Multilinear maps: is $\phi(av_1,v_2)$ always equal to $\phi(v_1,av_2)$?

Question

I am learning about multilinear maps by myself and the book I'm following gives a definition which is somewhat vague.

That's the definition: Given vector spaces $V_1,V_2,\dots,V_p,W$. A mapping $\phi:V_1\times\dots\times V_p \rightarrow W$ is multilinear if it is linear in each argument.

Let us consider the case in which $p=2$. Let $a \in \mathbb{K}$, $v_1 \in V_1$ and $v_2 \in V_2$. Are the following steps correct?

$$ \phi(av_1,v_2) = a\phi(v_1,v_2)=\phi(v_1,av_2) $$

Answer 1

Yes, they are correct. The first equality holds by the linearity in the first argument, and the second equality holds by the linearity in the second argument.