Dual space with $W=\ker T$ Let $f \in V^*$ with $V$ a vector space and $W=ker f $. If $v_0 \in V$ is a vector such that $f(v_0)\ne 0$ then for every $v \in V$ there exist unique $w \in W$ and scalar $c$  such that $v=cv_0+w$.
How can I prove this I don't understand it, please if someone can help me. Thanks for you time and help.

Let $n$ be the dimension of $V$.
It is an application of the classical direct sum decomposition $V=W \oplus W^{\perp}$ where $W$ is $n-1$ dimensional (an hyperplane, the kernel of linear form $f$) and $W^{\perp}$ is the one-dimensional subspace directed by $v_0$, thus $W^{\perp}=\mathbb{R}v_0$. Now use the fact that a direct sum means unique decomposition.
