Kernel of Linear Functionals Problem: Prove that for all non zero linear functionials $f:M\to\mathbb{K}$ where $M$ is a vector space over field $\mathbb{K}$, subspace $(f^{-1}(0))$ is of co-dimension one.
Could someone solve this for me?

If $V$ and $W$ are vector spaces and $T:V\rightarrow W$ is linear, the isomorphism theorem says that
$${V\over \ker T} \cong {\rm Im}(T) $$
In the case that $W$ is the base field and $T\not= 0$, $T$ must be onto, so 
the dimension of $V/{\ker(T)}$ is 1.  This does not depend upon finite-dimensionality.
