# Should we teach abstract affine spaces?

In France at least, there is quite an ancient tradition of teaching abstract affine spaces (e.g. as a triple $(\mathcal{E}, E, -)$ where $\mathcal{E}$ is a set, $E$ is a vector space and $-:\mathcal{E}\times\mathcal{E}\to E$ is a binary operation with the adequate properties) which somewhat continues.

I liked that kind of approach as an undergrad, by I more and more feel it is artificial, and that we should restrict to study affine subspaces of vector spaces.

Edit: to be more precise, I am not against explaining that an affine space is like a vector space without a origin, on the contrary; but my point is that such concepts can be explained by sticking to affine subspaces of vector spaces (the vector space itself being an affine subspace, and the origin loosing its meaning in that structure).

My first question is:

What are some arguments in favor of teaching abstract affine spaces ?

To explain more my reluctance, let me say I am turning more and more into a example-based mathematician and teacher; I am thus driven away from abstract affine spaces by the fact that I do not have a good answer to my second question:

What is an example of a "natural" affine space, which is not "naturally" an affine subspace of a vector space?

• May I ask why you're becoming more of an example based teacher? – Andrew Sanfratello Apr 2 '15 at 17:43
• I don't know whether 'natural' means "… to everyone" or "… to someone", but apartments (in the theory of buildings of reductive groups) are naturally affine spaces (under a real-ised character lattice) that do not arise naturally as affine subspaces of vector spaces. encyclopediaofmath.org/index.php/…. – LSpice Apr 2 '15 at 22:53
• @DagOskarMadsen, I think that Benoît Kloeckner means the binary operation to be not 'addition' but 'subtraction': to say that the image of $(a, b)$ is $x$ means that $a + x = b$. – LSpice Apr 2 '15 at 22:54
• I don't know whether this counts as an argument in favor, but I was not taught abstract affine spaces as an undergraduate, and when I first learned about them I felt like "oh!! Why did no one ever tell me about those before?" – Mike Shulman Apr 3 '15 at 5:06
• It might be worth comparing to the notion of studying a manifold rather than studying shapes in Euclidean space. – Hurkyl Apr 3 '15 at 12:19

As far as the first question is concerned I fully agree with Steven Gubkin's answer : the reason why abstract affine spaces are nice is because they match the intuition coming from elementary Euclidean geometry. The Euclidean plane is clearly not a vector space and clearly not naturally an affine subspace of a vector space.

One can also argue that undergrads should learn about affine spaces because they are the natural place where they should learn multi-variate calculus. In those space you can write $f(x+h) = f(x) + Df(x)(h) + o(h)$ with $x$ and $h$ having a very different nature (as they should have, and you certainly not draw them on the board as two arrows attached at the origin of a vector space).

About the second question, Andreas Blass's answer is of course the first one that comes to mind to differential geometers. But it obscures somehow its elementary nature by putting it into a differential setting.

Let $V$ be a linear space and $E$ a subspace of $V$. Let $S(E)$ be the space of linear subspaces $F$ of $V$ such that $V = E \oplus F$. Then $S(E)$ is naturally an affine space directed by the vector space $L(V/E, E)$ of linear maps from $V/E$ to $E$. If $\pi$ denotes the projection from $V$ to $V/E$ then the action is defined by $u \cdot F = \{f + u(\pi(f)); f \in F\}$.

The most famous example is when $E$ is a hyperplane. Then $S(E)$ is the complement in the projective space $P(V)$ of the projective hyperplane $P(E)$. That's why such complement are called "affine charts": they have a canonical (functorial) affine space structure.

This example also underlies the connection example as soon as you think in terms of Ehresmann connections.

• Your point about the derivative is excellent. I kept running into this problem in an online multivariable calculus class I wrote (ximera.osu.edu/course/kisonecat/m2o2c2/course/), but I did not think of the affine space formalism. I should probably go back and make a little more explicit in the presentation. I was always thinking tangent bundle with flat connection'' in my head, but affine space seems like a better choice at this level. – Steven Gubkin Apr 8 '15 at 21:28

The connections on a smooth vector bundle over a smooth manifold form an affine space with no canonical element to serve as $0$ for a vector-space structure.

