Intuitive understanding behind exterior algebra construction

Question

I'm trying to get deeper intuition into the exterior algebra construction on a finite dimensional $\mathbb{R}$-vector space.

Our accustomed notion of volume given by measure is neither multi-linear nor anti-symmetric, so I don't buy construction of a 'volume function' as an a priori motivation for an exterior algebra.

It's great that $v_1 \wedge \cdots \wedge v_n = \alpha \ e_1 \wedge \cdots \wedge e_n$ computes the (signed) volume $\alpha$ of the parallelotope spanned by these vectors.

But this fact seems rather arbitrary and a priori unexpected.

It would be nice to have a narrative as to why constructing an exterior algebra on a vector space is just the natural thing to do. For instance, generalizing from metric spaces to topological spaces is very natural once we realize that metrics just generate open sets, and that continuity of functions can be characterized by their behavior on open sets alone.

Is there any reason why one would intuitively anticipate beforehand that constructing an alternating algebra on a vector space would give a device to compute volumes, detect linear dependence etc.?

Or should the recognition of these facts just be considered a random encounter in the process of experimentation with mathematical constructs?

Answer 1

I had some of the same thoughts you're describing in this post about the whole philosophy behind alternating forms and exterior algebra. The thing that started making things click for me was reading Terry Tao's introduction to differential forms.

Basically at one point he describes how in the univariate calculus we get in school, three notions of integration are actually slurred together. For me, properties of one notion were actually interfering with my understanding of another.

While classical measure theory has us focus on nonnegative set functions to measure sets, differential geometry (or maybe I should say algebraic topology) chooses objects that retain information about orientation of $n$-dimensional volumes.

This is going to be badly explained, and experts are probably going to have a lot to say by way of correction, but here goes.

One intuition I have is that orientation and alternativity give you what you need to stack cells against each other and keep track of their surface area. For instance, you can think of the faces of two cubes joined on one side as having a "surface" consisting of a combination of oriented squares. Individually the cubes have six oriented squares on their surface, but together there are $10$ squares tiling the surface. The square shared on their common side has opposite orientations on each cube, so they cancel out when they make contact. These collections of squares are examples of chains (in this sense).

Answer 2

This is not rigorous, but hopefully addresses the intuitive understanding of the motivation for arriving at the exterior algebra. I am putting non-standard terminology that I am inventing for this answer in quotes.

In an affine geometry, relative locations of points is naturally described by vectors. For example, vectors giving the relative positions of a sequence of points naturally add together to give a resultant vector that gives the relative position of the last to the first point. For my purpose here, it is sufficient to consider the case of a collinear sequence of points: when we add vectors from the same subspace, the original and final vectors all belong to the same one-dimensional subspace (are all scalar multiples of a single vector). This holds independently of any length measure (e.g. a metric).

Similarly, in any plane of an $n$-dimensional affine space, we can have plane figures. Any linear transformation of the space keeps them related, and in some sense there is a measure of relative "affine area" of parallel plane geometric figures that is preserved by affine transformations of the space. A simple argument would show that any area must be a bilinear function of any two linearly independent vectors in the same plane that are transformed with the figure. My crucial premise: the "affine area" of any two plane figures in parallel planes must be related by a scalar multiplication just as two vectors giving offsets on parallel lines are related, and the scalar value is unaffected by affine transformations of the space. This holds for any higher-dimensional figures, such as the "affine volume" of parallel geometric figures in 3-dimensional affine subspaces: provided that the containing geometric $k$-flats containing the $k$-dimensional figures are parallel, the "affine $k$-measure" will be related by some scalar.

Multilinearity (bilinearity in the case of "affine area") together with this property of the "affine $k$-measure" of figures in parallel $k$-flats always being related by a scalar is sufficient to derive the exterior algebra as the unique algebra with the necessary properties. We can see this as follows in the 2-dimensional case (but the argument generalizes to any number of dimensions). Consider the "affine area" (i.e. "affine 2-measure") of a geometric figure in a plane (2-flat) in a possibly higher-dimensional affine space, along with two linearly independent vectors $u_1$ and $u_2$ and similarly of another pair $v_1$ and $v_2$ in the plane. The "affine area" of the figure is a bilinear function of each pair of vectors: $a = k_1 f(u_1, u_2) = k_2 f(v_1, v_2)$ for some scalar constants $k_1$ and $k_2$. However, being in the same plane, these two pairs of vectors are linearly related: $v_1 = c_{1,1}u_1 + c_{1,2}u_2$ and $v_2 = c_{2,1}u_1 + c_{2,2}u_2$. This leads us to the relationship $k_1 f(u_1, u_2) = k_2 f(v_1, v_2) = k_2 f(c_{1,1}u_1 + c_{1,2}u_2, c_{2,1}u_1 + c_{2,2}u_2)$. Using the bilinearity of the function $f$ and setting $u_2 = 0$, it follows that for all $u_1$, $f(u_1, u_1) = 0$. Thus we have that any reasonable "affine area" function $f$ of two vectors that are directly related to this area is necessarily alternating, i.e. equal to zero when two of its arguments are equal. I will not explicitly extend this to higher dimensions here.

Being an alternating multilinear function on its parameters is the defining characteristic of the exterior product: up to isomorphism, it is the unique "freest" or "most general" such function. That is, the exterior algebra is unique in having the simple geometric properties that we demanded when working from geometric intuition.