Where do the vector space axioms come from?

[JamesGold]

Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?

 

[Gamble93]

My linear algebra instructor tried to explained to us where the vector space axioms come from. He told us that the axioms are properties of common vector spaces such as \mathbb{R}^2 or \mathbb{R}^3 that people were used to working with. Then over time people started generalizing them to arbitrary sets of vectors over an arbitrary scalar field and thus came up with the now standard list of axioms. However I'm only going off of what he said. If that's wrong then I too would love to know the reasons behind the axioms.

 

[chiro]

Hey JamesGold.

Think about how to abstract the behaviour of things that can be represented as arrows and you'll get an idea of how it developed.

The benefits of doing this and having connections to linear algebra and matrices is that linear algebra is a well developed area with a lot of tools to decompose, transform, and analyze things that "behave like arrows".

The basic idea of linearity is expressed in two assumptions:

1) f(x+y) = f(x)+f(y) and
2) f(ax) = af(x)

This is the basic idea of linearity and the vector space axioms really formalize this core idea by introducing ten axioms that guarantee that they all behave like this (i.e. they act like "arrows").

Once things are shown to act like arrows and are able to converted to the constructs in Linear Algebra (like matrices, vectors, and so on) then you can look at things like decomposition.

Decomposition is just a way to take something and split it up and in the linear algebra framework, there are ways to to do this in many ways (i.e. by constructing bases) and by doing this, you can see how to take something and break it up in a way to get the right information you want depending on the problem and analysis you need.

Now when you apply this to infinite-dimensional spaces, you get the framework of the Integral Transforms like the Fourier decomposition, and other orthogonal decompositions and this is why for example the Fourier decomposition (i.e. decomposing into sines and cosines) actually works.

The only difference with this in comparison to normal vectors is that the "arrows" in are broken down into "sub-arrows" which correspond to cosines and sines instead of i's, j's, and k's but the idea is exactly the same.

So we have gone from arrows to signal processing and the core idea hasn't changed at all.

 

[Fredrik]

Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?

Consider the set $\mathbb R^2=\{(x,y)|x,y\in\mathbb R\}$ of ordered pairs of real numbers. If we define the addition of two arbitrary members of this set by
$$(a,b)+(c,d)=(a+c,b+d)$$ and the product of an arbitrary real number and an arbitrary ordered pair by
$$a(b,c)=(ab,ac),$$ then the vector space axioms are satisfied. This vector space, which is very useful in physics, can be thought of as the "prototype" for all the others.

 

[Stephen Tashi]

To what extent are they arbitrary?

From the modern mathematical point of view, axioms are arbitrary in the sense that they are assumed without proof.

Sometimes people develop axioms for a very basic kind of mathematical system and are able to use them to prove the axioms for a "higher level" mathematical system. If you want to find a lower level mathematical system that can be used to prove the axioms of vector space, I don't know of any work that has done that, but someone else on the forum probably does.

If you are asking if the axioms of a vector space are arbitrary in the sociological and cultural sense (as opposed to a purely mathematical sense), they are not arbitrary. The other posters have given you cultural reasons why human beings like the axioms. This is because some physical and geometric situations have a behavior that is summarized by the axioms. However, systems of mathematical axioms aren't developed just by writing down an axiom for every property of a physical system. Mathematicans attempt to write down only the smallest number of axioms. If something can be proven from other axioms, it is not needed as an axiom.

The quest to use the fewest number of axioms can be confusing to students of math because they may find it easier just to assume all the important properties and get it over with -instead of having to learn proofs. If you look at the way high school level algebra is taught versus how the same material would be covered in a very advance course in college, you can see that some of the things assumed in high school as "properties" and "laws" are proven as theorems in the advanced course. If you are studying vector spaces from an elementary textbook, you may find a similar thing happening.

There are various different ways to state the axioms for a vector space. The cultural explanation of what's in your particular text is complicated. It involves 1) The desire to state the properties of important physical and geometric systems 2) The desire to assume the fewest axioms that are needed 3) The desire to present facts to students in a way that simple enough for them to understand (which somewhat goes against 2) )

 

[Fredrik]

If you want to find a lower level mathematical system that can be used to prove the axioms of vector space, I don't know of any work that has done that, but someone else on the forum probably does.

The best one can hope to prove is that there is a vector space in the branch of mathematics defined by this other set of axioms. If you just supply the missing details from what I said about $\mathbb R^2$ above, you're almost done with such a proof. You could take the definition of the real numbers and a few set theory axioms as your starting point (your axioms need to e.g. guarantee the existence of functions and cartesian products), or you can take the axioms of ZFC set theory as your starting point and start by proving the existence of a Dedekind-complete ordered field (i.e. a set whose members have all the properties listed in the definition of the real numbers).