This is the problem: Let T be a complex linear space with a complex inner product <.,.>. Define T in L(V,V) to be Hermitian if <Tv,v> = <v,Tv> for all v in V. Show that T is Hermitian iff <Tv,w> = <v,Tw> for all v,w in V [Hint: apply the definition to v+w and to v+iw]. So this was my thought process: <T(v+w),v+w> = <v+w,T(v+w)> <Tv+Tw,v+w> = <v+w,Tv+Tw> <Tv,v> + <Tv,w> + <Tw,v> + <Tw,w> = <v,Tv> + <v,Tw> + <w,Tv> + <w,Tw> And the terms with the same variables cancel out by definition so this leaves <Tv,w> + <Tw,v> = <v,Tw> + <w,Tv> which doesn't really help. How do I go about doing this? Thanks.