Linear algebra proof help?
Suppose that V is finite dimensional and S,T are elements of L(V). Prove that ST=I if and only if TS=I.
My teacher suggested we use the rank + nullity theorem to prove this, but I have no idea how.
- 9 years agoFavorite Answer
I hope this hepls:
if ST=I then
note that (TS)^2=TSTS=TS, this is idempotent.
Then TS(TS-I)=0 therefore STS(TS-I)=S0, S(TS-I)=0.
The kernel of S is the rank of (TS-I). If S has a non trivial kernel then there is v non zero in V that Sv=0.
If Tw=v then STw=0 or if no point of w in V is such that Tw=v then STw is not v, not even if w=v; So ST cannot by the identity.
Hence S most have a trivial kernel this TS=I.