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The problem is:

Suppose $V$ is a vector space over a field $F$. Let $a \in F$ and $v \in V$. Show that if $av=0 \in V$, then $a=0 \in F$ or $v=0 \in V$ or both.

It's a pretty easy problem. But I've a little confusion. My approach is:

Suppose, for the sake of contradiction, $a \neq 0 \in F, v \neq 0 \in V$ and $av=0 \in V$. We'll show that these three cannot hold together. Now $av=0=a0=a\{v+(-v)\}=av+a(-v)=av+a(-1)v=av+(-a)v \implies (-a)v=0$, and so $av=(-a)v$. Is this enough to show that $a=0 \in F$ and $v=0 \in V$ cannot happen together if $av=0$? If $V=F^n$, sure. But what if it's some weird vector space? How to conclude this?

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When you approach problems such as yours in the first few times, it is worth being as explicit as possible about all the objects involved and axioms used. In particular, note that a vector space has a zero vector (which I'll denote by $0_V \in V$) and a field has the zero scalar (which I'll denote by $0_V \in \mathbb{F}$) and they are distinct objects.

You are asked to show that if $av = 0_V$ (this is the only possible interpretation, since $a$ is a scalar and $v \in V$ is a vector) then either $a = 0_{\mathbb{F}}$ or $v = 0_V$ (or maybe both).

Using this distinction, let us write your argument:

$$ av = 0_V = a \cdot 0_V = a (v - v) = av + a(-v) = av + a((-1)_{\mathbb{F}}\cdot v) = av + (-a)v. $$

By substracting $av$ from both sides we get

$$ (-a)v = 0_V $$

but you now reached pretty much the initial point, with $-a$ replaced by $a$.

Let's try a different approach. Assume that $av = 0_V$ and $a \neq 0_{\mathbb{F}}$ (otherwise, we are done). Then we can multiply both sides of $av = 0_V$ by $a^{-1}$ and obtain $$ a^{-1}(av) = (a^{-1}a)v = (1_{\mathbb{F}})v = v = a^{-1} \cdot 0_V = a^{-1}(0_V + 0_V) = a^{-1}0_V + a^{-1}0_V. $$

In particular, we have $a^{-1} \cdot 0_V = a^{-1} \cdot 0_V + a^{-1} \cdot 0_V$ so we can subtract $a^{-1}0_V$ from both sides of the equality and get $a^{-1}0_V = 0_V$ which then implies that $v = 0_V$ as required.

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    That helps. Thank you. And yes I was getting confused as the book I'm following doesn't distinguishes b/w the notations for additive identity for the field and the same for the vector space. – 2017-01-16