$n$-linear form: An Interpretation

Question

What is a good example of an $n$-linear form that is more familiar to a student learning at an elementary level?

EDIT:

I'm just trying to show that every $n$-linear alternating form on a vector space of dimension less than $n$ is the zero form.

Answer 1

Well, let's make it simple. Since you just want an example, consider the vector space $\mathbb{R}^n$ with usual operations. Then $\det : \mathbb{R}^n \times \cdots \times \mathbb{R}^n \to \mathbb{R}$ is one $n$-linear form. It's even one special case (one alternating $n$-linear form), that is it has more properties. But it doesn't matter, what it matters is that if you look at the usual definition of $\det$ you'll see that it's linear on each of it's arguments with the others held fixed.

You might ask: "what the hell you're talking about?! $\det$ is a function of a single matrix!" but if you think a little it's the same as being a function of $n$ vectors: you just have to think of the matrix as the result of joining $n$ vectors as the columns. So a function of a matrix is exact the same as a function of the $n$ columns when we think of each of them at a time.

Now if $v_1, \cdots, v_n$ are the columns of a matrix, you know that multiplying the $i$-th row $v_i$ by $k$ results the following:

$$\det(v_1,\dots ,kv_i,\dots,v_n)=k\det(v_1,\dots,v_i,\dots,v_n)$$

And if you sum some column $w$ to the column $v_i$ you have:

$$\det(v_1, \dots, v_i + w,\dots,v_n)=\det(v_1,\dots,v_i,\dots,v_n)+\det(v_1,\dots,w,\dots,v_n)$$

So, the determinant is really one $n$-form. And also it has a cool interpretation: volume. Indeed we usually define the determinant with axioms that grants him the property of being able to tell the the volume of the solid whose sides are precisely the $n$ vectors given to it. So, that it's properties reflect the fact that if $\{v_1, \dots, v_n\}$ is a linearly dependent set, it'll have $n$-volume zero, so that we get $\det(v_1, \dots, v_n) = 0$ as expected.

Also, you can think of the association of some $k$-linear alternating form of each point of the euclidean $n$-space. This is a function which we call differential form and it has very interesting properties (mainly when talking about integrals). So if you want a little more on this topic (which is too extensive to be treated here) you can see Spivak's Calculus on Manifolds or Munkres' Analysis on Manifolds. I hope this helps you. Good luck!