What is Spanning and its Relation to Dimensionality?

  • Thread starter Saggittarius
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In summary, the conversation discusses the concept of spanning and how a set of vectors can span a vector space of minimal dimension. It is explained that a 6-dimensional space cannot be spanned by 5 vectors, and that the dimension of M(2,3)=6. The conversation also discusses how to prove that span{u,v} = span{u,v,w} if w is an element of span{u,v} in R^n. The conversation ends with the suggestion to rearrange the expressions involved to show the desired result.
  • #1
Saggittarius
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I'm a little confused on exactly what spanning is. For example, It's not possible for a set of five vectors to span M(2, 3), but it is possible for a set of six vectors or seven vectors. Why is this? I understand the dimension of M(2,3)=6. I just need a little bit more information on what spanning is.
 
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  • #2


The span of a set of vectors is the vector space of minimal dimension that contains those vectors.

As you say you understand what dimension is - the size of a minimal spanning set - then it should now seem tautologous to say that a 6 dimension space cannot be spanned by 5 vectors: a set of 5 vectors can span a vector space of dimension *at most* 5.
 
  • #3


o ok.. thanks so much!
 
  • #4
Span

I have a problem. How do I prove that span{u,v} = span{u,v,w} if w is an element of the span{u,v), in R^n.
I don't know how to do this.
Any ideas anyone.
 
  • #5


You use the definitions:

(a,b,c,d,e,f,g represent elements of the base field)

span(u,v) is the set of things of the form au+bv
span(u,v,w) is the set of things of the form cu+dv+ew
w is in span(u,v) means w=...?
 
  • #6


Thanks for that.
I'm just having trouble getting started
 
  • #7


I can't seem to do it.
Damn it's quite hard.
Any help would be greatly appreciated.
 
  • #8


Have you written out what it is that you're trying to prove? You want to show that something that's in the span of {u,v} is in the span of {u,v,w} and vice versa.
 
  • #9


Yes I have done that.
 
  • #10


Well, what is left to say? The result follows simply by rearranging the expressions involved.
 

What is spanning?

Spanning refers to the process of connecting two points in a network or graph. It is a way to establish a path between two nodes or vertices, allowing for data to be transmitted or accessed between them.

What is the purpose of spanning?

The purpose of spanning is to establish communication and connectivity between two points in a network. This allows for the efficient transfer of data and information between nodes or vertices.

What are the types of spanning?

The two main types of spanning are data spanning and network spanning. Data spanning involves the process of organizing and transferring data between nodes, while network spanning involves establishing a physical connection between two points in a network.

How does spanning differ from bridging?

Spanning and bridging are similar concepts, but the main difference is that spanning involves connecting two points within a network, while bridging involves connecting two separate networks. Bridging is typically used to expand network coverage, while spanning is used to establish direct communication between two points.

What are some real-world applications of spanning?

Spanning is used in various applications such as computer networks, telecommunications, and transportation systems. It is also used in data storage and retrieval systems, as well as in the analysis of large datasets in fields like machine learning and artificial intelligence.

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