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Math Help - Linear Alegebra linear equations proofs

  1. #1
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    Linear Alegebra linear equations proofs

    Prove the following:

    A system of linear equations
    Ax = b with m < n either has no solution or has several different solutions.

    A system of linear equations
    Ax = b where A has rank m always has a solution.

    I think I'm making a little bit of headway but some help would be much appreciated
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by piophiad1990 View Post
    Prove the following:

    A system of linear equations
    Ax = b with m < n either has no solution or has several different solutions.

    A system of linear equations
    Ax = b where A has rank m always has a solution.

    I think I'm making a little bit of headway but some help would be much appreciated
    You need to define thoroughly what everything means here.
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  3. #3
    Senior Member Dinkydoe's Avatar
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    Suppose A has rank n, that is A is invertable. Then Ax = b has a unique solution x = A^{-1}b by uniqueness of the matrix-inverse.

    If A has rank m < n then it is row-equivalent with a matrix with at least one row zero's, say row i, 1 \leq i\leq n. Suppose it does have a solution x^* = (x_1,\cdots x_i,\cdots x_n). Then for any choice of x_i\in \mathbb{R} we have that x^* is a solution.
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