4. a) Let L:V → V be a linear transformation and let B = {v1,… , Vn} be a basis for V. Suppose that L(v;) is a linear combination of the vectors v1,…, v¡ for all 1 < i < n. (Equivalently, this says L(v») E Span(v1,..., vi).) Prove that [L]s is upper triangular. b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, µ E C are non-zero. 0 1 0 0 0 0 0 0 0 0 0 H 0 0 1 A = 0 0 0 0 0 0 1

4. a) Let L:V → V be a linear transformation and let B = {v1,… , Vn} be a basis for V. Suppose that L(v;) is a linear combination of the vectors v1,…, v¡ for all 1 < i < n. (Equivalently, this says L(v») E Span(v1,..., vi).) Prove that [L]s is upper triangular. b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, µ E C are non-zero. 0 1 0 0 0 0 0 0 0 0 0 H 0 0 1 A = 0 0 0 0 0 0 1

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

4. a) Let L: V → V be a linear transformation and let B = {v1,.…., Vn} be a basis for V. Suppose that
L(vi) is a linear combination of the vectors v1,..., vị for all 1 < i < n. (Equivalently, this says
L(v:) E Span(v1,... , vi).) Prove that [L]g is upper triangular.
b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, µ ɛ C are non-zero.
1 0 0 01
0 0 0 0
0 0
0 0 µ 0 0
A =
1
0 0 0 0 0 1
c) Give matrices B and C such that B is an upper triangular matrix and C is an invertible matrix
so that B = C-'AC.

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