(5) Let V be a finite dimensional vector space with basis {v1, ... , Un} and T E L(V) Show the following are equivalent: (a) The matrix for T is upper triangular. (b) T(vk) E Span(v1,.., vk) for all k. (c) Span(v1, ... , Vk) is T-invariant for all k.

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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(5) Let V be a finite dimensional vector space with basis {v1, ... , Un} andT E L(V).
Show the following are equivalent:
(a) The matrix for T is upper triangular.
(b) T(vk) E Span(v1, ... , vk) for all k.
(c) Span(v1, ... , Vk) is T-invariant for all k.
Transcribed Image Text:(5) Let V be a finite dimensional vector space with basis {v1, ... , Un} andT E L(V). Show the following are equivalent: (a) The matrix for T is upper triangular. (b) T(vk) E Span(v1, ... , vk) for all k. (c) Span(v1, ... , Vk) is T-invariant for all k.
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