Label the following statements as being true or false. (a) There exists a linear operator T with no T-invariant subspace. (b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristic polynomial of Tw divides the characteristic polynomial of T. (c) Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W' is the T-cyclic subspace generated by y, and W = W', then x y. (d) If T is a linear operator on a finite-dimensional vector space V, then for any xe V the T-cyclic subspace generated by x is the same as the T- cyclic subspace generated by T(x). (e) Let T be a linear operator on an n-dimensional vector space. Then there exists a polynomial g(t) of degree n such that g(T) = To. (f) Any polynomial of the form (- 1)"(a, + a,t + ·..+ a,-1t"-1 + t") is the characteristic polynomial of some linear operator. (g) If T is a linear operator on a finite-dimensional vector space V, and if V is a direct sum of k T-invariant subspaces, then there is a basis B for V such that [T], is a direct sum of k matrices.

Label the following statements as being true or false. (a) There exists a linear operator T with no T-invariant subspace. (b) If T is a linear operator on a finite-dimensional vector space V, and W is a T-invariant subspace of V, then the characteristic polynomial of Tw divides the characteristic polynomial of T. (c) Let T be a linear operator on a finite-dimensional vector space V, and let x and y be elements of V. If W is the T-cyclic subspace generated by x, W' is the T-cyclic subspace generated by y, and W = W', then x y. (d) If T is a linear operator on a finite-dimensional vector space V, then for any xe V the T-cyclic subspace generated by x is the same as the T- cyclic subspace generated by T(x). (e) Let T be a linear operator on an n-dimensional vector space. Then there exists a polynomial g(t) of degree n such that g(T) = To. (f) Any polynomial of the form (- 1)"(a, + a,t + ·..+ a,-1t"-1 + t") is the characteristic polynomial of some linear operator. (g) If T is a linear operator on a finite-dimensional vector space V, and if V is a direct sum of k T-invariant subspaces, then there is a basis B for V such that [T], is a direct sum of k matrices.

Name: Please see image Label the following statements as being true or false
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Description: Please see image Label the following statements as being true or false.(a) There exists a linear operator T with no T-invariant subspace.(b) If T is a linear operator on a finite-dimensional vector space V, and W isa T-invariant subspace of V, then t

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 38EQ

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