If T is an upper triangular matrix and Fk is the subspace of Rn defined as Fk = span{e1,...ek} where ei is the standard basis vector, show that Fk is an invariant subspace of T.

Then, if we restrict T to be a strictly upper triangular matrix (so the diagonal is also 0), then show that T(Fk) is contained in F(k-1).

I'm really just very confused about how to prove that something is an invariant subspace. Also, isn't the span of the standard basis vector just Rn?

I have that if v is in Fk, then v=sum(i=1 to k)(ai)(ei) for some scalars ai. Applying T and using linearity gives Tv=sum(i=1 to k)(ai)(Tei). I just don't know where to go from there.