Does the zero product property hold in vector spaces?

Question

Suppose $V$ is a vector space over a field $F$. Let $v \in V\setminus \{0\}$ and $\lambda \in F$. Does $\lambda v= 0$ imply $\lambda = 0$?

Answer 1

Of course, if $\lambda \ne 0$ then exist $\lambda ^{-1}$ so $$v = \lambda ^{-1}\cdot (\lambda v) = \lambda ^{-1}(0) =0$$ A contradiction, so $\lambda =0$.