For $U$ a $G-$invariant subspace of $V$ a $G-$representation, show that $T_g(U)=U$

Question

Prove that if the subspace $U$ of the space of the representation $T:G\to GL(V)$ of $G$ is invariant, then $T(g)U = U$ for all $g \in G.$

Could any one give me a hint how to solve this question ?

Answer 1

Well, I'm assuming $V$ is a $\mathbb{C}-$vector space and $T:G\to GL(V)$ is your representation. The fact that $U$ is $G-$invariant means that for each element $g\in G$ $T_g(U)\subseteq U$. However, we also know that $T_g$ is a nonsingular linear transformation, so $\dim T_g(U)=\dim U$ so, $T_g(U)=U$.