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Math Help - Matrix proof - augmented matrix - row reduction - column operation - proof

  1. #1
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    Matrix proof - augmented matrix - row reduction - column operation - proof

    If we 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 the the following 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
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  2. #2
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    Well, what have you done on this? You might think about how the last column of A relates to the first n columns in light of the fact that " A is the augmented m x (n + 1) matrix of a system of m linear equations
    with n unknowns".
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  3. #3
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    ok, since it is the augmented matrix then we can get a system of m linear equations, I'm not sure why exactly they are linear but we will use the fact that

    A has m rows and each column of A will have c _1 .... c _n we will make our linear equations but I think we can only achieve this if m=n?

    these are my first thoughts
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