Invariant Subspace Counterexample

Question

Can someone give an example:

Suppose $T \in L(V)$. If $V = W \bigoplus W'$ and if $W$ is T-invariant then $W'$ is not necessarily T-invariant.

Answer 1

Let $A:=\pmatrix{1&1\\0&1}$ as a linear map $\in L(V)$ for $V=\Bbb R^2$, and let $W$ be the $x$-axis and $W'$ the $y$-axis.