Hey I have this exercise where I have some questions
For a field $K$, let $x =\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}, y = \begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}, z = \begin{pmatrix}z_1\\z_2\\z_3\end{pmatrix} $be elements in $K^3$.
Which of the following mappings $µ: K^3 × K^3 × K^3 → K$ are $K$-multilinear, which are $K$-multilinear and alternating?
(a) $µ(x, y, z) := y_2$
(b) $µ(x, y, z) := x_1y_2z_3 − x_1x_2y_3$
(c) $µ(x, y, z) := x_3y_1z_2$
(d) $µ(x, y, z) := x_1y_2z_3 + x_3y_1z_2 + x_2y_3z_1 − x_3y_2z_1 − x_2y_1z_3 − x_1y_3z_2$
So I know that "a multilinear map is a function of several variables that is linear separately in each variable".
So what I need now to prove for every exercise is that
$µ(x+\alpha j, y, z)=µ(x, y, z)+\alphaµ( j, y, z)$
$µ(x, y+\alpha j, z)=µ(x, y, z)+\alphaµ( x, j, z)$
$µ(x, y, z+\alpha j)=µ(x, y, z)+\alphaµ( x, y, j)$, right?
As you can see it can be take a lot of time to do all these steps.
My question is: is there a faster method to show that a map is multilinear? (for example using matrices)
$\mu:(K^n)^m\to K$ is $m$-linear iff $\mu(v_1,\dots,v_m)$ ($v_i\in K^n$) is a linear combination of monomials of the form $$v_{1,i_1}\dots v_{m,i_m}$$ i.e. monomials where each of the $m$ vectors contributes exactly once in the product, by one of its coordinates.
E.g. in your exercise, it is $3$-linear in cases c, d but not in cases a, b (in b, $x$ contributes twice).