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Homework Help: Linear Algera ?Dependent Rows?

  1. Feb 19, 2012 #1
    If the rows of A are linearly dependent, are the rows of AB also linearly dependent? Justify answer.

    I don't completely understand this question because (so far) my instructor and my text book has not discussed what it means for rows to be linearly dependent. There is a similar question in my text book that asked "if the columns of B are linearly dependent, show that the columns of AB are also."

    Does this have something to with transverses? Can I just think about it the same way I have been with columns or does "row" dependencs mean something different?
     
  2. jcsd
  3. Feb 19, 2012 #2
    (AB)T = BTAT, so yes, they're more or less the same problem.

    Vectors v1, …, vn are linearly independent if k1v1 + … + knvn = 0 (the zero vector) implies the scalars ki must be equal to 0.

    They are linearly dependent if one of the vectors can be written as a linear combination of the others (or is contained in their span). This is equivalent to saying they're not linearly independent.

    (Sorry for the multiple edits, a bit rusty.)
     
    Last edited: Feb 19, 2012
  4. Feb 19, 2012 #3

    HallsofIvy

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    In particular, the rows of matrix A are linearly independent if and only if A is invertible. But then [itex]A^T[/itex] is also invertible and its rows are A's columns.
     
  5. Feb 19, 2012 #4
    This is true only for square matrices. I believe the question is asking about matrices in more generality.

    I would write the components of a row vector in AB in terms of the dot products of a row vector in A and column vectors in B. Which row vector in AB can be written as a linear combination of the other row vectors in AB?
     
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