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Exactly what spanning is

  1. Mar 16, 2009 #1
    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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  3. Mar 16, 2009 #2

    matt grime

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    Re: Span

    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.
     
  4. Mar 16, 2009 #3
    Re: Span

    o ok.. thanks so much!
     
  5. Mar 24, 2009 #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.
     
  6. Mar 24, 2009 #5

    matt grime

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    Re: Span

    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=....?
     
  7. Mar 24, 2009 #6
    Re: Span

    Thanks for that.
    I'm just having trouble getting started
     
  8. Mar 25, 2009 #7
    Re: Span

    I can't seem to do it.
    Damn it's quite hard.
    Any help would be greatly appreciated.
     
  9. Mar 25, 2009 #8

    matt grime

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    Re: Span

    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.
     
  10. Mar 25, 2009 #9
    Re: Span

    Yes I have done that.
     
  11. Mar 25, 2009 #10

    matt grime

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    Re: Span

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