# Why teach linear algebra before abstract algebra?

Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra?

I'm asking this because personally it seems to be much easier to understand the architecture behind linear algebra as supposed to simply solving problems after a course in abstract algebra.

• Possibly because it is simply easier?
– user122283
Mar 19, 2014 at 2:06
• Why give babies baby food before giving them steak? Mar 19, 2014 at 2:24
• And because field theory and ring theory build on linear algebra. Mar 19, 2014 at 2:37
• I took abstract algebra and real analysis before I took linear algebra, because I had (effectively) no advisor and linear algebra wasn't listed as a prerequisite. I can tell you it was definitely mistake and linear algebra was really boring. Mar 19, 2014 at 16:20
• I think the most honest answer would be: Because this has been done for decades and few teachers dare to try something new. In the linear algebra class I attended, a considerable portion of module theory has been developed, in order to give a concise proof of the Jordan normal form. The catch is that nobody could compute it. Sep 8, 2014 at 2:25

I have to provide a counterpoint to the rather cynical answers already present. To be fair, almost everyone seems to have interpreted the question to mean "what is the rationale for the current system of putting linear algebra first", whereas I would like to take the perspective that there is a good pedagogical and mathematical rationale for doing it this way, regardless of historical precedent or the needs of service classes.

The worst way to teach math is in historically-correct order: history is rife with epic intellectual struggles to find the correct generalization from within the context of an existing (possibly quite unfamiliar to us) perspective on math, previous partial generalizations and poorly-understood (possibly incorrect!) foundations. I had a professor once who said that he'd taken an abstract algebra class that proceeded from Lagrange's work on solvability of polynomials, and that the most he got out of it was that it's very difficult to think like Lagrange.

The second-worst way is in logically-correct order. That's not to say that there is no place for a rigorous development of mathematics; obviously, that is necessary at some point in the presentation of any of its fields. But you can't just sit people down, not even interested and intelligent people, and say "now we are going to learn the axiomatic development of mathematics". They have no personal reason to buy into this narrative and will probably become bored and confused.

And yet, mathematics was developed in a particular order historically, and later formalized in a particular order logically, so there is something to these choices. To teach a novice effectively you have to create a faux history for them and then develop the material internally consistently within the history. Sometimes this involves lying if that means getting across a useful piece of intuition; or omission, if that means simplifying some deeply technical preparations; or repetition, if that means that the first round of lessons makes more sense in the context of what came before than the second, complete round would have. I am reminded here of a dictum, possibly due to Littlewood, that in explaining math, a single triviality omitted is a trivial gap to fill, but two trivialities in a row can be deep. If to get from lecture 1 to lecture 3 requires the students keeping in mind a whole house of cards of connections before reaching the payoff, then there should be a payoff inside the house that is given in lecture 2.

Linear algebra has this exact relationship to abstract algebra. It starts with something that anyone with a little experience using math is familiar with: solving equations. It proceeds through something that, while apparently complicated, is also familiar: defining notation and some odd operations (matrices and row operations, multiplication, etc). This is, after all, part of solving equations too. Finally, it can lead to really abstract concepts such as vector spaces (= free modules), abstract linear spaces (= modules), change of basis (= conjugation, surely something to be learned before taking group theory!), and the first isomorphism theorem (the "rank-nullity theorem"). Problems in linear algebra can be written first to sound semi-physical or geometric, and then to make reference to concepts that were taught in such problems, and then to be fully abstract. By the end of a linear algebra course, students should have at least some foundation for thinking abstractly, as well as a big list of familiar references that will recur in abstract algebra.

So my response to your question is necessarily, "why not teach linear algebra before abstract algebra?" I think it's unfortunate that more algebra books do not use it as motivation.

• What a beautiful answer! Precise, clear and right on the money. Mar 20, 2014 at 2:05
• @RyanReich I found abstract algebra to be easier: not too many theorems and calculations to remember, and it ties more directly into the field axioms and results learned in Algebra I (so when I ignored the Professor, I could still come up with things on my own).The lower-dimensional objects around me found linear algebra easier however. Also if you need motivation to study abstract algebra you are not a true believer in Cantor's Paradise within the Platonic Realm and should be excommunicated accordingly. Nov 21, 2015 at 19:09
• @Jacob I'm adept enough at detecting irony on the internet to avoid being offended, but only just. Calling your classmates "lower-dimensional objects" is quite arrogant, and you should hew away from staking out uncompromising positions on mathematical abstraction: it's immature. Nov 21, 2015 at 21:16

Because the number of consumers of the linear algebra course (math majors, engineers, physics majors, economists, almost everyone else) is much much bigger than the number of consumers of abstract algebra course.

• that could be the case precisely because of the order of the curriculum?
– user135562
Mar 19, 2014 at 2:22
• No, it could not. While the ability to solve a system of linear equations is necessary for almost everyone (which is basically the subject of the first course in linear algebra), to know about such abstract mathematical concepts as groups, fields, and rings are required only for those studying mathematics and mathematically heavy fields. Mar 19, 2014 at 2:25
• I see. I just think that abstract algebra emphasizes more on understanding as supposed to computing, and the motivations are easier to understand. Thanks for the clarification.
– user135562
Mar 19, 2014 at 2:39
• @JunichiKoganemaru I wrote none of these conclusions that are now in your comment. Mar 19, 2014 at 2:40
• Those are just my personal opinions after my linear algebra course. Sorry if I made it seem like it was derived from your comment.
– user135562
Mar 19, 2014 at 2:42

There are several aspects that contribute to the decision to invest quite a lot of time on linear algebra before introducing abstract algebra.

