Expectation of inner product of random vector $\mathbb{E}_{{\bf{\epsilon}}}[\langle {\bf{x}}, {\bf{\epsilon}}\rangle] = ?$

Question

Suppose I have a random vector ${\bf{\epsilon}}$ such that $$\mathbb{E}_{{\bf{\epsilon}}}[{\bf{\epsilon}}] = 0$$ I want to find the expectation of the inner product between another vector which is constant/deterministic and the random vector, i.e. $$\mathbb{E}_{{\bf{\epsilon}}}[\langle {\bf{x}}, {\bf{\epsilon}}\rangle] = ?$$

I have no clue how to go about this.

Answer 1

$E\epsilon=0$ means $E\epsilon_i=0$ for each $i$. Hence $E \langle x, e \rangle =E \sum x_i \epsilon_i=\sum x_i E \epsilon_i=0$.