How do we prove this:
Btw, here are the properties of inner products:
<u,v>=<v,u> (
symmetry property)
<u+v,w>=<u,w>+<v,w> (a
dditivity property)
<ku,v>=k<u,v> (
homogeneity property)
<v,v> ≥ 0 and <v,v>=0 if and only if v=0 (
positivity property)
So, I
think to show that a mapping is linear we must show that it satisfies the additivity and homogeneity conditions. But I don't know how to prove these conditions here but here's what I think so far
So, for vectors
and scalars a and b,
T(x)= <x,u> v becomes
By the additivity property we get
I don't know what to do after this point. Can anyone help?