Question

If V is a finite dimensional vector space and v not equal to 0 is a vector in V, show that there is a linear functional f E V* such that f(v) not equal to 0

Answer 1

Given any non-zero vector we can extend it to a basis. So we extend $v$ to a basis of $V.$ Let the basis be $\{v_{1}(=v),v_2, \cdots,v_n\}$. Now we define an element $f$ of $V^*$ as $f(v_1)=1$ and $f(v_i)=0$ for all other $i.$ Then we extend $f$ by linearity since we are working on a basis. So we got a functional $f$ such that $f(v)=1\neq 0.$ This is a functional since any map from the basis set extends uniquely to a functional on the entire space.