Let $f_1,f_2,\dots,f_n$ and $f$ be linear functionals on a finite dimensional linear space $E$ over the field $K$. Let $$N=\{x:f_1(x)=f_2(x)=\cdots=f_n(x)=0\}$$ Then I need to show that (b) implies (a).
(a) There exists scalar $c_1,c_2,\dots,c_n$ such that $f= c_1f_1 + c_2f_2 +\dots+ c_n f_n$.
(b) $f(x)=0$ for all $x\in N$.
In fact both are equivalent. I have been able to show (a) implies (b), but how do I show (b) implies (a).
Any help would be appreciated.
Choose a linear independent subset of $f_1,\dots,f_n$ (say $f_1,\dots,f_k$) and complete it to a basis (say $f_1,\dots,f_k,g_1,\dots,g_l$). Find a dual basis $e_1,\dots,e_{k+l}$ of $E$. Then $c_i=f(e_i)$. (Show that $e_{k+j}\in N$ for all $j\in\{1,\dots, l\}$.)