The question is from Hoffman Kunze pg 106,Q14.
Let F be a field of characteristic zero and let V be a finite-dimensional vector space over F. If there exist m finitely many vectors in V, each different from the zero vector, prove that there is a linear functional f on V such that
f(a)=0 for all a in V.
This is equivalent to showing that there exist a nonsingular linear functional for a finite dimensional vector space, as i can extend m to any arbitrary no of vectors. I understand that we will map the basis of the subspace generated from above to some nonzero 1d vectors. After that,how can be show that all non zero vectors are not zero as we have only less generators in comparison to the entire V. Please help.
