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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 43EQ
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