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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51fc6b6d-0522-4a79-901e-2932bb102a96%2Ffc256d3c-22b3-4c91-b220-04030ba242ca%2F3vipwc6_processed.png&w=3840&q=75)
Transcribed Image Text: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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