Linear Transforms, direct sum

Question

Suppose $T\colon V \rightarrow F$ is linear. Prove that if $ v \in V $ is s.t. $v \notin \ker(T) $ then $$ V = \ker(T) \oplus \{\alpha v: a \in F\} $$

This is a question I got in an exam. I'm trying to use the Rank Nullity theorem to prove it but am getting stuck. I also tried using the property that for $ V = U \oplus W $ to be true , $ v = u + w $ should be true where $ v \in V u \in U$ and $w \in W $ but not reaching any conclusion. Kindly help me out.

Answer 1

You have to show that $\ker(T) + Fv = V$ and you have to show that $\ker(T) \cap Fv = 0$. The second one is easy. For the first let $x \in V$ and write $x = (x - T(x)v) + T(x)v$.

Answer 2

Let $w\in V$. Suppose $T(v)=c\in F\setminus \{0\}$. Then $$T(w-T(w)c^{-1}v)=T(w)-T(w)c^{-1}c=T(w)-T(w)=0$$

Hence $w-\alpha v=u\in\ker T$. I let you verify that $vF\cap\ker T=\{0\}$.