a basis for N(A – 11)², and then extending to a basis for K1 = N(A – 1I)³, if necessary. 1 by first (b) Find the matrix [TB of T relative to the basis B and find a matrix P such that P-'AP = [T]B.
Recall that A = [T]E, where E is the standard basis, hence P is a matrix such that P-'[T]pP= [T]B.
Transcribed Image Text:Recall that A = [T]E, where E is the standard basis, hence P is a matrix such that P-'[T]pP= [T]B.
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