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Math Help - Consistent System and Null Space

  1. #1
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    Consistent System and Null Space

    Let A be an n m matrix. Prove that if b is a non-zero vector in the nullspace of A transposed, then the systemof linear equations Ax = b has no solution.


    So I'm not exactly sure how to go about this.

    I know rank(A) = rank(A transposed)
    which implies dim(col(A))=dim(col(A transposed))
    And for the system to be consistent b must belong to the col(A), which implies rank(A) = n.
    But wouldn't the rank(A transposed) = m for the system A(transposed) b = 0 to be consistent which implies rank(A) = m.
    But then Ax=b would have no solution right?
    Last edited by turbozz; November 13th 2013 at 07:22 PM.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Consistent System and Null Space

    correction:

    "...b must belong to the col(A), which implies rank(A)=m is greater than. or equal to, n"
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  3. #3
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    Re: Consistent System and Null Space

    Quote Originally Posted by turbozz View Post
    Let A be an n m matrix. Prove that if b is a non-zero vector in the nullspace of A transposed, then the systemof linear equations Ax = b has no solution.


    So I'm not exactly sure how to go about this.

    I know rank(A) = rank(A transposed)
    which implies dim(col(A))=dim(col(A transposed))
    And for the system to be consistent b must belong to the col(A), which implies rank(A) = n.
    But wouldn't the rank(A transposed) = m for the system A(transposed) b = 0 to be consistent which implies rank(A) = m.
    But then Ax=b would have no solution right?
    you're making it harder than it is.

    if b is a non-zero vector in the null space of AT, then ATb=0, that's what it means to be in the null space.

    So now suppose for contradiction that there does exist a solution to Ax=b

    then ATAx = ATb = 0

    So either x is 0 or ATA is 0. ATA is |A|2 and is only 0 if A=0.

    So there are no non-trivial solutions to Ax=b.
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