• Nice one. This both answers my second question, and kind of makes my point as far as undergrad education is concerned. – Benoît Kloeckner Apr 2 '15 at 20:38
• @BenoîtKloeckner, an honest question that is not meant to be rude: what is your point about undergraduate education? Is it the general point that undergraduate education focusses too much on abstract theories at the expense of understanding of examples, or the specific point that there are no 'natural' undergraduate examples of affine spaces? If the latter, then surely the existence of one *non-*undergraduate example does not imply the *non-*existence of an undergraduate example! – LSpice Apr 2 '15 at 22:56
• @LSpice: sure, there is no such logical implication. But that the first really satisfying answer to my second question I've seen is so high-level is an evidence of my point, which is that affine subspaces of vector spaces cover most if not all undergrad examples (from which I tend to conclude that teaching abstract affine spaces might be uncalled for). – Benoît Kloeckner Apr 3 '15 at 8:01

I am sure you know this, but the Euclidean plane is a prime example.

The Euclidean plane starts as just a collection of points together with a group of isometries. There is no natural origin. However, there is a natural associated vector space consisting of equivalence classes of pairs of points. Identify $(p_1,q_2)$ with $(p_2,q_2)$ there is a translation of the plane taking $p_1$ to $p_2$ and $q_1$ to $q_2$. These are the "vectors as floating line segments" we learn about in high school.

This turns the Euclidean plane into an affine space via $p-q = [(p,q)]$.

I am not sure that you will get too many really "exciting" examples, since as soon as you pick an origin you have a vector space. There is no natural origin to Euclidean space.

Another example is time. Assuming that there it has no beginning or end there is no natural origin. It is an affine space for the vector space $\mathbb{R}$. Of course, as soon as we pick an origin (like the death of a certain historical figure) it becomes a vector space. But without human interference it is more naturally an affine space.

• Ok, you have a point, notably if one teaches geometry based on geometric axioms. – Benoît Kloeckner Apr 2 '15 at 20:44
• @BenoîtKloeckner, without meaning to speak for Steven Gubkin, I think that this point of view (with which I agree) means to emphasise that, although $\mathbb R^2$ and (what I would call) $\mathbb E^2$ are isomorphic (as affine spaces), they are not equal (or even naturally isomorphic). Thus, for example, I live (at least locally) in $\mathbb E^3$, but not in $\mathbb R^3$, because every point has an equally good claim to being $(0, 0, 0)$. – LSpice Apr 2 '15 at 23:00
• @LSpice I give you permission to speak for me in this matter. That is exactly what I am trying to say. – Steven Gubkin Apr 2 '15 at 23:21
• That being said, I consider it a valid point that when it comes to plane and 3-space, the abstract point of view can be made reasonably motivated. – Benoît Kloeckner Apr 3 '15 at 7:55
• @BenoîtKloeckner ("I agree with that …"): by analogy, for me, $\mathbb Z/n\mathbb Z$ is a ring, whose underlying group I call $C_n$, because I think that using the same symbol for both encourages one psychologically to equip the group with more structure than it already has. Similarly, even though there is definitely a close relationship between $\mathbb R^n$ and $\mathbb E^n$, I think that using the former notation for both encourages us to think that $\mathbb E^n$ 'really' has an origin, but that we are just being coy and pretending that we don't see it. – LSpice Apr 15 '15 at 20:08

The set of solutions to a nonhomogeneous linear differential equation form an affine space. (The underlying vector space is the set of solutions to the associated homogeneous equation.)

