Let V be a finite dimensional vector space over a field F. Denote by n the dimension of V, and suppose that we are given a vector v $\displaystyle \neq$0 in V.

Show that there exits a linear map T: V-->F such that T(v)$\displaystyle \neq$0.(We view F as a one dimensional vector space over itself).

Show that any such T is onto, and that the kernel of T must have dimension n-1