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Linear algebra proofs

  1. Sep 29, 2009 #1
    1) prove that for any five vectors (x1, ..., x5) in R3 there exist real numbers (c1, ...., c5), not all zero, so that BOTH

    c1x1+c2x2+c3x3+c4x4+c5x5=0 AND c1+c2+c3+c4+c5=0

    2)Let T:R5-->R5 be a linear transformation and x1, x2 & x3 be three non-zero vectors in R5 so that
    T(x1)=x1
    T(x2)=x1+x2
    T(x3)=x2+x3

    prove that {x1, x2, x3} are three linearly independent vectors.

    any help would be greatly appreciated, thank you!
     
  2. jcsd
  3. Sep 29, 2009 #2
    I've thought up a proof for the first one but it might be too complicated. I'll try to think of a simpler one if somebody else doesn't.

    As for the second, assume that you have a linear combination of the three equal to zero. Map it under the matrix and see if something cool happens. Then see if it happens again. There's probably a contradiction with the assumptions in there somewhere ;)
     
  4. Sep 29, 2009 #3

    Office_Shredder

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    For question 1)

    Extend a vector in R3 to one in R4 by adding a 1 in the fourth entry.
     
    Last edited: Sep 29, 2009
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