f:VxV -> F

we call f a withered form if there's a vector for whichf(v,u)=0 for all u in V.

Could f be not-withered, while there still is a for whichf(u,v)=0 for all u in V?

(BTW : for any base B of V)

Printable View

- January 11th 2010, 08:47 AMadam63Bilinear Forms
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) - January 11th 2010, 10:54 AMtonio

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 - January 11th 2010, 11:28 AMadam63
- January 11th 2010, 11:15 PMNonCommAlg
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.