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Construction of a Basis

  1. Jul 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Prove that any collection of vectors which includes [tex]\theta[/tex] (zero vector or null vector)is linearly dependent. Thus, null vector cannot be contained in a basis.


    3. The attempt at a solution
    Well, I know that in order for a collection of vectors to to be linearly dependent, one vector can be expressed as a linear combination of other vectors such as:
    let s be some non-zero scalar
    let v be vectors

    s1v1 + s2v2 + .... + skvk = 0

    but lets say that v2 was a zero vector (is this what the question is asking?),
    -s2v2 = s1v1 + s3v3 + ... + skvk?
    I don't quite get the phrase "any collection of vectors which includes 0(theta)"
     
  2. jcsd
  3. Jul 13, 2009 #2

    Office_Shredder

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    Start off by proving the set containing ONLY the zero vector is linearly dependent. It's easiest to use the direct definition here: A set of vectors v1,..., vn is linearly dependent if and only if there exist coefficients s1,..., sn such that not all the si are zero (but some of them are allowed to be) and that

    s1v1 + .... + snvn = 0

    If you only have the zero vector, what s1 can you pick to satisfy this? Then consider if you add additional vectors... what coefficients can you pick for them?
     
  4. Jul 13, 2009 #3
    Okay thanks, I got that one. How about this one?

    1. The problem statement, all variables and given/known data
    Prove that if the vectors b1, b2,...bm are linearly dependent, then any collection of vectors which contains the b's must also be linearly dependent

    3. The attempt at a solution
    So to be linearly dependent,
    s1b1 + s2b2 +.... + smbm = 0
    given that the scalar s is not equal to zero.

    How should I go about proving this one?
     
  5. Jul 13, 2009 #4

    HallsofIvy

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    There is NO "scalar s" in what you wrote! You mean that "at least one of the s1, s2, ..., sm is not 0."

    If a collection of vectors includes the b's, take the coefficients of the addtional vector to all be 0.
     
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