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Math Help - proof involving rank of a matrix

  1. #1
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    proof involving rank of a matrix

    I have to prove the following statement, or provide a counterexample:
    If the coefficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution.
    My thought is that this statement is true, and I think the proof would use the following theorem:
    Let Ax=b be a system of linear equations. Then the system is consistent (the solution set is nonempty) if and only if rank(A) = rank(A|b).

    Is it true that if an mxn matrix has rank m as assumed that the rank of the corresponding augmented matrix (A|b) would ALWAYS have the same rank?
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    I have to prove the following statement, or provide a counterexample:
    If the coefficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution.
    My thought is that this statement is true, and I think the proof would use the following theorem:
    Let Ax=b be a system of linear equations. Then the system is consistent (the solution set is nonempty) if and only if rank(A) = rank(A|b).

    Is it true that if an mxn matrix has rank m as assumed that the rank of the corresponding augmented matrix (A|b) would ALWAYS have the same rank?

    The claim is false and as an easy example take the map \mathbb{R}\rightarrow\mathbb{R}^2 defined by r\rightarrow\left(\begin{array}{c}r\\r\end{array}\  right) . Clearly the system:

    \left(\begin{array}{c}1\\1\end{array}\right)(r)=\l  eft(\begin{array}{c}b_1\\b_2\end{array}\right)

    has no solution if b_1\neq b_2

    Tonio
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