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Math Help - another linear dependence problem

  1. #1
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    another linear dependence problem

    Let the set {v1, v2, v3} be linearly dependent. Prove that {v1, v1+v2, V3} is also linearly dependent.

    Your help would be greatly appreciated. thank you!
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  2. #2
    Senior Member Shanks's Avatar
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    the Given two vector set are equavalent, thus if the second set is linear independent, then the first set is also linear independent.
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  3. #3
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    following up Shanks words, let's use p\implies q\equiv\sim q\implies\sim p.

    av_1+bv_2+cv_3=(a+b)v_1+bv_2+cv_3-bv_1, but \{v_1,v_1+v_2,v_3\} is a linear independent set, thus a+b=b=c=0, and a=b=c=0, so \{v_1,v_2,v_3\} is a linear independent set.
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  4. #4
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    Smile

    Quote Originally Posted by Krizalid View Post
    following up Shanks words, let's use p\implies q\equiv\sim q\implies\sim p.

    av_1+bv_2+cv_3=(a+b)v_1+bv_2+cv_3-bv_1, but \{v_1,v_1+v_2,v_3\} is a linear independent set, thus a+b=b=c=0, and a=b=c=0, so \{v_1,v_2,v_3\} is a linear independent set.
    i think he's trying to prove that the system \left \{ v_{1},v_{1}+v_{2},v_{3} \right \} is linearly dependent.
    non ?
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  5. #5
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    no, he's trying to prove that the second set is linearly dependent.
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  6. #6
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    the system \left \{ v_1,v_1+v_2,v_3 \right \} is linearly dependent if there's some scalars  \lambda_1,\lambda _2 and \lambda_3 not all zeros such that,
    .
    which give us,

    setting

    and since the system is linearly dependent,the scalars aren't all zeros,therefore the system \left \{ v_1,v_1+v_2,v_3 \right \} is linearly dependent.
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