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Math Help - are these vectors linearly independent?

  1. #1
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    are these vectors linearly independent?

    I need to know if these vectors are linearly independent or not- According to answer at least two should be but I don't know how to find out please help
    {(1,-1,1) (3,1,5) (1,1,2)}
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  2. #2
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    Re: are these vectors linearly independent?

    See Wikipedia for an example.
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  3. #3
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    Re: are these vectors linearly independent?

    Quote Originally Posted by rohan03 View Post
    I need to know if these vectors are linearly independent or not- According to answer at least two should be but I don't know how to find out please help
    {(1,-1,1) (3,1,5) (1,1,2)}
    It's always good to start with the definitions. A set of vectors, [tex]\{v_1, v_2, v_3, \cdot\cdot\cdot, v_n\}[/itex] are "independent" if and only if the only way we can have
    a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_nv_n= 0 is to have a_1= a_2= \cdot\cdot\cdot= a_n= 0.

    In this example, you would have
    a_1(1, -1, 1)+ a_2(3, 1, 5)+ a_3(1, 1, 2) = (a_1+ 3a_2+ a_3, -a_1+ a_2+ a_3, a_1+ 5a_2+ 2a_3)= (0, 0, 0).

    That is, you have the three equations:

    a_1+ 3a_2+ a_3= 0
    -a_1+ a_2+ a_3= 0
    a_1+ 5a_2+ 2a_3= 0

    If the only solution to those three equations is a_1= a_2=a_3= 0 then the vectors are independent. If there are other solutions (a system of n homogeneous linear equations in n unknowns has a single unique solution, or has an infinite number of solutions) the vectors are dependent.
    Last edited by HallsofIvy; July 23rd 2012 at 07:29 AM.
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