How do I conceptualize a vector space

Question

I have read a number of articles and posts about vector spaces I am still left unconfident that my conceptual model of a vector space is correct.

Here is an example to illustrate how I am thinking about them currently. To simplify things lets put a constraint on our vector space and only focus on a subset of the Real numbers. Imagine a 2d space with points (-5,5) (5,5) (-5,-5) (5,-5) that create a bounding area for our vector space. We can assume all the axioms of a vector space apply to this space (essentially scalar multiplication and vector addition).

Would the vector space be the set of all possible vectors that can fit in that box? Or would it be the space in which the vectors live (the 2d box)? Wikipedia describes a vector space as "a collection of objects, called vectors", which makes me think it is the former.

Answer 1

Here are my thoughts on what are vector spaces (and mathematical objects in general)...

Vector spaces aren’t a thing. As in, the idea of a vector space is not so much one object, but rather a description. What is it a description of? Well, anything that satisfies its axioms. Whenever you have a set of things that can be:

• Added together reasonably (i.e. Addition satisfies closure, identity, inverse, associativity, plus commutativity)
• Scaled by numbers of sorts (be it the ring of integers, the field of rational numbers, real numbers or complex numbers)

With the extra requirement that:

• Scaling interacts with addition in a distributive way.

So in a way, a vector space is more of an abstract object that symbolizes all things that fits the vector description. If you think of a vector space like this, as a description, rather than a noun refering to a specific object, then you will be less prone to being trapped in thinking in $\mathbb{R}^n $ (which is a real shame, because you’re missing out on the amazing things one could do on many other vector spaces).

You can conceptualize other mathematical objects in this way too, from the “everyday” objects to the not so everyday objects. E.g.

• What is a triangle? Anything that fits the description of something that has three sides.
• What is a function? Anything that fits the description of something that sends elements from one set to elements in another set in a reasonable way (i.e. all elements in the domain is mapped to something).
• What is a group? Anything that satisfies the description of something that contains things which can be combined reasonably (i.e. a binary operation that is closed, has an identity, has inverses, and is associative).

To me, this also answers why mathematics is so applicable everywhere, as well as why it is so abstract. It is because its theories are not about anything in particular like forces or atoms or cells, but rather, anything that fits a certain description.

I think the following analogy sums up neatly my response to your question:

Asking what is a vector space is like asking what is something that is red. Well, it is anything that is red.

I hope this is of some help.

EDIT: To address your example of a subset of $\mathbb{R}^2$, the set of pairs of real numbers that are added component-wise and scaled by real numbers in an appropriate way.

Note that a vector space must be closed under addition, so it must necessarily be the entire space of $\mathbb{R}^2$, since you may “fall out” of a subset by repeated adding some vectors. The point of the closure axiom of a vector space is that so that you can’t “fall out of it” by addition.

Also, note that I don’t refer to $\mathbb{R}^2$ as “the” 2D space. It’s because there are many other vector spaces out there that are also “2D”. E.g. The space of polynomials

$$ \{ c_1 x + c_0 | c_1,c_0 \in \mathbb{R} \} $$

You may be able to see intuitively that this space should be 2D, but then that begs the question: How does one define dimension of a vector space when the vector space is anything that fits the vector description?

That’s the motivation for defining bases of a vector space (plural of basis), which in turn requires the idea of linear combination, span, linear independence.

Answer 2

A vector space is not just a collection of vectors; it's a collection of vectors with an addition operation and a multiplication by a number operation defined on it.

An $n$-dimensional vector space is:

• a point when $n = 0$,
• a line when $n = 1$,
• a plane when $n = 2$,
• ordinary space when $n = 3$,
• analogous spaces when $n \geq 4$, though not as easy to visualize.

The square you describe is only part of the plane it lies in, not the whole plane, so it's not a vector space.

Technically, you can prove this by saying that the vector $v = (5,5)$ belongs to the square, but $2v = (10,10)$ doesn't. (I am presuming that the usual multiplication rule is intended.) The square doesn't satisfy the axioms of a vector space, because the operation $2 \cdot v$ is undefined within it.

Answer 3

I think the essence of this question is not really about the specifics of vector space axioms, but more generally, whether the "vector space" is some kind of container (like some background or stage) or the collection of all things that can be put into said container (set of vectors).

Such philosophical questions are rarely discussed. The short answer is: things in math are whatever people have defined them to be. How to best think about them intuitively is a separate question.

Over the last century or so, set theory has become the basis of most math. Even for primary school math. For example, in school (depending on country etc.) we learn that geometric shapes, like circles "are" sets of points.

Vector spaces are sets with operations defined on them, fulfilling certain axioms. However, the distinction often isn't clearly made between the set and the vector space, even (or especially) in academic texts. The vector space may be denoted with the same letter as the underlying set. But this is just an abuse of notation to reduce clutter for experienced readers (although it confuses beginners who like to see things written out explicitly and less ambiguously).

I think in practice, most people intuitively work with the "vector space as background stage" mental model, though, because it's hard to visualize or imagine all the elements (vectors) populating the space at the same time. I usually imagine the vector space as just sitting there, and focus my attention on a few particular vectors under analysis as if they were floating in there in an empty space. But again, this is just how my monkey brain deals with it, this has no bearing on what a vector space mathematically is.

Answer 4

Commenters have already observed that your box is not a vector space. To answer the question anyway, in a way that I hope will help for future examples which are vector spaces: there is no difference between the box and the set of vectors that live in it. A vector space is conceptualized as a collection of arrows from the origin to an endpoint, but mathematically, a vector is identified completely with its endpoint. So the answer is, "both."