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Another method? - Matrix & Linear Independence

  1. Apr 16, 2012 #1
    Question :
    Let A be a 7 × 4 matrix. Show that the set of rows of A is linearly dependent.



    Answer:
    The row vectors of a matrix are linearly independent if and only if the rank of the matrix is equal to the number of rows in the matrix.
    Since rank (A) = 4 , and the number of rows in the matrix is 7, the row vectors are linearly dependent.



    I am sure that my answer is right, but I was considering or wondering if there was an alternative method, because the actual answer is worth 4 marks, so was wondering if there was a more mathematical sound proof possible to prove the answer?
     
  2. jcsd
  3. Apr 16, 2012 #2

    HallsofIvy

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    The rows of a 7 by 4 matrix are members of R4 which has dimension 4- any set of more than 4 vectors must be dependent.

    Your basic statement that a 7 by 4 matrix has rank 4 is NOT correct. The matrix
    [tex]\begin{bmatrix}1& 1 & 1 & 1\\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \\1& 1 & 1 & 1 \end{bmatrix}[/tex]
    does not have rank 4.
     
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