• True, but this perfectly covered by teaching affine subspaces of vector spaces: here the set of solution is an affine subspace of the vector space of $C^k$ functions where $k$ is the order of the equation. I would not confuse the direction of an affine subspace and the "underlying" vector space that "naturally" contains the affine space being considered. – Benoît Kloeckner Apr 3 '15 at 7:58
• The space of $C^k$ functions is so large as to be practically useless for intuition in this situation. – Alexander Woo Apr 3 '15 at 18:20
• I have barely seen any geometrical intuition related to the affine structure of the set of solutions of linear differential equations. It seems more about the algebraic aspects of it, and then the addition and scalar multiplication in $C^k$ are pretty clear. And again, I agree that the direction (aka set of solution of the associated homogenous equation) is of prime importance, but this example stills perfectly fits into the "affine subspaces of vector spaces" point of view. – Benoît Kloeckner Apr 3 '15 at 21:02

In a vector space, you have an origin, addition of vectors, and scalar multiplication. Using a vector space structure to study something conveys the impression that the origin, addition, and scalar multiplication are actually meaningful things about the object of study.

But sometimes that simply isn't true. And when it isn't, it's handy to have a way to apply linear algebra without giving the misleading impression that certain things are meaningful when they are not.

"Forgetting" information is actually a pretty important thing in mathematics. For example, the whole field of differential geometry got started as a way to study what properties of a shape were intrinsic to a shape, and what properties were accidents of how they were drawn in Euclidean space.

• I agree with this, but it can be dealt with by defining an affine subspace of a vector space, observe that a vector space is an affine subspace of itself and that seeing it that way consist precisely in forgetting the origin. No real need to define abstract affine subspaces. – Benoît Kloeckner Apr 3 '15 at 12:47

Temperatures live in (part of) an affine space of dimension 1, and there seems to be some disagreements about what "the" origin should be...

• This is not quite true, since there is an absolute zero. Likewise, my example of time is not quite right since there is a big bang. It would be nice to think of a natural one dimensional quantity which does not have this limitation. – Steven Gubkin Apr 2 '15 at 22:38
• Although @StevenGubkin is right that temperature isn't a perfect example, I think that time very nearly is (although one might claim that Big Bang-type universes have an "absolute 0 time"). – LSpice Apr 2 '15 at 22:50
• Oops, sorry, @StevenGubkin already made just this point at matheducators.stackexchange.com/a/7781/2070 (and I just missed the edit window for my previous comment). – LSpice Apr 2 '15 at 22:58
• It is true that there is an absolute zero, but 99.99 percent of the world population seems to ignore this fact and the Farenheit-Celsius battle is not going to end any time soon. – user4990 Apr 3 '15 at 1:26
• @StevenGubkin a fun, and weird tidbit from thermodynamics perhaps brings your simple analogy new mathematical breadth, however, at the cost of physically familiar material. It is true that temperatures are in bijective correspondence with $\mathbb{R}$; see en.wikipedia.org/wiki/Negative_temperature brought to you from the bizarre world of statistical mechanics. Probably 99.9999 percent of the world ignores this. – James S. Cook Apr 9 '15 at 19:31

One answer is to regard this question as an instance of a more general question about whether to build embedding theorems into the foundations of a subject. Should we define abstract manifolds, or only submanifolds of Euclidean space? Should we define abstract groups, or only subgroups of permutation groups? I think in all cases there is something to be gained by making the abstract definition and then proving the embedding theorem, because it makes clear what aspects of a notion are "intrinsic" and independent of a chosen embedding.

As mentioned in other answers, time is an excellent 1D example, and of course space is a higher-dimensional one. In your comments on these answers you say that since we always measure these quantities with numbers, it makes your point about using affine subspaces of vector spaces. But I would argue that these examples nevertheless do answer your second question, because the choice of numbers with which to measure them is not natural but rather arbitrary.

This is really another viewpoint for answers already given. In statistics and data analysis, whenever you have a quantity which you can measure, and for which (arithmetic) means are meaningful, but addition is not, then those quantities form (at least a part of) some affine space. Important examples are already mentioned, temperature and time. And this also answers the objection about absolute zero or the big bang origin of time: For most mundane purposes where we use statistics, those origins are rather besides the point, and the affineness of the spaces presents itself as the fact that averages are meaningful, but sums are not. That point have much larger significance than those far-away origins!

