Proving Hermitian Transformations: A Simple Approach

[eridanus]

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.

 

[AKG]

Follow the entire hint.

 

[eridanus]

Follow the entire hint.

I feel really dumb for asking this, but what? Simultaneously? I'm so confused

 

[topsquark]

I feel really dumb for asking this, but what? Simultaneously? I'm so confused

I think what he's getting at is to do the "v+iw" part and compare the two answers.

-Dan

 

[dextercioby]

Simply add the 2 results and then simplify by 2.

Daniel.