Suppose φ∈L(V,F)\varphi \in \mathcal L(V,\mathbf F). Suppose u∈Vu \in V is not in null φ\text{null }\varphi. Prove that

V=null φ⊕{au:a∈F}.V = \text{null } \varphi \oplus \{au : a\in \mathbf F\}.

TODO