What is the sound of one argument permuting?

Question

A function $M:\left(\mathbb{R}^{n}\right)^{k}\to\mathbb{R}$, written $M\left[\mathfrak{a}_{1},\dots,\mathfrak{a}_{k}\right]$; where $\mathfrak{a}_{i}\in\mathbb{R}^{n}$ is said to be k-multilinear on $\mathbb{R}^{n}$ if it is linear in each of its arguuments. It is said to be alternating if

$$M\left[\dots,\mathfrak{a}_{i}\dots,\mathfrak{a}_{i},\dots\right]=0.$$

That is, if any pair of arguuments are equal. Or, equivalently, if interchanging a pair of arguments reverses the the arithmetic singn of the function. That is, if $$M\left[\dots,\mathfrak{a}_{i}\dots,\mathfrak{a}_{j},\dots\right]=-M\left[\dots,\mathfrak{a}_{j}\dots,\mathfrak{a}_{i},\dots\right].$$

For example, $D:\left(\mathbb{R}^{n}\right)^{n}\to\mathbb{R},$ the determinant of an $n\times n$ matrix written as a function the column vectors

$$D\left[\mathfrak{a}_{1},\dots,\mathfrak{a}_{n}\right]=\left|\begin{bmatrix}a_{\cdot1}^{1} & \dots & a_{\cdot n}^{1}\\ \vdots & \ddots & \vdots\\ a_{\cdot1}^{n} & \dots & a_{\cdot n}^{n} \end{bmatrix}\right|,$$

is such a function.

Edwards condescends:

Notice that every linear function on $\mathbb{R}^{n}$ is automatically alternating.

How do I interchange or make equal a pair of arguments in the function $L\left[\mathfrak{a}\right]\in\mathbb{R}$?

Or, why should I conclude that a "1-multilinear function" is althernating?

Answer 1

There is no pair of distinct arguments, so the alternating hypothesis is vacuously satisfied.

Answer 2

A multi linear map is alternating for all possible inputs $v_1, \ldots, v_k$, if $v_i = v_j$ for $i \ne j$, then $M(v_1,\ldots,v_k) = 0$. For $k=1$, the hypothesis of this statement is always false so the statement that every linear map is alternating is vacuously true.