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Math Help - Subspace problem

  1. #1
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    Subspace problem

    Hi, I've been trying to figure out this problem, but I'm not quite sure how to go about it.

    Problem:
    For any matrix A(m,n) show that the set of right hand sides b in R^m for which Ax = b is solvable is a subspace of R^m

    Attempt at a solution:
    I know how to prove the solvability of a specific linear system, but how can I show that is true for any matrix? (Assuming I need to)

    Any help/advice will be much appreciated!
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  2. #2
    Junior Member bondesan's Avatar
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    Post

    Let S\subset\mathbb{R}^m be the set of all b's which make Ax=b solvable.

    Consider that for an arbitrary x1 and x2 we have A(x1) = b1 and A(x2) = b2.

    If we take the sum A(x1)+A(x2)=b1+b2 \Rightarrow A(x1+x2)=(b1+b2) \Leftrightarrow (b1+b2) \in S.

    In the same way, we can have, for k\in\mathbb{R}, kA(x1) = kb1 \Rightarrow A(kx1)=kb1 \Leftrightarrow kb1 \in S.

    I hope this is what you're looking for.
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  3. #3
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    Yes this helps a lot! Thanks!
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  4. #4
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    Quote Originally Posted by MEMPHIS View Post
    Hi, I've been trying to figure out this problem, but I'm not quite sure how to go about it.

    Problem:
    For any matrix A(m,n) show that the set of right hand sides b in R^m for which Ax = b is solvable is a subspace of R^m

    Attempt at a solution:
    I know how to prove the solvability of a specific linear system, but how can I show that is true for any matrix? (Assuming I need to)

    Any help/advice will be much appreciated!
    You're given A is mxn.
    You might give some consideration to the column space of A, which we (should) know is a subspace of R^m.

    Can such a b fail to reside in the column space of A?
    Is there a vector in the column space of A, say b, where we find that there is no solution set in R^n for the system Ax=b?
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