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-1 4 6 2 0 -1 0 0 Let A = so that A4(t) = (t+ 1)²(t – 2)². Define T : Rª → Rª by 2 2 1 6 0 2 T(x) = Ax. (a) For each eigenvalue A, find a basis B, for the generalized eigenspace K, by first finding a basis for E, = N(A– AI) and extending to a basis for KA = N(A–AI)². (b) Let B = B_1 U B2. Find the matrix [T]B of T relative to the basis B and find a matrix P such that P-'AP = [T]B.

-1 4 6 2 0 -1 0 0 Let A = so that A4(t) = (t+ 1)²(t – 2)². Define T : Rª → Rª by 2 2 1 6 0 2 T(x) = Ax. (a) For each eigenvalue A, find a basis B, for the generalized eigenspace K, by first finding a basis for E, = N(A– AI) and extending to a basis for KA = N(A–AI)². (b) Let B = B_1 U B2. Find the matrix [T]B of T 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 4 6 2 0 -1 0 0 2 2 1 6 0 2 Let A = so that Aa(t) = (t + 1)²(t – 2)². Define T : R' → R“ by T(x) = Ax. (a) For each eigenvalue A, find a basis B, for the generalized eigenspace K, by first finding a basis for Ex = N(A- AI) and extending to a basis for K = N(A-AI)². (b) Let B = B-1U B2. Find the matrix [T|B of T relative to the basis B and find a matrix P such that P-AP = [T\B.
Transcribed Image Text:-1 4 6 2 0 -1 0 0 2 2 1 6 0 2 Let A = so that Aa(t) = (t + 1)²(t – 2)². Define T : R' → R“ by T(x) = Ax. (a) For each eigenvalue A, find a basis B, for the generalized eigenspace K, by first finding a basis for Ex = N(A- AI) and extending to a basis for K = N(A-AI)². (b) Let B = B-1U B2. Find the matrix [T|B 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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