Definition:0 whenever i > j. (So, the onlyA square matrix A = [a;,j]1 a) Let L : V –→ V be a linear transformation and let B = {v1,..., vn} be a basis for V. Suppose thatL(v;) is a linear combination of the vectors v1,. .. , V; for all 1 < i < n. (Equivalently, this saysL(v;) E Span(vı, ..., vi).) Prove that [L]g is upper triangular.b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, u E C are non-zero.4.1A =11c) Give matrices B and C such that B is an upper triangular matrix and C is an invertible matrixso that B = C-lAC.

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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, ..., v;).) Prove that [L]B is upper triangular. b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, µ E C are non-zero. 1 A = 1 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-1AC.

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, ..., v;).) Prove that [L]B is upper triangular. b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, µ E C are non-zero. 1 A = 1 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-1AC.

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

Definition:
0 whenever i > j. (So, the only
A square matrix A = [a;,j]1<i,j<n is called upper triangular if a;,j
non-zero elements are in the triangular region on or above the diagonal.)

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(vı, ..., vi).) Prove that [L]g is upper triangular.
b) Find the characteristic polynomial PA(t) of the matrix A given below, where a, u E C are non-zero.
4.
1
A =
1
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-lAC.

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