What does the notation $T_{|\text{inv}(v)}$ means in linear algebra?

Question

I'm working on algebra and I've been seeing this notation: let $T: V \rightarrow V$ be a linear map and take $v \in V$ such that $\text{inv}(v) = V$.

What does the notation $T_{|\text{inv}(v)}$ means?

In my case, I know that $T_{|\text{inv}(v)} = T$.

Thank you!

Answer 1

Most generally, if $f:A \rightarrow B$ is a map of sets, and $C \subseteq A$, then the notation $f|_C$ is the restriction of $f$ to $C$. It just means restrict the domain of $A$ to the subset $C$. In your case, $T|_{\text{inv}(v)}$ means the restriction of $T$ to the invariant subspace of $V$ generated by $v$.

In a specific linear algebra setting, it is possible to write a statement like if $T:V \rightarrow W$ is a linear map and $U$ is a subspace of $V$, consider $T|_U$. It is the linear map restricted to the subspace $U$.

Edit: You asked for another specific example, you've talked about invariant subspaces so if $T:V \rightarrow V$ is a linear map and $W$ is a $T$-invariant subspace of $V$, then $T|_W$ is a linear map $T:W \rightarrow W$.