A vector v in an n-dimensional vector space V is said to be a cyclic vector for a linear transformation T : V → V if the vectors (v, T(v),T 2(v),...,T n-1(v)) form a basis for V. (a) If there is a cyclic vector for T, show that the minimal polynomial of T has degree n. (b) Give an example of a linear transformation on a vector space V that does not admit a cyclic vector.

A vector v in an n-dimensional vector space V is said to be a cyclic vector for a linear transformation T : V → V if the vectors (v, T(v),T 2(v),...,T n-1(v)) form a basis for V. (a) If there is a cyclic vector for T, show that the minimal polynomial of T has degree n. (b) Give an example of a linear transformation on a vector space V that does not admit a cyclic vector.

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 26EQ

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