Existence of spanning set for every vector space

In summary, the question of whether every vector space has a generating set depends on the specific requirements placed on the set. While a set consisting of all vectors in the vector space can trivially span the space, the question becomes more complex when specific properties are demanded. Additionally, the statement that every vector space has a basis is equivalent to the axiom of choice.
  • #1
Mr Davis 97
1,462
44
I know that the span of any subset of vectors in a vector space is also a vector space (subspace), but is it true that every vector space has a generating set? That is, the moment that we define a vector space, does there necessarily exist a spanning set consisting of its vectors?
 
Physics news on Phys.org
  • #2
Mr Davis 97 said:
That is, the moment that we define a vector space, does there necessarily exist a spanning set consisting of its vectors?

Trivially, you can take all vectors in the vector space as a spanning set. To make the question interesting, you must put more requirements on the kind of spanning set that you want.

Some textbooks declare that by the phrase "vector space", they will mean a "finite dimensional vector space". How do your course materials use the phrase "vector space"?
 
  • #3
Are we assuming axiom of choice?
 
  • #4
A vector space can be finite or infinite dimensional, and we are assuming the axiom of choice.
 
  • #5
Mr Davis 97 said:
A vector space can be finite or infinite dimensional, and we are assuming the axiom of choice.

The set consisting of all vectors in the vector space spans the vector space. The question becomes more complicated if you demand a set that spans the space and has special properties. For example, if there are an uncountably infinite number of vectors in the vector space then does there exist a countably infinite set of vectors that spans the space?
 
  • #6
The statement "every vector space has a basis" is equivalent to AC. So just use that.
 

1. What is a spanning set?

A spanning set is a set of vectors that can generate every vector in a vector space through linear combinations. In other words, every vector in the vector space can be created by adding a scalar multiple of each vector in the spanning set.

2. Does every vector space have a spanning set?

Yes, every vector space has a spanning set. This is because a vector space, by definition, is a set of vectors that can be added and multiplied by scalars to create new vectors. Therefore, there must exist a set of vectors that can generate every vector in the space.

3. How do you know if a set of vectors is a spanning set for a given vector space?

To determine if a set of vectors is a spanning set for a given vector space, you can use the span operation. If the span of the set of vectors includes all vectors in the vector space, then the set is a spanning set. Another way to check is to see if the set of vectors is linearly independent. If the set is linearly independent and contains the same number of vectors as the dimension of the vector space, then it is a spanning set.

4. Can a vector space have more than one spanning set?

Yes, a vector space can have more than one spanning set. In fact, there are infinitely many spanning sets for any given vector space. This is because a spanning set can have any number of vectors as long as they are linearly independent and can generate all vectors in the vector space.

5. Is a basis for a vector space the same as a spanning set?

A basis for a vector space is a special type of spanning set. A basis is a set of linearly independent vectors that can generate all vectors in the vector space. So, while all bases are spanning sets, not all spanning sets are bases. A basis is considered the "minimal" spanning set, as it contains the fewest number of vectors needed to generate all vectors in the space.

Similar threads

I Vector subspace and basis vectors in the context of data science
    • Linear and Abstract Algebra
Replies
6
Views
886
I Row space, Column space, Null space & Left null space
    • Linear and Abstract Algebra
Replies
8
Views
886
I Terminologies used to describe tensor product of vector spaces
    • Linear and Abstract Algebra
Replies
7
Views
255
I Finding the orthogonal projection of a vector without an orthogonal basis
    • Linear and Abstract Algebra
Replies
3
Views
1K
I Non orthogonal basis and the lines of its coordinate grid
    • Linear and Abstract Algebra
Replies
9
Views
205
I Visualization of complex vectors and dot product
    • Linear and Abstract Algebra
Replies
14
Views
651
I Standard basis and other bases...
    • Linear and Abstract Algebra
Replies
9
Views
583
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
    • Linear and Abstract Algebra
Replies
3
Views
1K
A How to visualise complex vector spaces of dimension 2 and above
    • Linear and Abstract Algebra
Replies
4
Views
2K
I Question about an eigenvalue problem: range space
    • Linear and Abstract Algebra
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top