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 the characteristic polynomial of Twdivides the characteristic polynomial of T.(c) Let T be a linear operator on a finite-dimensional vector space V, and letx 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 forany 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 thereexists 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 Vis a direct sum of k T-invariant subspaces, then there is a basis B for Vsuch that [T], is a direct sum of k matrices.

Name: Please see image Label the following statements as being true or false
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
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