- #1
MathematicalPhysicist
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let V be a vector space with inner product, and T:V->V linear trans.
then for V on R, prove that for every v in V, <v,T(v)>=0 iff T*=-T.
now i got so far that: from <v,T(v)>=0 we have <v,(T+T*)(v)>=0 for every v, here I am stuck, i guess if it's for every v, if i were to write (T+T*)(v)=av for some scalar a, then i would get: a<v,v>=0 for every v, so a=0, and we get what we wanted, but I am not sure that T and T* are eigen functions i.e in the form I've given.
any pointers?
then for V on R, prove that for every v in V, <v,T(v)>=0 iff T*=-T.
now i got so far that: from <v,T(v)>=0 we have <v,(T+T*)(v)>=0 for every v, here I am stuck, i guess if it's for every v, if i were to write (T+T*)(v)=av for some scalar a, then i would get: a<v,v>=0 for every v, so a=0, and we get what we wanted, but I am not sure that T and T* are eigen functions i.e in the form I've given.
any pointers?