Prove that $-(-v) = v$ for every $v \in V$
By definition, $-v + -(-v) = 0$. We also know that $v + (-v) = 0$. Since inverses are unique, $v = -(-v)$.
Suppose $a \in F, v \in V$ and $av = \vec{0}$. Prove that either $a=0$ or $v=\vec{0}$.
If $a=0$, then we are done.
If $a \neq 0$, then we have
$$ v = 1 \cdot v = a^{-1}a \cdot v = a^{-1} \cdot (av) = a^{-1} \cdot \vec{0} = \vec{0} $$
Suppose $v,w \in V$. Explain why there exists a unique $x \in V$ such that $v + 3x = w$.
Let $x = \frac{1}{3}(w - v)$. Then
$$ v + 3x = v + 3(\frac{1}{3}(w - v)) = v + w - v = w $$
To show uniqueness, assume another $y \in V$ exists such that $v + 3y = w$. Then we get that
$$ v + 3x = v + 3y\\ 3x = 3y\\ x=y $$
The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in the definition of a vector space (1.20). Which one?