• I am all in for teaching and using affine properties, maps, ideas and so on. How does this calls for abstract affine spaces instead of mere affine subspaces of vector spaces? – Benoît Kloeckner Apr 3 '15 at 20:58
• It seems to me that the answer explains perfectly why: in an affine space you can take barycenters, but not multiply by a scalar (it's twice as hot today as it was yesterday has no meaning). So this is a different concept than a vector space and it is good to be aware of its existence. Of course one can always add a dimension and view an abstract affine space as a translate of a vector space in a vector space, but this makes the concept less clear. – user4990 Apr 4 '15 at 21:49
• I still fails to see how this kind of facts, e.g. that sometimes averages make sense but not addition, is better expressed at undergrad level by defining abstract affine spaces that in defining affine subspaces of vector spaces (no need to add dimension, a vector space is an affine subspace and this corresponds to forgetting about its origin). My question is not whether affine spaces should be used, I am sure they should; its about the "set acted upon by a vector space in a certain way" versus "subset of a vector space with certain properties" points of view. – Benoît Kloeckner Apr 5 '15 at 14:53
• In fact, the examples of time and temperature are rather good to explain why going with affine subspaces of vector space is a good thing: we always measure time and temperature by numbers, i.e. we do represent them in $\mathbb{R}$; so explaining why forgetting about the vector structure (the $0$) and looking at objects that still make sense in the mere affine structure (e.g. arithmetic mean) in this case seems mandatory to me. – Benoît Kloeckner Apr 5 '15 at 14:57
• Thinking more about this, I think you are right. In practice, we calculate the mean by a formula such as $\frac1n\sum x_i$, which is using the non-meaningful addition! So for the calculation, we need the vector space structure, we just must assure us that we only use it do calculate quantities which have meaning in the affine structure. – kjetil b halvorsen Apr 5 '15 at 19:53

My opinion is, that no, you shouldn't teach that to undergrads in your mathematics course. The main reason for this is that it makes simple things complicated.

I agree that the affine $d$-dimensional real space and the linear $d$-dimensional real space are different, and also different from $\mathbb{R}^d$. However, introducing these three (or any two of them, for that sake) as a different mathematical concept is unnecessary for any calculus result you present, and also unnecessary for any exercises they're going to work on in the problem sessions.

If any undergrad course should explain the differences, it should be the one that needs it, which is the Mechanics course. It's also much easier for the students to grasp the idea there, because it makes sense in mechanics that position $0$ can be anywhere, but distance $0$ or velocity $0$ is dependent.

As a mathematician, you can point this out at an appropriate place in the course; the students should simply realize that it corresponds to their physics experience, but shouldn't be bothered by you not distinguishing the two concepts.

As a side note: I was taught the "complicated" way, but when I got this question during my state exams, I made it the "simple" way. Two of the committee members than thanked me for making it easy and nice.

I would like to give an answer, which is not completely my own but that stems from other answers given here. It still seems work writing because I don't think it appears explicitly.

One compelling reason to teach in one form or another abstract affine spaces, or at least planes, is to be able to do basic non-analytic geometry (e.g. study triangles and quadrangles in a light formalism). More precisely, in undergrad education we have to teach geometry to future high-school teachers, and we have to do so in a way that relates to what they will have to teach; and they won't be able to say to the pupils "just consider $\mathbb{R}^n$"! I don't think we can do that by avoiding completely abstract affine spaces. Now, there is a subsequent question that I will ask separately: how do we teach abstract affine spaces in a simple way that clearly relates to high-school geometry?

Finally I would like to thank all users who answered or commented. Even if I have often commented back not so positively, I was partly playing the devil's advocate, and all other answers have made me think about this issue.

The point of affine spaces is that they deal with "free vectors" as encountered in Dynamics to represent forces. But that they are worth the effort is indeed debatable.

Similarly, physicists have no trouble with the Dirac and Heaviside "functions" and the fact that these are distributions à la Schwartz does not seem to add to their comfort. Which is not to say that the latter are not beautiful. They are.

But, somehow, affine spaces seem less so to me.

• Are free vectors used to represent forces, O RLY? – Incnis Mrsi Apr 21 '15 at 8:25