Label the following statements as being true or false. (a) There exists a linear operator...

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.

(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.
Let ${\displaystyle W=\{x,T\left(x\right),T^2\left(x\right),\dots \}=\{x,y\}}$
Since, ${\displaystyle T\left(y\right)=-x}$
${\displaystyle T(-x)=-y}$
Which means that,
${\displaystyle W^{\prime }=\{y,T\left(y\right),T^2\left(y\right),\dots \}=\{x,y\}}$
Thus, ${\displaystyle W=W^{\prime }}$ but ${\displaystyle x\ne y}$Hence, the given statement is false