Vector space/Simple Properties/Fact/Proof/Exercise

Let ${}K$ be a field and let ${}V$ be a ${}K$-vector space. Show that the following properties hold (where ${}\lambda \in K$ and ${}v\in V$).

1. We have ${}0v=0$.
2. We have ${}\lambda 0=0$.
3. We have ${}(-1)v=-v$.
4. If ${}\lambda \neq 0$ and ${}v\neq 0$ then ${}\lambda v\neq 0$.

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