Relation between invariant subspaces and direct sums

Question

Let $V$ be a vector space, Let $T$ be an operator $T: V \to V$

Let $U$ be a $T$-Invariant subspace of $V$.

Does it necessarily mean that there is a $T$-Invariant subspace ($W$) of $V$ s.t.

$V = U \oplus W?$

Thanks in advance!

Answer 1

The linear transformation generated by the matrix $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ has an invariant subspace without an invariant complement.