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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.

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# Hermitian Transformations

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