Question: **The problem**. 

Let $T:V \rightarrow \mathbb{F}$ be a linear map. $null\{T\}=\{v \in V : Tv=0\_{\mathbb{F}}\}$. Let $u \in V$ such that $u \notin null\{T\}$. To show that $$V=null\{T\} \oplus \{au:a \in \mathbb{F}\}$$

**My approach**.

First let us show that $V \supset null\{T\} \oplus \{au:a \in \mathbb{F}\}$. Since both $null\{T\}$ and $\{au:a \in \mathbb{F}\}$ are subspaces of $V$, the claim follows from the definition of vector space.

Now let us try to show that $V \subset null\{T\} \oplus \{au:a \in \mathbb{F}\}$. I want to show that given $v \in V$, I can decompose it as $v=u\_1+u\_2$ such that $u\_1 \in null\{T\}$ and $u\_2 \in \{au:a \in \mathbb{F}\}$. Applying $T$ to both sides and using linearity, we have $Tv=Tu\_2$. Since $u\_2 \in \{au:a \in \mathbb{F}\}$, $\exists a^\* \in F$ such that $u\_2=a^\*u$. Hence $Tv=T(a^\*u)=a^\*Tu$. So given $v \in V$, I need to find $a^\* \in \mathbb{F}$ such that $Tv=a^\*Tu$. If $F=\mathbb{R}$ or any one-dimensional field, sure. But can I always do this?

Any help would be greatly appreciated.

Answer: 
Your sentence is constructed to have the property that
$$ H \leftrightarrow (\mathsf{PA}\vdash H) $$
In particular, then, PA proves
$$ (\mathsf{PA}\vdash H) \to H $$
This is the premise for [Löb's theorem](https://en.wikipedia.org/wiki/L%C3%B6b's_theorem) which then concludes that PA proves $H$ itself.

So $H$ is true!
According to the [Stanford Encyclopedia of Philosophy](https://plato.stanford.edu/entries/goedel-incompleteness/#RefPriLbsThe), this question was raised by [Leon Henkin](https://en.wikipedia.org/wiki/Leon_Henkin) in 1952:

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> L. Henkin, Problem, *J. Symbolic Logic* 17 (1952), 160.
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[Georg Kreisel](https://en.wikipedia.org/wiki/Georg_Kreisel) pointed out that "this depends vitally on how provability is expressed; with different choices, one gets opposite answers":

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> G. Kreisel, On a problem of Henkin's, *Proc. Netherlands Acad. Sci.* 56 (1953), 405–406.
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[Martin Löb](https://en.wikipedia.org/wiki/Martin_L%C3%B6b) showed that, if the arithmetized provability predicate satisfies certain natural conditions, then the Henkin sentence $H$ is provable:

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> M. H. Löb, Solution of a problem of Leon Henkin, *J. Symbolic Logic* 20 (1955), 115–116.
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