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Homework Help: Need help with linear independence proof!

  1. Mar 7, 2006 #1
    Hi,

    I don't know how to do the following proof:

    If (v1, ...vn) are linearly independent in V, then so is the list (v1-v2, v2-v3, ...vn-1 -vn, vn).

    I can do the proof if I replace 'linearly independent' with 'spans V' ...so what connection am I missing?

    Thanks much!
     
  2. jcsd
  3. Mar 7, 2006 #2

    0rthodontist

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    Prove it by contradiction. If the second set of vectors was not linearly independent, then you can write 0 as a linear combination of those vectors. Then simply expand out each vector to show that this implies v1...vn are also linearly dependent.
     
  4. Mar 7, 2006 #3
    Great thanks. Here's another one for you...Prove that if V is finite dimensional with dim V > 1, then the set of noninvertible operators on V is not a subspace of L(V)
     
  5. Mar 7, 2006 #4

    AKG

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    What have you done for that second problem dyanmcc? Start by thinking about matrices.
     
  6. Mar 8, 2006 #5

    0rthodontist

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    Also be sure to prove that there is still a nonzero coefficient when you expand the vectors out. (look at the FIRST nonzero coefficient before expansion)
     
    Last edited: Mar 8, 2006
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