3x3 Matrices are Too Big

by rrusczyk, May 21, 2009, 7:19 PM

I'm working on the 3-dimensional vector space chapter for the Precalc book. I'm at 50 pages, with 1.5 sections to go, and no diagrams yet. All because 3x3 matrices and 3-d column vectors eat great piles of space.

In general, we've gone about linear algebra a bit differently with the AoPS curriculum. We haven't introduced vectors and matrices (except in sidebars and extras) until precalculus, rather than giving early algebra students algorithms to memorize without giving them much reason why they are true or why to memorize them. On the former point, I think very few curricula make any effort to teach where the cross product comes from, or why matrix multiplication works the way it does, or why determinants work the way they do. We include that, but it eats a ton of space. I'd be curious to hear AoPSers' input on this -- are you content with the "here are the formulas and the properties" approach that's typically offered?

One downside of taking a more natural (to me) approach to developing the concepts is that while the tools and results are simple, the algebra that sits on a page to get there can be quite ugly. Of course, that's the whole point of linear algebra to some extent -- to perform (and hide) a ton of grindwork with much simpler notation.

I also struggle with how to motivate these tools. Why do we care about the cross product? It's tough to get into that without physics. In my ideal world, all this material would be integrated into a physics class. In some places it is, but usually the physics approach is, "here are the formulas; they work, so we use them". Or, worse yet, "here's the thing you already know, even if you don't". A nontrivial part of the reason I feel compelled to add this material to the Precalc book is to make sure students have a very good idea why these tools work the way they do when they hit science classes that will assume they already have this down cold. Another reason I like putting so much of this material in the Precalc is to give students a small diversion on their death march to calculus into other areas of math (and it's a diversion that will greatly help them while they're on that death march through calculus when they hit multivariable calculus).

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Of all the things in the high school math curriculum, it took matrices the longest to make sense to me. I think they should be motivated as linear transformations.

Actually, I once wrote a short paper defining a matrix as a linear transformations between (finite-dimensional) vector spaces. I defined the matrix product as the composition of transformations, and then I introduced the normal notation for matrices. It was then a not-too-difficult exercise to verify that the traditional method for computing the product of matrices well. It's more difficult to learn, but afterwards at lot of seemingly ad-hoc stuff makes a lot of sense. For instance, it's obvious that matrix multiplication is associative, with this definition, and it's a lot easier to make sense of things like the representation of complex numbers as 2-by-2 matrices.

How are you doing determinants? I haven't seen an explanation of them that both made sense and wasn't modern-algebra-heavy. You can give a formal justification for the fact that they're multiplicative, but they still seemed very ad-hoc to me until I got through an article by Skip Garibaldi called "The Characteristic Polynomial and Determinant are Not Ad Hoc Constructions". (Here's a non-JSTOR preprint.) And they still seem a little ad hoc.

by Boy Soprano II, May 21, 2009, 7:51 PM

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I'm introducing matrices as linear transformations, pretty much. I start from a linear system of equations, and show that we can represent it as this "thing" that, when you multiply a vector by it, you get another vector. Then I go through various special matrices: the do-nothing matrix (I), a rotation matrix, a matrix that is equivalent to multiplying by $a+bi$ , etc.

Determinants are really tough. You *can* start from three properties (linearity, det(I), and multiplicative), and derive their forms (I think), but that's very heinous, even if you go a bit hand-wavy. Instead, I start by deriving Cramer in 2x2, and then define the determinant to be that form that pops out. Then I go on to show that it has all sorts of cool properties and geometric significance. I do pretty much the same on 3D, using 2D Cramer to climb up to 3D Cramer (and inspire expansion by minors), then prove properties, then do cross product and the geometric significance of everything.

I then plan to have a short chapter pointing at higher dimensions (I don't want to write a whole linear algebra book!), and then a couple shorter (I hope) chapters on applications to 2D and 3D geometry.

by rrusczyk, May 21, 2009, 8:10 PM

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Oh, cross products are very sneaky beasts. Even after calculus, it's easy to miss that they really don't belong in the same space you started with. It never makes sense to add a cross product to a vector that didn't come from a cross product.
It took a manifolds class and exposure to ideas like wedge products and Lie brackets for me to understand them as I do now.

