I am wondering about exercise 3.15 on page 25.
If $f$ is a 3-linear function on a vector space $V$ and $v_1,v_2,v_3 \in V$, what is $(Af)(v_1,v_2,v_3)$?
I have by far that $$ (Af)(v_1,v_2,v_3) = -f(v_2,v_1,v_3)-f(v_1,v_3,v_2)-f(v_3,v_2,v_1)+f(v_2,v_3,v_1)+f(v_3,v_1,v_2)+f(v_1,v_2,v_3). $$ Is this correct?
$Af$ is defined to be $(Af)(v_1,...,v_k) = \sum_{\sigma \in S_k}f(v_{\sigma(1)},...,v_{\sigma(k)})$, or in short, $Af = \sum(sgn\sigma)\sigma f$.
$\sigma$ represents all the permutations.
I also don't get the significance of those operators introduced in the book, can someone help me with the importance of those operators?