Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, wedefine Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v E Vbe any non-zero vector.i. Prove that there is a positive ksn such that T k(v) = aoT o(v)+a1T 1(v)+ …+ak-1Tk-1(v) forsome ao,a1,...,ak-1EF.ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive naturalnumber n such that T k- (v) = aoT o(v) + aiT 1(v) + … + ak-iT kr-1(v) for some ao,a1,..,ak-1EF.Prove that a .= T 0(v),T 1(v),...,T kv-1(v) is linearly independent.

Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, we define Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v EV be any non-zero vector. i. Prove that there is a positive ksn such that T k(v) = aoT 0(v)+a1T 1(v)+ ·…+ak-1Tk-1(v) for some ao,a1,..,ak-1 EF. ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural number n such that T k- (v) = aoT o(v) + aiT 1(v) + ·… + ak-1T kr-1(v) for some ao,a1,..,ak-1 E F. Prove that a = T o(v),T 1(v),.,T kv-1(v) is linearly independent.

Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, we define Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v EV be any non-zero vector. i. Prove that there is a positive ksn such that T k(v) = aoT 0(v)+a1T 1(v)+ ·…+ak-1Tk-1(v) for some ao,a1,..,ak-1 EF. ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural number n such that T k- (v) = aoT o(v) + aiT 1(v) + ·… + ak-1T kr-1(v) for some ao,a1,..,ak-1 E F. Prove that a = T o(v),T 1(v),.,T kv-1(v) is linearly independent.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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