Can Vector Spaces be Considered Spans?

[evilpostingmong]

Hello, very new to vector spaces, it seems like they take some getting used
to. Anyway, since spans are sets of all the linear combinations of vectors
contained within subspaces, I wonder whether or not vector spaces
which contain elements (or vectors) that follow the ten axioms can be
considered spans.

 

[Tac-Tics]

You seem to be mixed up in your terminology.

"Span" by itself doesn't really mean anything. You always talk about the span of a set of vectors.

The span of a set of vectors is the set of all vectors that can be made from linear combinations of those vectors. This is the definition. It is easy to prove that a span of a set of vectors is a linear space itself.

I'm not sure what "ten axioms" you're talking about.

 

[evilpostingmong]

Oh so it doesn't just apply to subspaces. Yeah those ten axioms are
rules that determine whether or not a set of vectors are contained
within a vector space such as wheter or not they are closd under
addition. Thank you for the response, very informative!

 

[Ignea_unda]

Oh so it doesn't just apply to subspaces. Yeah those ten axioms are
rules that determine whether or not a set of vectors are contained
within a vector space such as wheter or not they are closd under
addition. Thank you for the response, very informative!

I believe that you are referring to these ten:

Addition:

1. u + v is in V. (Closure under addition)
2. u + v = v + u (Commutative property)
3. u + (v + w) = (u + v) + w (Associative property)
4. V has a zero vector 0 such that for every u in V, u = 0 = u (Additive identity)
5. For every u in V, there is a vector in V denoted by -u such that u + (-u) = 0 (Additive inverse)

Scalar Multiplication
6. cu is in V. (Closure under scalar multiplication)
7. c(u + v) = cu + cv (Distributive Property)
8. (c + d)u= cu + du (Distributive Property)
9. c(du) = (cd) u (Associative Property)
10. 1(u) = u (Scalar Identity)



^From Elementary Linear Algebra by Larson, Edwards, and Falvo 5th edition.

 

[Count Iblis]

Yes, every vector space has a basis, i.e. a set of linearly independent vectors such that every element of the vector space is a linear combination of the set of basis vectors. This is true also for infinite dimensional vector spaces. There always exists a so-called Hamel basis which is a set of vectors such that every element of the vector space is a finite linear combination of the basis vectors.