Codimension of the null space

Dear Colleagues,

Could you please help me in the following problem:

Let $\displaystyle f\neq0$ be any linear functional on a vector space $\displaystyle X$ show that in the quotient space $\displaystyle X/N(f)$ the codim$\displaystyle N(f)=1$. Here $\displaystyle N(f)$ denotes the null space of $\displaystyle f$, and codim means the dimension of $\displaystyle X/N(f)$.

Remark: two elements $\displaystyle x_{1}, x_{2}\in X$ belong to the same element of the quotient space $\displaystyle X/N(f)$ if and only if $\displaystyle f(x_{1})=f(x_{2})$.

Regards,

Raed.