Why use the vague notion of "vector" when you have $\mathbb R^2,\mathbb R^3,\mathbb R^4,\ldots$?

Question

I'm reading an introductory course on groups. In this course, the author illustrates concepts using the vectors of the plane. For example, "the set of vectors in the plane(or in space) is a group for $+$". Thus, it refers to the notion of vector as taught in secondary education. However, everyone knows that this concept is quite complex. Why not settle for the additive group of real vector space $(\mathbb R^2,+,.)$ and a fortiori $\mathbb R^3$? Even if it means illustrating the elements of $\mathbb R^2$ with vectors of the plane to refer to what the student already knows. It's mathematically satisfying, as well as enriching for the beginner focused on group theory.

To put it simply, we could just as easily illustrate the notion of an abstract set by talking about the set underlying a group $(G,.)$ such as $\mathfrak{S}_3$. Why do we do with the complex geometrical notion of "vector" what we would not reasonably allow ourselves to do elsewhere?

Answer 1

In the euclidian spaces vectors and points are the same thing (there is a bijection between them and everything works as it should).

In physics and in differential geometry vectors and points have different roles.

You will lose nothing in an introductory group theory course if you understand vectors as corresponding to points. However, a physicist taking the course might indeed want to think about vectors, as adding them and scalar multiplication makes sense.

Answer 2

I don't think the concept of vector addition of geometric vectors is especially more complicated than the concept of $\mathbb{R}^2$. In particular, the tip-to-tail addition and scalar multiplication of vectors is geometrically natural and it is for that reason it is likely given as an example. In terms of ease of use, formulation and concreteness, sure $\mathbb{R}^2$ is easier.

But, in the same way many authors find it interesting to illustrate the dihedral group in terms of physical motions to defining objects. Such rotations and reflections are stupidly complicated, especially as the number of sides increases. Why not just teach the generators and relations formulation of $D_n$? I'm guessing the motivation is to ground the students intuition in the idea that groups can be geometric in their origin. In other words, groups are not just a formal system with neat and tidy rules. Rather, a group may arise from a nasty and somewhat complicated geometric question which is inconvient in it's formulation and certainly far from minimal in terms of the economy of its logic. Yet we teach the geometry. Why ? Because there exist geometers.

Answer 3

Because vectors are familiar.

Sure, $\mathbb{R}^n$ is simpler to construct "from scratch". But when teaching you're not constructing things from scratch: you're doing it from the ideas students already understand. And with those as your primitives, "vectors" are simpler than "$\mathbb{R}^n$, equipped with the operation of addition of corresponding components" -- because the concept of "vectors, which get combined with vector addition" is already available for you to use!

This is especially true when "equipping a set with an operation" is still a hazy concept for most students: having a familiar set, with an operator that comes pre-equipped, is very effective in helping students "get their footing".

Answer 4

Perhaps the more general context of an affine space can be helpful. When we discuss $\mathbb R^n$, we typically think of the origin as a special point. We think of vectors as ordered tuples of real numbers. The we can add on the image of an arrow that has its tail at the special point (origin) and its tip at the point corresponding to the ordered tuple.

But in physics, we tend to think of vectors as arrows that go between any two points, and consider two such entities as equal if they are just translations of one another. So the vector tailed at $(1,1)$ with tip at $(4,2)$ is equal to the one tailed at $(3,-1)$ and tipped at $(6,0)$. We go from the first to the second by translating the plane by isometry $(x,y)\rightarrow (x+3,y+1)$.

Notice that the translation $(x,y)\rightarrow (x+3,y+1)$ is technically not a transformation of ${\mathbb R^2}$ as a vector space because it does not preserve the identity for addition, i.e., it moves the origin. The transformation group is the general linear group $GL(2,{\mathbb R})$, which can be represented as the 2x2 matrices with non-zero determinant.

So what is going on? The collection of arrows that are vectors for physicists form what is called an affine space, as opposed to a vector space. The geometer Marcel Berger said "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." The definition of an affine space is kind of tricky. nLab might give you some ideas. One characterization found there is that an affine space is simply a vector space, but with different morphisms; an affine linear map is a function that is the difference between a linear map and a constant function. There are many ideas on the nLab site, with varying degrees of abstraction.

In Newtonian physics, we inhabit a "big cube" that is three-dimensional. Sometimes we over-step and think of it as the vector space ${\mathbb R^3}$. But in Newtonian physics, there is no distinguished point like the origin, e.g., the "me reference frame" which puts me at the center of the universe. So the big cube of Newtonian physics is really an affine space, not a vector space. The symmetry group of Newtonian physics is larger than the symmetry group for an associated vector space.