Let T: V →W be a linear transformation. Prove the following results. (a) N(T) = N(−T). (b) N(Tk) = N((−T)k). (c) If V = W (so that T is a linear operator on V) and λ is an eigenvalue of T, then for any positive integer k N((T −λIV)k) = N((λIV−T)k).
Let T: V →W be a linear transformation. Prove the following results. (a) N(T) = N(−T). (b) N(Tk) = N((−T)k). (c) If V = W (so that T is a linear operator on V) and λ is an eigenvalue of T, then for any positive integer k N((T −λIV)k) = N((λIV−T)k).
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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Let T: V →W be a linear transformation. Prove the following results.
(a) N(T) = N(−T).
(b) N(Tk) = N((−T)k).
(c) If V = W (so that T is a linear operator on V) and λ is an eigenvalue of T, then for any positive integer k
N((T −λIV)k) = N((λIV−T)k).
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