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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
100%
![-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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fc9e6bd5c-2c66-4a52-9427-032dca81a0aa%2F31i5qjl_processed.png&w=3840&q=75)
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fc9e6bd5c-2c66-4a52-9427-032dca81a0aa%2Ftf0y7kk_processed.png&w=3840&q=75)
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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