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!