I just, just, started reading about invariant subspaces, but I don't think I'm getting a really concrete idea of what they are. Could someone try to explain to me more advanced examples of this?
This is what I know so far; If we let $V$ be a the $F$-vector space and let a linear transformation exist that maps onto itself, i.e $T \in L(V)$. Then, a subspace $w$ of $V$ is invariant under $T$ if $T(w) = w$ for all $w \in W$.
I think a very generalized way to say this would be that if you take a linear operator and apply it to a subspace where the subspace doesn't change, then it is invariant.
I get this definition, but how does this apply to a span example? Or an upper triangle example? Or any non-trivial examples, i.e not just $ $?