The most natural way I have of looking at determinants is as volume operators. A determinant tells you how the volume of a region scales when you apply a linear transformation; that obviously works well with composition. Linearity in a column is nice: you're just stacking the two volumes, and then shearing into place with Cavalieri's principle. For the alternating properties: well, if there's a linear dependence relation, everything fits inside a plane with zero volume.

by jmerry, May 21, 2009, 8:36 PM

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I'll be including the effects on area and volume in the book, but everything else you mention is, well, a good bit beyond the scope of the book

by rrusczyk, May 21, 2009, 10:45 PM

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I don't know much about what the previous commenters have said, but it might be best if the ugly algebra is derived once so we know why it works.

by ThinkFlow, May 22, 2009, 1:31 AM

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Agreed with jmerry: After learning differential forms and wedge products, cross products took on a whole new meaning to me. In that light, it also becomes interesting to consider the analogs in four or more dimensions and how they (don't) work. (Even dot products break down in 4-D--or at least can't be represented the same way as 3-D dot products!)

by Sly Si, May 22, 2009, 7:35 AM

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Of course, you can't do any of that in a precalculus book . . .

by rrusczyk, May 22, 2009, 5:18 PM

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Sly Si wrote:

Even dot products break down in 4-D--or at least can't be represented the same way as 3-D dot products!

Derailing the conversation even further: What do you mean dot products in 4-D can't be represented the same way as 3-D products??

In 3-D, $\vec{v} \cdot \vec{w} = \sum_{i = 1}^3 v_iw_i$
In 4-D, $\vec{v} \cdot \vec{w} = \sum_{i = 1}^4 v_iw_i$

Back to the main conversation: I like to first introduce matrices as (linear) geometric actions on vectors (rotation, scaling, and/or reflection). I get students to learn how to write matrices that perform such actions (or combinations of actions).

I then introduce determinants as signed areas (volumes) of parallelograms (parallelopipeds) spanned by vectors. This allows students to visualize their action, and understand why when one vector is a linear combination of other vectors, the determinant of the matrix containing those vectors is zero. The properties and applications of determinants can be worked out from there.

One way to get around the nasty algebra of matrix operations is to introducuce Einstein notation (or some simplified variant of it) and the Levi-Civita Symbol. Not saying it's necessarily a good idea, just a possibility to consider.

by gauss202, May 22, 2009, 8:53 PM

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Yeah, I've thought about Einstein notation and decided that I'm not going far enough into the linear algebra material to merit it -- the complexity of the nasty algebra I'll be doing is no worse than trying to teach the machinery of Einstein notation. I'll be sidebarring Einstein notation, at the very least.

I went back and forth a great deal over whether or not to start with the geometric approach to the determinant. I think my many discussion with Dave here in the office finally convinced me to lead with the algebra since the seeing the tie between matrices and linear systems of equations is marginally more important than the tie between them and geometry. (That said, the geometric approach has a strong appeal to me, since I see things geometrically much easier than I do algebraically. I may be overcompensating in fighting that when I define the determinant with an appeal to linear systems of equations.)

I wish I had a way to test which is the best way to introduce it. It may not really matter, as long as both are presented. (After showing the tie to area for 2D determinants, I then ask them to rederive all the various properties.) So, maybe if I just go through both, it doesn't really matter in which order I present them.

For 3D, it's magic. I kind of blend the two approaches by starting with a geometric approach to an expression for the cross product, then algebraically march through the determinant, and see that the same expression pops out, and then go into the box product and geometry.

by rrusczyk, May 22, 2009, 10:05 PM

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My attitude toward Einstein notation: the up/down indexing is very useful if you do it consistently, but I'm always leaving the summation signs in. Even if you don't specify the boundaries, the visual reminder that this is a sum over certain indices is powerful.
You can't be universal with it, because of the conflict between superscripts and exponents. Talking about the Vandermonde matrix $A_i^j=a_i^j$ is just hopelessly confusing.

Yes, I can see starting with linear equations working well. They are the standard first problem, and the determinant appears as that bit you have to divide by at the end when solving the system. I think the geometric approach is better for developing the determinant in a vacuum, but systems of liner equations are better for linear algebra in general.

by jmerry, May 22, 2009, 11:58 PM