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Vectors - Spanning sets

  1. Nov 23, 2011 #1
    1. The problem statement, all variables and given/known data
    Consider a set of vectors:

    S = {v[itex]_{1}[/itex], v[itex]_{2}[/itex], v[itex]_{3}[/itex], v[itex]_{4}[/itex][itex]\subset[/itex] ℝ[itex]^{3}[/itex]

    a) Can S be a spanning set for ℝ[itex]^{3}[/itex]? Give reasons for your answer.
    b) Will all such sets S be spanning sets? Give a reason for your answer.


    3. The attempt at a solution

    a) Yes, because a linear combination of these vectors can form any given vector in ℝ[itex]^{3}[/itex].

    b) Yes, don't really know a reason besides something similar to the one above, I can't see why not.

    I'm not really sure of these answers, can anyone confirm this, or given any insight to understand this better?

    Cheers
     
  2. jcsd
  3. Nov 23, 2011 #2

    HallsofIvy

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    What about, {(1, 1, 0), (1, 2, 0), (3, 1, 0)}?
     
  4. Nov 23, 2011 #3
    Well I guess since the 3rd elements are zero, you can't form a vector in ℝ[itex]^{3}[/itex] which has a non-zero 3rd element.

    So part b is definitely a no. However part a "can" be since they haven't explicitly defined the vectors, so it's possible given that zero/non-zero condition, is that the only reason?
     
  5. Nov 24, 2011 #4

    HallsofIvy

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    In general, a set with fewer than n vectors cannot span a vector space of dimenson n but a set with n or more vectors may. A set with more than n vectors cannot be independent but a set with n or fewer may. Only with sets with exactly n vectors is it possible to both span and be independent (a basis).
     
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