1. ## Dimension, onto, kernels

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

2. Originally Posted by studentmath92
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
Let v1,v2,....,vn be basis of V

v = a1v1 + a2v2 + .......... anvn
as v $\displaystyle \neq$0 not all ai's are = 0. Let a1 $\displaystyle \neq$0

Define T as such
T(v1) = 1
T(vi) = 0 for all other vectors in the basis.

This T satisfies all the properties you mentioned. You can check them one by one.