Showing a feature about a linear transformation $T:V\to V$

Question

Let $V$ be a vector space and $\lambda$ a constant real number. Suppose that $T:V\to V$ is a linear map and $\alpha:V\to \mathbb{R}$ is a linear functional on $V$.

Suppose for every $v\in Ker(\alpha)$ we have $T(v)=\lambda v$. Then prove there exists a vector $v_0\in V$ such that $$\forall v\in V\quad\quad T(v)=\lambda v\,+\,\alpha(v)v_0$$

I defined $H:V\to V$ such that $H(v)=T(v)-\lambda v$.

It's obvoius that $H$ is also a linear map and $Ker(H)=Ker(\alpha)$, and now we have to show for all vectors in $V-Ker(\alpha)$ there is a $v_0$ such that $H(v)=\alpha(v)v_0$ holds.

I don't know what to do from here, any hints would be appreciated

Answer 1

Hint: as the codimension of $\ker\alpha$ is $1$, $V = \ker\alpha\oplus\Bbb R v_0$ for some $v_0\in V$. Can you continue?