# property of vector-space

I don't know how can I to prove the following (this isn't homework, i'm just flipping through a book):

if a ∈ F, v ∈ V, and av=0

then a=0 or v=0

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Assume $av = 0$. If $a = 0$ you're done; otherwise, by the field axioms there is $b \in F$ with $ba = 1$; then the vector space axioms imply that $v = 1v = (ba)v = b(av) = b(0) = 0$ and you're done. [The fact that $b0 = 0$ may itself require proof.] – anon Mar 12 '11 at 4:48
anon, thank you wish i could give you rep for your reply. from your answer i was able to see how to begin proving such problems – Krolique Mar 14 '11 at 2:40

If it hasn't been proven already, try proving the preliminary facts that $0_Fv=0_V$ and $a0_V=0_V$ for $a\in F$ and $v\in V$. This follows by using that fact that $0=0+0$ (for both the zero element of $F$ and of $V$), and using distributivity.
If $a=0$, you are done. So suppose $a\neq 0$, and hence has a multiplicative inverse. Try multiplying both sides of your equation by the inverse of $a$, and recall the axiom that $1v=v$ where $1$ is the multiplicative identity of $F$, to get your conclusion.