Invariant subspace for a linear representation

Definition with symbols
Let $$G$$ be a group and $$\rho: G \to GL(V)$$ be a linear representation of $$G$$. An invariant subspace for $$\rho$$ is a linear subspace $$W$$ of $$V$$ such that for any $$x \in W$$, we have $$\rho(x) \in W$$.

Given an invariant subspace of $$V$$ for $$\rho$$, we can define a subrepresentation of $$\rho$$ on this invariant subspace (in fact, subrepresentations correspond to invariant subspaces. In other words, we can define an action of $$G$$ on $$W$$ be restricting the action on $$V$$.