This answered question of mine explains that by saying consider $$\dot x =Ax,$$ then the stable and unstable subspaces are invariant with respect $A$ and therefore also with respect to $\exp (tA)$ one means the definition of "invariant set" saying that a set $S\subset \mathbb{R}^n$ is invariant w.r.t. the map $A:\mathbb{R}^n \rightarrow\mathbb{R}^n$ if $A(S)\subseteq S$ - not the definition of "invariant" saying that solutions starting in $S$ stay in $S$ for all future and past times as long as they exist!
My question is: So that I can understand this distinction better, could someone please provide me with a counterexample that shows that when one reads the italic statement using the second definition,the italic statement is false ?