What is the difference between algebraic structure and space

[mikeeey]

Hi All
A mathematical structure : is A set with an Object ( structure ) and there are generally two types of mathematical structure , which are algebraic structure and space ( geometric structure )
Eaxmples of algebraic structure are rings , fields , modules vector spaces ... act
Examples of spaces are topological space , metric space ... act
My question is the vector space is an algebraic structure ? And why is it called a space ( considered as space ) ?.

Does it mean because vector space can be used to represent spaces in geometry ?! So its called a space ! As it is an algebraic structure also !
Thanks

 

[micromass]

I'd say that a vector space is both an algebraic structure as a geometric space. So yes, it has a definite geometric meaning, but it is inherently algebraic.

 

[aikismos]

@mikeey,

I'd have to agree with micromass that's it's both. I believe that the space in vector space comes from the fact that 3-dimensional space was first modeled in three variables in algebraic equations. Remember that vector space is nothing more than a more advanced construct of seeing space as a neutral set of points in three dimensions. The difference of course is that vectors have directions and therefore are a more convenient mathematical construct for modeling phenomena in physics. Like all things mathematical, naturally, three is too few, and therefore it becomes abstracted to $n$ dimensions. What makes vector space familiar is that it is an algebra, and who hasn't learned calculations primarily through algebra as a road to understanding mathematical systems?