Firstly, there is the historical aspect. Linear algebra came first, and groups, rings and the rest of the gang came (considerably) later. University curricula change rather slowly and we still see the left-overs of past centuries historical developments.

There is also a question of practicality. Linear algebra services numerous fields and while abstract algebra is certainly of great importance, it can be argued (successfully) that linear algebra equips one with plenty of immediate tools for use in many areas.

Then there is the general fear of the abstract. Since many students find abstract material to be very difficult (for whatever psychological reasons), universities tend to cater to the 'needs' of the students by postponing the more abstract concepts till later. I know that I personally would have loved to have learned vector spaces as a special case of modules, and deduce plenty of the theory of linear spaces from the more general theorems of module theory, but it seems to not be a preferred path for most students since linear algebra is amenable to geometric visualizations, while general module theory is not.

• Maybe it's just me, but I find it much easier to reason about abstract structures if I have a few (preferrably simple) concrete examples to toy around with.
– fgp
Mar 19, 2014 at 16:06
• Also, linear algebra is broadly used by other major fields such as engineering and finance. Abstract algebra is not often used by practitioners in those fields unless they are specialists in some specific topic. Pragmatism dictates that a linear algebra course be offered to as many people in add many disciplines as possible, and that usually means making it a lower level math department course. Sep 7, 2014 at 23:29

What confuses me is the very premise. At my school, I took linear algebra and abstract algebra in the same semester, and I wasn't skirting any prerequisites. Sure, there is what "most students do" but to say there is really a curriculum at the undergraduate level would be mistaken at many places.

Aside from that, a big reason is that linear algebra is taught to not only math majors, so the course is easier although considered higher level than calculus. Thus, it is seen as a natural progression in maturity needed (proofs to be understood and written, abstraction, etc.) Additionally, linear algebra can be used for these other majors if a math major decides to quit. None of these answers seem so compelling to the motivated undergraduate mathematician, but the normal course of things is not designed (where it is designed at all) for him/her.

I think a part of the consideration must be that the fundamental concept of linearity and linear operators, which are basic building blocks for the theory of differential equations, it's treated write extensively in linear algebra.

If we look at linear algebra add an engineering class, it adds a lot of extremely valuable intuition fire people going on to ODE and PDE worlds, where formal group theory is not always of immediate use, but linear algebra is fundamental.

In addition, if one is taught linear algebra as a sort of precursor of Hilbert spaces, with enough abstraction, it becomes nut just an intuition builder but a fundamental tool in subjects like Fourier analysis.

Linear algebra is a topic used in many applications, that many students in engineering, physics, statistics, etc. will therefore want to learn how to use without necessarily knowing all of the logic involved, which they cannot afford to learn because they have many other things they need to do. Learning a fairly concrete proof of the basic properties of determinants without learning all of the characterization theorems of determinants that hold in all fields, rings, etc. is often what is appropriate for non-math majors.

In France at one time (until reforms in the mid-90's, I think), students learned about normal subgroups and ideals of rings before they ever saw a matrix.

The reason this happened in France is that engineering school entrance exams are very competitive there, and mathematics weighs more heavily than other subjects in the selection. (Most mathematicians, physicists, etc., go through the same system as people preparing for engineering school.) Since all students were going to study abstract algebra anyway at that time, it made sense to present vector spaces as just another structure. Now they've reduced the amount of abstract algebra in the curriculum.

From the standpoint of mathematics majors, I think doing abstract algebra first would be preferable, but of course the content of the abstract and linear algebra courses would have to be changed somewhat compared to what engineering students did.

In places like the U.S., Artem's answer explains the situation perfectly.

• Even from the perspective of mathematics majors, I don't think it's good pedagogy to put abstract algebra before linear algebra. Me might ask, "why not teach ring theory before we introduce the integers?" That's a mistake of the same flavor, although different magnitude. I'm really happy I had linear algebra before abstract: linear algebra was probably what convinced me to do graduate work in abstract algebra! Sep 7, 2014 at 23:39
• There's plenty about groups of permutations, congruences and arithmetic that's interesting before you need to get to linear algebra. And of course there's a lot about abstract algebra that clarifies linear algebra - e.g., you've had a chance to learn what an inverse is in a simple context. The "abstract" part of things can in many cases help students for the same reason it helps all of us - you can go straight to the essence of some phenomenon without being concerned with extraneous considerations specific to your context.
– Dave
Sep 7, 2014 at 23:47
• I think most people learn what an inverse is in an even simpler context: the integers and real numbers. There is no burning need to crack open group theory for that purpose. Teaching from "most general to more specific" is not usually good pedagogy. It's like teaching swimming where lesson 1 is being thrown into the deep end of the pool. There is enormous motivating value in getting a good dose of linear algebra before abstract. For many students, it's one of the first classes one learns proof writing. Sep 8, 2014 at 10:24
• And please do note that I'm not trying to say "teach everything in increasing order of abstraction." I think a better picture is "first give them a lot of stepping stones to reach the abstract thing." I think there are a lot of valuable stepping stones in abstract algebra. Sep 8, 2014 at 10:28
• I think it depends in part on someone's intelligence and motivations for doing higher math. I got into higher math because of the almost religious feeling and freeness of Cantor's set theory. I did not care too much about solving linear equations (except for the fact that the condition of my brain meant I could do them faster than most of the students around me). Nov 21, 2015 at 18:38