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# 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.

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- 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.

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