Can Vector Spaces be Considered Spans?

In summary, vector spaces are sets of elements that follow ten axioms, including closure under addition and scalar multiplication. The span of a set of vectors is the set of all vectors that can be created from linear combinations of those vectors, and it is proven to be a linear space. This concept applies to all vector spaces, not just subspaces. Additionally, every vector space has a basis, which is a set of linearly independent vectors that can be used to create all elements in the vector space through linear combinations.
  • #1
evilpostingmong
339
0
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.
 
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  • #2
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.
 
  • #3
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!:smile:
 
  • #4
evilpostingmong said:
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!:smile:

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.
 
  • #5
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.
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and two operations, vector addition and scalar multiplication, that satisfy certain properties. It is a fundamental concept in linear algebra and is used to study linear transformations, systems of linear equations, and other mathematical structures.

2. What does it mean for a vector space to span a set of vectors?

A vector space spans a set of vectors if every vector in that set can be written as a linear combination of the vectors in the vector space. This means that the vectors in the set can be generated by scaling and adding the vectors in the vector space.

3. How do you determine if a vector space spans a set of vectors?

To determine if a vector space spans a set of vectors, you can use the span definition and check if every vector in the set can be written as a linear combination of the vectors in the vector space. Another method is to check if the set of vectors is linearly independent, which means that none of the vectors in the set can be written as a linear combination of the other vectors.

4. Why is it important for a vector space to span a set of vectors?

If a vector space spans a set of vectors, it means that the vectors in the set can be generated using the vectors in the vector space. This is important because it allows us to simplify calculations and solve problems by using the properties of vector spaces. It also helps us understand the relationships between different vectors and how they can be combined to represent more complex systems.

5. Can a set of vectors span more than one vector space?

Yes, a set of vectors can span more than one vector space. This can happen if the set of vectors contains linearly independent vectors, which means that they cannot be generated by scaling and adding the vectors in any single vector space. In this case, the set of vectors can span multiple vector spaces simultaneously.

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