Thread: basic bilinear forms

Let f:V × V → F be a bilinear form on a vector space V, where F is the field of scalars. 3 conditions must be fulfilled:
1. f(u1+u2,v)=f(u1,v)+f(u2,v)
2. f(u,v1+v2)=f(u,v1)+f(u,v2)
3.f(u,tv)= f(tu,v)=tf(u,v)

If F=Q (as the set of rational numbers) then if f fulfills 1. and 2. then it must fulfill 3.
I was wondering how to prove that?
Another question is to find an example of a function/map that is not a bilinear form (and precise on which field it is) because it fulfills 1. and 2. but not 3.

Thanks a lot for any help or suggestions!

Hey there,

I had an exam question on this the other day! To show that 1.,2. imply 3. on it's easiest to show it for first, then , and then . I'll give you some hints!

(n) Try induction to show that . This is obvious for , and so assume it is true for . Then From here you can use 1. and the inductive hypothesis. The symmetric case, i.e. for follows from 2.

(z) Then for , it suffices to show that (why?). Consider and use 1. Again use 2. for the symmetric case.

(q) Now for , it suffices to show (why?). Now consider and use (n).

Hope this is helpful, write back if you get stuck

(I don't know why I just used a smiley I think they're stupid but I was endeared by the yellowness.)