# Vector geometry as a prelude to linear algebra

When I was in grade 11, I was fortunate enough to attend a high school that offered an optional course in vector geometry. The course was taught out of the book Analytic Geometry with an Introduction to Vectors and Matrices by Murdoch. The high school course covered the first $$2/3$$ of the book, so it didn't really deal with matrices.

After that, my next contact with linear algebra was in a highly abstract form as an undergraduate in Europe. (To give an example, the proofs of the theorems on kernels and images of linear mappings appealed to the analogous facts about groups. Matrices were introduced after linear mappings, quotient and dual spaces, etc.) This was made possible in part by the fact that my European classmates had all studied vector geometry the way I had as part of their normal school curriculum, and perhaps even more in depth.

In any case, I never really experienced the typical beginning linear algebra course most Americans do, so when I eventually had to teach one a number of years back, it was something foreign to me. I taught a college linear algebra course out of Elementary Linear Algebra by Anton, the compulsory textbook. I was shocked that my students' first contact with linear algebra could possibly be so ungeometric. The chapters on matrices and determinants came before the chapter on vectors. Had I not made significant changes to the presentation of the material, much of the work on linear equations, matrices and determinants would have been unsupported by what I thought were indispensable geometric interpretations.

My questions are as follows.

• Do college teachers agree with my feeling (a hunch, really) that the typical approach to linear algebra in the U.S. that deemphasizes vectors until a late stage is likely to compromise non-specialist students' ability to use linear algebra in later studies and in real-world applications?

• If so, what is the impediment in the U.S. to shifting vector geometry to an earlier stage of the high school or college curriculum? Could there ever be an AP Vectors?

• Could it be that the study of mechanics acts as a kind of back door to facility with vectors for science students, but students going on to economics or social science are handicapped because they never get an equivalent opportunity?

• Germany calling in: During my Linear Algebra course (in my first semester at uni, studying CS which had a very strong side-dish of maths back then) there was not even a hint that LA had anything whatsoever to do with geometry. I mean sure, if you were so inclined you could guess that the vector spaces or matrices of LA could somehow be related to geometry, but it was never expounded on, or heaven forbid any lines drawn in a coordinate system on the whiteboard. You could have gotten through it without having the slightest idea. Not sure if that helps you any, but you're not alone. ;)
– AnoE
Commented Aug 24, 2021 at 16:13
• @AnoE Thanks for this. Things obviously vary from country to country. I am certain that there were many problems I could never have solved without thinking geometrically, not least those that involved topological considerations.
– Dave
Commented Aug 24, 2021 at 17:20
• This is a nice book (but I haven't taught from it): Linear Algebra Through Geometry. Banchoff & Wermer. Springer. The 1st two chapters introduce vectors. Commented Aug 24, 2021 at 20:43
• @JosephO'Rourke That book looks great! Thanks for pointing it out.
– Dave
Commented Aug 24, 2021 at 21:55
• You might like Strang's linear algebra, he has a lot more geometry in proofs than most other books. Personally I view it as a bad thing since I don't want to use geometry to prove something which is really algebraic. That is the other force you're battling, the algebraists who don't see geometry where you see it. Great question, good luck understanding the logic of our current curricular set-up. It has much more to do with pragmatism than logic. Commented Aug 24, 2021 at 22:20

"You fight with the army you brought." Nobody is going to rearrange the standard HS prep courses so that you can run a non-standard (in the US) linear algebra course. You'd be better off teaching a standard course and concentrating on the actual student interactions--do a good job within your constraints--rather than dreaming about alternate scenarios. And for God's sake, please don't derail things with trying to remediate [what you see as] a gap at the beginning of the course, thus abandoning the scaffolding of the text!

Is it possible that some economists end up weak on vectors? Sure. Especially the lower ability, back of the class, not heading to grad school, tranche. But so what? They need vectors a hell of a lot less for their standard undergrad courses than the naval architects need vectors for their standard undergrad courses. And the Narchs will get what they need in HS/college physics class as well as statics, dynamics, strength of materials, etc. And heck, I run into econ professors (especially the chartist, time series obsessed, macro types) that make sunk cost or basic supply and demand logic errors. So, there's like just the econ itself that they need to learn!

Life is a linear programming optimization problem. With real constraints. Time, money, IQ, etc. Yes, there will be some gaps in people's knowledge. Try to go for the easy wins and the quickest, biggest improvements. Not obsessing on "oh there's a gap".

I would also be a little bit wary of having to have geometric intuition first. Yes, it is nice to have intuitions, when they can be QUICKLY and EASILY given ("derivative is like a tangent line"). But if it takes too much time to give--and asking for an AP course, or changing the first few weeks of course order, IS too much time--then you are actually increasing the cognitive load, not decreasing it. You might be better off teaching some symbol pushing FIRST and then developing the insights LATER.

One of the big reasons why Feynman Lectures suck versus Halliday and Resnick is that the they make more sense as a second exposure than the initial one. But the mechanical cannonball questions are a more powerful, time-efficient and cognitive load efficient way to "build your physics muscles" first. (Even Feynman, who was very intuition driven and liked thinking things out on first principles, realized this at times. For example, when he needed to understand E&M better for some work he was doing, he got a standard book and mechanically worked every exercise.)

I would also be wary of extrapolating what worked for you (clearly by profession an outlier in terms of aptitude) or even Europe (with traditionally a more elite/smaller college prep "track"), with what is best for the population in question here (US students). Perhaps you're even right. But consider that there's a reason why different materials are used with different tracks. At least be uncertain.

• Thank you for your perspective. In fact, the official syllabus of that course did emphasize vectors more than most. It just seemed the book was ill-suited to that. But the book gave me a window into what the experience of most American students was at this stage of their learning. Looking into it further, in the country I went to university in, perhaps 25% of the school population have contact with, for example, scalar and vector products, equations of lines and planes, etc. (A larger number still have used vectors in plane geometry.) So if anything, it's less "elite" than AP calculus.
– Dave
Commented Aug 24, 2021 at 14:42
• Exposure to basics of vectors (addition, components) is a normal part of the high school classwork. I still would be hesitant to derail the normal order, within LA, of matrices first, just for the benefit of more supposed intuition. Maybe it's simpler that way. I see a bad tendency here of people wanting to "do everything at once". At least, I would be wary of doing so, that it may increase the cognitive load. It's not like the standard order is always better. But you ought to at least be a bit wary that there may be good reasons for it. Commented Aug 24, 2021 at 15:17
• Bold of you to think life is a linear problem. Commented Aug 24, 2021 at 16:31
• @Dave: in the country I went to university in, perhaps 25% of the school population have contact with, for example, scalar and vector products, equations of lines and planes, etc. --- I think the typical first contact with these notions for U.S. students is either in a 1st year college physics course or the 3rd semester of an elementary calculus sequence (when multivariable calculus is taken up). The topics used to be covered in college analytic geometry courses in the U.S., which were gradually phased out during the 1950s and 1960s, with the analytic geometry moving into (continued) Commented Aug 24, 2021 at 18:03
• I'm not really sure about this, but my personal experience goes back to 1976, and at least in the few colleges/universities I was aware of near where I lived, a basic 2nd year linear algebra course was not as common as a calculus 3 course (many smaller private colleges did not offer the former then), but places that offered both tended to have students take them at roughly the same time (2nd year sometime). Commented Aug 24, 2021 at 21:18