= Solution
{c}
\b[No.] Over any <field> $F$, take $V=F^2$, let $T=I$ be the <identity operator>, and choose the <vector subspace>
$$
U=\operatorname{span}\{e_1\},\qquad e_1=(1,0),\quad e_2=(0,1).
$$
Both $U$ and the <quotient vector space> $V/U$ have <dimension> one. A one-dimensional <vector space> has no nonzero <proper vector subspace>, so neither can have a nonzero proper <invariant subspace>. The <induced linear map on a quotient vector space> is again the <identity operator>.
However, $L=\operatorname{span}\{e_2\}$ is another nonzero <proper vector subspace> of $V$, and $T(L)=L$. Hence
$$
\boxed{L\ne U\text{ is also a nonzero proper }T\text{-invariant subspace}.}
$$
The hypotheses do not prevent a <direct sum> $V=U\oplus L$ whose two summands are <invariant subspaces>.
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