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Math Help - [SOLVED] I don't know

  1. #1
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    [SOLVED] I don't know

    what to do for this one:

    Given: 3 vectors, a, b, c, are linearly independent in a vector space V
    Asked: Show that 3a, 2a-b, a + c are also linearly independent
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  2. #2
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    Quote Originally Posted by Noxide View Post
    what to do for this one:

    Given: 3 vectors, a, b, c, are linearly independent in a vector space V
    Asked: Show that 3a, 2a-b, a + c are also linearly independent
    Assume 3a, 2a-b, a+c are dependent. Then, there exist scalras x_1,x_2,x_3, not all zero, such that x_1(3a) + x_2(2a-b) + x_3(a+c) = 0 \Rightarrow 3x_1a + 2x_2a -x_2b + x_3a + x_3c = 0  \Rightarrow (3x_1+2x_2+x_3)a + (-x_2)b + (x_3)c = 0

    If x_2=x_3=0 then x_1 \neq 0. Therefore not all the coefficients are zero.

    Let \lambda_1 = 3x_1+2x_2+x_3
    \lambda_2 = -x_2
    \lambda_3 = x_3

    Then \lambda_1a + \lambda_2b + \lambda_3c = 0 but not all of the coefficients are zero, in contradiction to the fact that a,b,c are linearly independent... ==> the assumption can not hold, therefore 3a, 2a-b and a+c are linearly independent.
    Last edited by Defunkt; October 23rd 2009 at 06:40 AM. Reason: fixed typos
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  3. #3
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    wow, that's a really awesome proof by contradiction!

    thanks for the help
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