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!