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Number of vectors needed to Span R^n

  1. Nov 3, 2012 #1
    I am reading my linear algebra book, and in the chapter on Spanning, I got the impression that for a set to span R^n, it must contain at least n vectors. I confirmed that by searching through the forums.

    However, I've reached the chapter on Linear Independence and in one of the examples it shows that a set containing 2 vectors can span R^3. So now I'm totally confused. Any help will be appreciated.

    Example in the book:

    S1= {[1,0,1], [0,1,1]}, S2 = {[1,0,1], [0,1,1], [3,2,5]}. Book says S1 and S2 span R^3, and prefers S1 since its "more efficient".
     
  2. jcsd
  3. Nov 3, 2012 #2

    lurflurf

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    S1 and S2 have the same span, but it is not R3. For example [1,1,0] is not in their span.
     
  4. Nov 3, 2012 #3
    Thanks for the help. I just read this page like 100 times, and after reading your answer I read it again and missed an important detail.

    The book is describing a subset of R^3 where W = [a, b, a+b] is a subspace of R^3. I still don't understand why they claim that S1 spans W though.
     
  5. Nov 4, 2012 #4

    lurflurf

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    It does
    W=span(S1)=Span(S2)
    what is
    a*[1,0,1]+b*[0,1,1]?
     
  6. Nov 4, 2012 #5
    that equals [a, b, a+b] which is in W. I see that it works, but what's confusing me is the R^3. Even though W is a subspace of R^3 shouldn't W need 3 vectors in order to be spanned? How come 2 vectors are able to span it when its in R^3?

    I wish they left out the R^3 and just put down R^2, then it would make sense to me. Maybe I'll edit that page in my book lol :smile:

    Thanks for your patience. I'm sure this is something that should be simple to understand.
     
  7. Nov 4, 2012 #6

    lurflurf

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    Our vectors are in R3. A subspace need not span the space

    {[1,2,3,4,5,6,7,0,0,0,0,0,0,0,0,0,0,0,0,0],[1,2,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]}
    is a subspace of R20, but only has dimension 2 like W
     
  8. Nov 4, 2012 #7

    Fredrik

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    Looks like you need to look at the definition of "subspace" again. A subspace is just a subset that's also a vector space. Your W is a subspace of ##\mathbb R^3## because a) it's a vector space, and b) it's a subset of ##\mathbb R^3##.

    The set ##\{(x,y,z)\in\mathbb R^3|y=z=0\}## ("the x axis") is a subspace of ##\mathbb R^3## that's spanned by a set containing only one vector: ##\{(1,0,0)\}##.
     
  9. Nov 4, 2012 #8
    Got it! I see what you mean. That example actually clicked...finally. Thanks so much!
     
  10. Nov 4, 2012 #9
    Did look over it again. Your example really helped clear things up some more. Thank you so much!
     
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