f:VxV -> F
we call f a withered form if there's a vector for which f(v,u)=0 for all u in V.
Could f be not-withered, while there still is a for which f(u,v)=0 for all u in V?
(BTW : for any base B of V)
f:VxV -> F
we call f a withered form if there's a vector for which f(v,u)=0 for all u in V.
Could f be not-withered, while there still is a for which f(u,v)=0 for all u in V?
(BTW : for any base B of V)
Perhaps I'm missing something, but I'm afraid your question is nonsensical. You're given a definition (If A then B) and then you ask: could it be that not B but still A? Of course not! The definition tells you that if A (=exists a non zero vector s.t. f(u,v) = 0 ...etc.) then B(=the bil. form is called withered)...!!
Tonio
the answer is no. to see this, suppose and identify with then for some and all let be the vector with in the th row
and everywhere else. see that is the th row of and, for any the th row of is now suppose there exists such that for all so
for all thus and so but then and hence for some thus and so for all i.e. is withered.