Why is $U$ $T$-invariant?

Question

Let $V$ a finite dimensional vector space and two sub-spaces, $U, W$ such that $V = U \oplus W$. Let's assume $T$ is a linear operator such that $W$ is $T$-invariant.

Why is it true that $U$ is also $T$-invariant?

Answer 1

Not true. For instance, let $V=\mathbb C^2$, and $$ U=\left\{\begin{bmatrix}a\\0\end{bmatrix}:\ a\in\mathbb C\right\},\ \ W=\left\{\begin{bmatrix}0\\b\end{bmatrix}:\ b\in\mathbb C\right\}. $$ Put $$ T=\begin{bmatrix}1&0\\1&1\end{bmatrix}. $$ Then $TW=W$, while $TU\not\subset U$.