Could you help me with this task in linear-algebra? I do not know what to do to solve this math problem.I should prove that there is no function $m: \mathbb R^3 \times \mathbb R^3 \to \mathbb R^3$ so that the following conditions apply:
- $m((a,0,0),(b,0,0)) = (ab,0,0) \forall a,b\in \mathbb R$
- $m(av,w) = am(v,w) = m(v,aw) \forall v,w \in \mathbb R^3$ $a \in \mathbb R$
- $\mathbb R^3$ is a field in terms of vector addition and multiplication $m$
I would appreciate it, if you would explain to me the solution in detail or give me a hint, because I want to try to understand the task. I tried my best to translate the task from German into English.