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Linear Algebra - augmented matrix proof

  1. Feb 20, 2013 #1
    1. The problem statement, all variables and given/known data
    Let A be the augmented m x (n + 1) matrix of a system of m linear equations
    with n unknowns. Let B be the m x n matrix obtained from A by removing the last
    column. Let C be the matrix in row reduced form obtained from A by elementary
    row operations. Prove that the following four statements are equivalent.
    (i) The linear equations have no solutions.
    (ii) If c_1,......, c_(n+1) are the columns of A, then c_(n+1) is not a linear combination of
    c_1,......, c_(n+1)
    (iii) Rank(A) >Rank(B).
    (iv) The lowest non-zero row of C is (0 0 ... 0 0 1)

    2. Relevant equations

    3. The attempt at a solution

    Now, i have so far assumed linear dependence, so a_1c_1 + a_2c_2 + ...... + a_nc_n = a_n+1c_n+1.

    I then converted this into a matrix form, as c_i are the columns of matrix A, and then i have row matrices with m rows, and n unknowns. From here i think i need to make the deduction that c_n+1 is NOT a linear combination of c_1,.....,c_n, but not sure how to jump to that conclusion, as once i show this, i can show that (i) is true and the linear equations have no solutions.

    part (iii) and (iv) are for later, not concerned about those at the moment, just i and ii. Sorry about the lack of LaTeX.
  2. jcsd
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