a) Given statement: there exists a linear operator T with no T-invariant subspace. Since, subspace {0} is a T-invariant for every linear operator T, that means, for linear operator T(0)=0. Hence, the given statement is false.

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 characteristics polynomial of \(\displaystyle{T}_{{W}}\) divides the characteristics polynomial of T.

The provided statement is the direct theorem.

So, the given statement is true.

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 characteristics polynomial of \(\displaystyle{T}_{{W}}\) divides the characteristics polynomial of T.

The provided statement is the direct theorem.

So, the given statement is true.