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Let $V$ be a vector space and $\ell: V \rightarrow \mathbb{R}$ be a linear map. If $z \in V$ is not in the nullspace of $\ell$, show that every $x \in V$ can be decomposed uniquely as $x=v+c z$, where $v$ is in the nullspace of $\ell$ and $c$ is a scalar. [MORAL: The nullspace of a linear functional has codimension one.]
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