What is Spanning and its Relation to Dimensionality?

[Saggittarius]

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.

 

[matt grime]

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.

 

[Saggittarius]

o ok.. thanks so much!

 

[squenshl]

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.

 

[matt grime]

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=...?

 

[squenshl]

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

 

[squenshl]

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

 

[matt grime]

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.

 

[squenshl]

Yes I have done that.

 

[matt grime]

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