Definition

Let $T\in\mathscr{L}(V)$. A subspace $U\subset V$ is called invariant under $T$ if, for every $u\in U$, $T(u)\in U$

Notation

IF $T\in\mathscr{L}(V,W)$ and $U\subset V$ is a subspace, then we can restrict $T$ to get a linear map $T|_U\in\mathscr{L}(U,W)$ defined by $T|_U(u)=T(u)$ for every $u\in U\subset V$.