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Math Help - Gaussian elimination help

  1. #1
    Newbie
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    Jun 2011
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    Unhappy Gaussian elimination help

    Question:The spanning set

    [(2,2,1), (4,1,0), (2,-1,-1), (2,0,1)]

    of R3 is not independent.

    Explain why, and identify a

    maximal independent set among these vectors.
    -----------------------------------------------------
    I'm having trouble with this question and i was wondering if someone could help me out and show me what I have to do to answer this question.
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  2. #2
    Senior Member
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    Feb 2008
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    Re: Gaussian elimination help

    Quote Originally Posted by kashmoneyrecord View Post
    Question:The spanning set

    [(2,2,1), (4,1,0), (2,-1,-1), (2,0,1)]

    of R3 is not independent.

    Explain why, and identify a

    maximal independent set among these vectors.
    -----------------------------------------------------
    I'm having trouble with this question and i was wondering if someone could help me out and show me what I have to do to answer this question.
    Well the dimension of \mathbb{R}^3 is three, so any set of four vectors is going to have to be linearly dependent.

    Let v_1=(2,2,1),v_2=(4,1,0),v_3=(2,-1,-1),v_4=(2,0,1). Then notice that

    e_2=\frac{1}{2}(v_1-v_4),

    e_1=\frac{1}{4}(v_1+v_3-e_2), and

    e_3=v_4-2e_1,

    where e_1,e_2,e_3 are the standard basis vectors. So \{v_1,v_3,v_4\} spans \mathbb{R}^3 and hence must be linearly independent.
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  3. #3
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    Jun 2011
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    Re: Gaussian elimination help

    yo bro is that suffiencient enough to answer the question or is there more to it? and what does maximal independent set mean?
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  4. #4
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    Jun 2011
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    Re: Gaussian elimination help

    okay got it thanks a lot
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