Should the cross-product in $\mathbb{R}^3$ be discussed in Linear Algebra?

Question

I have not yet taught Linear Algebra, but I teach Computer Graphics regularly, which uses linear algebra at many junctures, and uses concepts such as the cross product. I have often been disappointed to learn that even students who took Linear Algebra and did well, have little (or no) familiarity with the cross product in $\mathbb{R}^3$.

Q. Does the cross product properly belong to linear algebra, or is it a geometric diversion from the main thrust of linear algebra?

My sense is that mathematics departments prefer to keep linear algebra abstract, to shy away from geometry in favor of formal rigor, often the first such rigorous course for students. Whereas in my experience, I never fully "grokked" linear algebra until I saw orthogonal transformations (etc.) geometrically. But that is my personal bias.

I think it can be instructive to explain (at some level) that the cross product can only exist in its familiar form in in $\mathbb{R}^3$ and in $\mathbb{R}^7$:

Massey, W. S. "Cross products of vectors in higher dimensional Euclidean spaces." American Mathematical Monthly (1983): 697-701. (JSTOR link)

(My question is prompted by the recent question on determinants.)

Answer 1

The problem is that there are multiple main thrusts of linear algebra.

1. For many students (especially pure math majors), linear algebra is their first introduction to abstract algebra. When a course is designed with this in mind, rigor and formalism takes higher precedence as we want as much of the material to be reusable in the context of modules over rings and also multilinear algebra

2. On the other hand, even in the world of pure mathematics, linear algebra is fundamental in the development of differential geometry, functional analysis ("infinite dimensional linear algebra"), and differential equations (including dynamical systems, stability analysis, and PDEs). With these applications in mind, it would only make sense to introduce to the students appropriate geometric intuition behind linear transformations.

3. Outside of pure mathematics, linear algebra is often viewed as synonymous with "stuff having to do with matrices". For these applications the crucial points of understanding are how to manipulate matrices to facilitate analysis and computation (think PCA and SVD, computationally extracting information from the matrix without inverting it).

If you are somehow able to teach a class addressing all of the above, more the power to you. But the trends that I have been familiar with is to have linear algebra classes designed with different applications in mind: at the very least often times math majors end up taking a different version of the class compare to the engineers and economists.

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Going back to your question about cross-products: using a strict definition of linear algebra as the study of linear transformations of vector spaces, then cross products do not belong to linear algebra. It requires necessarily some multilinear input, be it a quadratic form (an inner product, so we can talk about orthogonality), or a volume form (to convert from two-vectors to one-vector by Hodge dual). On the other hand, these topics may already be covered in a linear algebra class (which usually do not restrict itself to the strict definition anyway), and in this case cross product may be mentioned as an application.

But if the cross product is mentioned, care must be given to the fact that it behaves "weirdly" under coordinate transformations. (The result of a cross product is a pseudovector.) One of the themes of linear algebra (at least for the first two "thrusts" that I describe) is the possibility of thinking about things in a manner that is coordinate independent (which should certainly be part of any "geometric intuition"). The formula used to compute the cross product, as it is usually taught, is certainly not coordinate independent. (I have seen students trying to compute the cross product in a spherical coordinate system.) There are a lot of subtle issues involved that may bring the discussion in conflict with some of the earlier themes of the class, and care must be given to resolve them.

When I taught linear algebra, I did not talk about the cross product in any great detail (it was not on the centrally-set syllabus; a student brought it up when we talked about computing determinants for 3-by-3 matrices). Looking at the above, it would be hard to fit it into an already tight schedule: to do justice to the subject it will have to be at least one full lecture just on its properties.

I suspect somehow that this is the heart of your problem: many of your students must have seen the cross product before. But the amount of geometry related to the cross product and its properties for a computer graphics course is likely a bit more involved than the cursory discussion the cross product is afforded in any other introductory level mathematics class.

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As a side remark: the first time I saw cross products was not in a math class at all! This was in a high school physics class where it naturally came up in the discussion of torque and angular momentum. (Yes, American high school.) But I wouldn't claim to have a good understanding of the cross product until having studied multilinear algebra and differential geometry.

Answer 2

One of the main reasons why the cross product is seldomly (to my experience) taught in linear algebra is not only the fact that this particular vector product only works well in $\mathbb{R}^3$ (a specific vector space which isn't the main focus of linear algebra), but that the arsenal of linear algebra provides already a tool to find vectors which are orthogonal to a given set of vectors:

The Gram-Schmidt algorithm.

If you take $n-1$ vectors $v_1, v_2, \dots, v_{n-1} \in \mathbb{R}^n$ and want to know a vector which is perpendicular to each of these, i.e. the normal vector on the hyperplane spanned by them, then you find some vector $v_n$ such that it completes the set to a basis. Then you can start the Gram-Schmidt procedure: Subtract orthogonal projections, normalize, repeat. You end up with an orthonormal basis $w_1, \dots, w_n$ of $\mathbb{R}^n$ such that $$\text{span}\{v_1, \dots, v_i\}=\text{span}\{w_1, \dots, w_i\}, i=1, \dots, n.$$ In particular, $w_n$ is orthogonal to all $w_1,\dots,w_{n-1}$, and, thus to $v_1,\dots,v_{n-1}$.

From a computational point of view, this might not be very effective, but from a mathematical point of view, there is a tool which does what the cross product does, and it's constructive and I think has an easier geometric interpretation than the cross product.

As a final note, there are a number of generalizations from the cross product in different areas of abstract algebra (I am no algebraist, so I cannot tell how popular or important these are).

Answer 3

Our eyes and visual cortex are fabulous tools for modeling scenarios, processing information, and communication. It's a disservice to students to ditch the visual approach because it's a "geometric diversion"

2 space and 3 space are familiar and can be visualized. Presenting coordinates as vertices of polygons or polyhedra can give a visually oriented student a handle on what the numbers can represent. Showing that matrices can scale, rotate, flip or skew these objects can engage the interest of a student that might otherwise see linear algebra as abstract concepts with little application.

To answer your question, if vector representation of 3-space is used, it's a good idea to talk about cross products. Not only is the cross product a fun and useful tool for making stuff, but it also gives an opportunity to talk about determinants.

Answer 4

Where I am working now (EPFL, Switzerland), first year students have a dedicated two semester geometry course in parallel with the usual calculus, analysis, and linear algebra sequence. I have not taught these courses but they seem to do a lot of plane and space geometry using the various tools at hand (inner product, cross products, determinants, angles...). They also cover some elementary spherical geometry and geometry of curves and surfaces.

Answer 5

The cross product is an important element of "determinants," which is a topic normally covered in linear algebra.

As such, it belongs in most courses of linear algebra. Except one that is "light" on determinants (and heavier in vector spaces and mappings).