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Linear independence

  1. Aug 26, 2005 #1
    Hi, can someone help me with the following question?

    Q. Show that if [tex]\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}[/tex] is linearly independent and [tex]\mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}[/tex] then [tex]\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to ,\mathop {v_{k + 1} }\limits^ \to } \right\}[/tex] is linearly independent. Use this to prove that the non-zero rows of a matrix in row-echelon form are linearly independent.

    Here is my attempt.

    Write [tex]\alpha _1 \mathop {v_1 }\limits^ \to + .... + \alpha _k \mathop {v_k }\limits^ \to + \beta \mathop {v_{k + 1} }\limits^ \to = \mathop 0\limits^ \to ...\left( 1 \right)[/tex]

    [tex]
    \beta \mathop {v_{k + 1} }\limits^ \to = - \left( {\alpha _1 \mathop {v_1 }\limits^ \to + .... + \alpha _k \mathop {v_k }\limits^ \to } \right)
    [/tex]

    If [tex]\beta \ne 0[/tex] then [tex]\mathop {v_{k + 1} }\limits^ \to = - \left( {\frac{{\alpha _1 }}{\beta }\mathop {v_1 }\limits^ \to + ...\frac{{\alpha _k }}{\beta }\mathop {v_k }\limits^ \to } \right)[/tex] but this is impossible since [tex]\mathop {v_{k + 1} }\limits^ \to \notin span\left\{ {\mathop {v_1 }\limits^ \to ,...,\mathop {v_k }\limits^ \to } \right\}[/tex]

    So beta is equal to zero and equation one reduces to [tex]\alpha _1 \mathop {v_1 }\limits^ \to + .... + \alpha _k \mathop {v_k }\limits^ \to = \mathop 0\limits^ \to [/tex] where all of the a_i are equal to zero by hypothesis. Is that enough to show the given result?

    I can't think of a way to tackle the second part with the matrix. Seeing as that's the case I'll just write out whatever I can think of.

    I think the key idea is that in row echelon form, each time I 'move up' one row, the vector(represented by a row in the matrix) has at least one additional non-zero component. So let A be the n by k (n columns and k rows) matrix in row echelon form whose rows are the vectors v_i where i = 1,...,k and each of the vectors has at least one non-zero component.

    Starting at the bottom of the matrix and moving up to the first non-zero row I a vector which has c non-zero components call it v_1 and {(v_1)} is linearly independent since it consists of a non-zero single vector. Moving up to the next row I get another vector call it v_2 which has at least c + 1 non-zero components. Since v_2 has more non-zero components than v_1 then {v_1, v_2} is linearly independent. From here I'd probably just continue with the same argument. The problem is that what I've said is a pretty clumsy explanation. I wasn't really sure how to do this question either. So can someone please help me with this?

    Edit: Ok my attempt for the second part is completely incorrect because I could have something like v_1 = (0,0,1,0,0) and v_2 = (1,0,0,0,0). Help would be appreciated.
     
    Last edited: Aug 26, 2005
  2. jcsd
  3. Aug 26, 2005 #2

    lurflurf

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    Homework Helper

    Do the second part by induction.
    n=1 case
    the vector is not zero so is linearly independent
    n+1 case
    n vectors are independent
    if (n+1)st vector is in span(first n vectors) the matrix is not in row-echelon form
    therfore (n+1)st vector is not in span
    thus n+1 vectors are linearly independent
     
  4. Aug 27, 2005 #3
    Thanks for te help lurflurf. However, I am not sure how my answer should be worded. The question refers to a matrix so would I need to make some reference to the matrix? If so how would I do it in a clear and concise manner? For example, do I need to mention the position of the vectors(represented as rows) in the matrix?
     
  5. Aug 27, 2005 #4

    lurflurf

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    Homework Helper

    The matrix is a representation. You can consider the set of the nonzero row vectors. The position of the vectors is not important.
    Just say something like
    let v1,...,vn be the nonzero row vectors
     
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