Why is it Important that something is a vector space?

[MAGNIBORO]

hi

I am studying algebra and i have a question.

why is important that something is a vector space?, i mean, what implications have?

matrix, complex numbers , functions , n-tuples.
What do these have in common, apart from being a vector space?

why is so important that a certain set of "vectors" over a field satisfy some axioms?

thanks

 

[fresh_42]

hi

I am studying algebra and i have a question.

why is important that something is a vector space?, i mean, what implications have?

matrix, complex numbers , functions , n-tuples.
What do these have in common, apart from being a vector space?

why is so important that a certain set of "vectors" over a field satisfy some axioms?

thanks

It is a natural tool to describe geometric properties, which was then generalized to fit more applications.
You have directions, which can be added, stretched and compressed, and every pilot can tell you that the length of such a direction is also important (wind). We try do add almost everything and all the time, and if those added quantities have the additional property, that they can be scaled, then you have a vector space. And there are really many occasions where these conditions hold.

 

[mfb]

There are thousands of proofs for properties of vector spaces. If you can show that something is a vector space, you know you have all these properties without having to prove everything again.
For finite-dimensional vector spaces many results are quite intuitive - they are similar to our intuition from physical space. But some are less intuitive, and then you can rely on all the things shown for vector spaces.

 

[member 587159]

In algebra, the goal is to classify certain structures. The most common ones are semigroups, monoids, groups, semirings, rings, modules, vector spaces (which is a module over a field). We can even classify these structures further (for example, a ring can be a domain, with or without unique factorisation, or it can be a field, or a division ring, ...). This abstract classification gives us a lot of advantages. By working out theories like linear algebra, group/ring theory, ... , we can say a lot of things about a particular object just by saying things like "it is a vector space of dimension n". For example, we know that such a vector space is just a copy of the standard vector space $\mathbb{R}^n$, because finite dimensional vectorspaces are isomorphic iff they have the same dimension. Another example: "That structure is a group with $23$ elements" tells us that the given group is (a copy of) $\mathbb{Z}_{23}$

Now, this is a general discussion why algebra is useful, but it doesn't answer entirely why vector spaces are useful.

They are the underlying structure on which entire linear algebra is based (well, most of linear algebra can be done with modules over division rings, but let us not consider this). Linear algebra is the language of geometry and coordinate transformations (which are expressed with linear mappings), things that physicist use all the time.

 

[fresh_42]

... we can say a lot of things about a particular object just by saying things like "it is a [real] vector space of dimension n". For example, we know that such a [real] vector space is just a copy of the standard vector space $\mathbb{R}^n$ , ...

Sorry for nitpicking.

 

[member 587159]

Sorry for nitpicking.

No need to apologise. This is very relevant. I should have written that if $V$ is a vector space over $K$ and $dim(V) =n$, then $V \cong K^n$

 

[FactChecker]

A general mathematical vector space (in physics, a vector means more than it does in math) allows you to talk about "twice as much of vector A" and "A + B = C". Without that, there is not much you can do. That is a basic start. Add a norm and you can start talking about distance. Add an inner product and you can start talking about angles. With each addition, you come closer to something that is intuitively like Euclidean space. Take anyone of those away and strange (but interesting) things can happen.

 

[Stephen Tashi]

why is so important that a certain set of "vectors" over a field satisfy some axioms?

As a vague generality:

It often turns out that some physical phenomena L(v) whose result depends on another phenomena "v" satisfies properties that are described as "linearity" or "the principle of superposition". (e.g. the superposition principle for certain types of waves, phenomena described by 'linear' differential equations etc.) The algebraic description of these properties involve L(v + w) = L(v) + L(w) and L(kv) = kL(v) where k is a number. The "+" operations in these equations need not be the addition of ordinary arithmetic, they can be a special type of operation defined on the particular phenomena involved. (For example, if you "add" two waves, you get a wave not a number.)

For the algebraic treatment to work out nicely, if v and w are phenomena of a given type, v+w should be one of the same type and kv should be one of the same type. So you can see why some of the axioms of an abstract vector space need to apply for things to work out.

Matrices, functions, numbers, and n-tuples are often used to represent phenomena.

From this point of view, the importance of vectors spaces (in applied math) is an empirical result that follows from frequent encounters with phenomena that obey linearity and superposition. In an imagined world (or field of study) where no important phenomena obeyed these algebraic patterns, vector spaces might be unimportant.

 

[MAGNIBORO]

thanks all of you for your comments , i understand that is important that something is a vector space because then can discribe a certain physical phenomenan
so i can interpret that all vector space are equivalent in some way ( correct my if I wrong) but other are better to describe certain phenomenan that others
(also as @Math_QED said the goal is classify certain structures)
so in applied math and physics that something is a vector space is a useful description (I imagin that also in pure maths this have some important implciation as well)

thanks to all of you =D

 

[fresh_42]

so i can interpret that all vector space are equivalent in some way

All vector spaces of the same finite dimension with the same scalars. Any field can supply the allowed scalars: $\mathbb{Q}\, , \,\mathbb{R}\, , \,\mathbb{F}_2\, , \, \mathbb{C}(t)$ or whatever. There are also many vector spaces, which are not finite dimensional, e.g. continuous functions.

 

[member 587159]

thanks all of you for your comments , i understand that is important that something is a vector space because then can discribe a certain physical phenomenan
so i can interpret that all vector space are equivalent in some way ( correct my if I wrong) but other are better to describe certain phenomenan that others
(also as @Math_QED said the goal is classify certain structures)
so in applied math and physics that something is a vector space is a useful description (I imagin that also in pure maths this have some important implciation as well)

thanks to all of you =D

The notion of equivalent vector spaces is made rigorous using the concept "isomorphism". You will certainly encounter this if you start learning linear algebra.