Question from an exercise
Let $V$ be a vector space over a field $F$ with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi \colon V \rightarrow F$ by $(\varphi\cdot \psi)(v) := \varphi(v)\cdot\psi(v)$ .
1) Prove that $\varphi\cdot\psi$ is a quadric form;
2)Show that every quadric form $g:V\rightarrow F$ can be written as sum of quadric forms in the form $\varphi \cdot\psi$ for some linear functionals $\varphi$ and $\psi$ in $V^{\vee}$.
I proved the first one, which was pretty easy. But I have problem proving the second one.
My idea was basically showing that there is ismorphism between the vector space $Q=\{g:V\rightarrow F| g \text{ is a quadric form} \}$ and the vector space $Z=\{\varphi \cdot \psi|\varphi,\psi\in V^{\vee}\}$
In other words I created a linear mapping $\sigma:Q\rightarrow Z$ and tried to show that it is linear and bijective. I succeeded showin that it is linear and injective but I couldn't show that it is surjective. I could show that $\operatorname{Im}\sigma = Z$ , however I would have to show that $\dim Q=\dim Z$. And I couldn't do it. Can someone give me a hint?
Thank you for the feedback!