MHB Can a vector space also be a set?

[ognik]

Wiki says "A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context." To me the term (linear) Vector Space has always seemed a little mysterious ... how far wrong would I be in thinking of a vector space as a set? For example if it was the vector space of polynomials of degree 2, could I also say the set of polynomials of degree 2 (or lower of course)?

If not, what is superior about a Vector Space as opposed to a set?

 

[Fantini]

A vector space is a set, combined with an algebraic structure. A mere set does not satisfy a vector space properties. Whenever a set has certain operations defined on it and behave in a certain way, it is called a vector space. :)

 

[ognik]

Thanks Fantini. So if I have a set of vectors and I say they can be linearly combined, then I have a linear vector space?

I suppose there are mathematical spaces that have other than vectors? And other than linear algebra associated?

 

[Fantini]

Thanks Fantini. So if I have a set of vectors and I say they can be linearly combined, then I have a linear vector space?

I suppose there are mathematical spaces that have other than vectors? And other than linear algebra associated?

If you have a set and they can be linearly combined, then they are vectors because they live in a set that is a linear vector space. :) You can only call them vectors if you have a vector space.

There are other mathematical structures different from vector space. You can have a group, a monoid, a module, a ring, a field...there are ample possibilities (though I don't know enough to explain them in detail).