Let V = R^3. Consider the matrix, A = 11 0 11 7 -3 30 4 -8 25 Define T : V →V as Tv = Av. (iii) Show that A −λI is a nilpotent matrix, i.e. there is a positive integer n such that (A −λI)^n = 0. Find the smallest such n. (iv) Find v ∈V such that (A −λI)^(n−1)v =/ 0. (v) For 1 ≤i ≤n, define v_i = (A−λI)^(n−i)v. Prove that B = {v_1,...,v_n} is a basis of V .
Let V = R^3. Consider the matrix, A = 11 0 11 7 -3 30 4 -8 25 Define T : V →V as Tv = Av. (iii) Show that A −λI is a nilpotent matrix, i.e. there is a positive integer n such that (A −λI)^n = 0. Find the smallest such n. (iv) Find v ∈V such that (A −λI)^(n−1)v =/ 0. (v) For 1 ≤i ≤n, define v_i = (A−λI)^(n−i)v. Prove that B = {v_1,...,v_n} is a basis of V .
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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Question
Let V = R^3. Consider the matrix,
A =
| 11 | 0 | 11 |
| 7 | -3 | 30 |
| 4 | -8 | 25 |
Define T : V →V as Tv = Av.
(iii) Show that A −λI is a nilpotent matrix, i.e. there is a positive
integer n such that (A −λI)^n = 0. Find the smallest such n.
(iv) Find v ∈V such that (A −λI)^(n−1)v =/ 0.
(v) For 1 ≤i ≤n, define v_i = (A−λI)^(n−i)v. Prove that B = {v_1,...,v_n}
is a basis of V .
(vi) Find the matrix of T with respect to B i.e. [T]^B_B.
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