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Math Help - Column and Null Space?

  1. #1
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    Column and Null Space?

    This was a question on my test and I couldn't figure it out.

    Suppose A is m x n matrix, Prove the following equality:

    dim Col(A) +dim Nul(A^T) = m

    I had no clue how to do this. I know rank(A)=rank(A^T) but that's it.
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  2. #2
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    Re: Column and Null Space?

    do you know the rank nullity theorem?
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  3. #3
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    Re: Column and Null Space?

    What's that? Is it when If matrix A hasn columns, then rankA+ dimNulA = n?
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  4. #4
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    Re: Column and Null Space?

    that is one version of it, yes.
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  5. #5
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    Re: Column and Null Space?

    That's the only theorem we have. I don't know what else other than rankA=rank(A^T)
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  6. #6
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    Re: Column and Null Space?

    if A is an mxn matrix, then A^T is an nxm matrix.

    apply the rank nullity theorem to A^T:

    m = rank(A^T) + dim(null(A^T)).

    now, what is your definition of rank(A)?
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