3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n × n matrix A. а) Prove ker T C kerT² C kerT³ C ….. Note: ker T" = nullspace of A" b) If kerT = kerT², show that in general we must have ker T" = ker T"+1 for all n> 1. c) Use this result ( i.e., ( b) ) to show that there is no 3 x 3 matrix X such that Го 1 01 x2 = 0 0 1 0 0 0
3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n × n matrix A. а) Prove ker T C kerT² C kerT³ C ….. Note: ker T" = nullspace of A" b) If kerT = kerT², show that in general we must have ker T" = ker T"+1 for all n> 1. c) Use this result ( i.e., ( b) ) to show that there is no 3 x 3 matrix X such that Го 1 01 x2 = 0 0 1 0 0 0
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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![3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n x n matrix A.
a)
Prove ker T C kerT² C kerT³ C …..
Note: ker T" = nullspace of A"
b) If kerT = ker T2, show that in general we must have ker T" = ker Tn+l for all n> 1.
c) Use this result ( i.e., ( b) ) to show that there is no 3 × 3 matrix X such that
0 1 0]
0 0 1
0 0 0
x²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F707cac12-b30c-449e-8c7f-ec7366c048b1%2F0ea250e2-dc90-4374-8495-5fc2fdb7062b%2Futl6kdp_processed.png&w=3840&q=75)
Transcribed Image Text:3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n x n matrix A.
a)
Prove ker T C kerT² C kerT³ C …..
Note: ker T" = nullspace of A"
b) If kerT = ker T2, show that in general we must have ker T" = ker Tn+l for all n> 1.
c) Use this result ( i.e., ( b) ) to show that there is no 3 × 3 matrix X such that
0 1 0]
0 0 1
0 0 0
x²
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