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Math Help - Matrix Rank

  1. #1
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    Matrix Rank

    I) n = rank + nullity
    1) x = y + z, Ay ≠ 0, Ax = 0, n = rí + ν
    2) Ax = Ay
    3) Ay = col space of A, dim r (rank)
    4) y = sol space of A, dim rí
    5) y1 ≠ y2 → Ay1 ≠ Ay2 → y1 ≠ y2 → rí = r
    6) n = r + ν

    II) row rank = column rank
    1) Ax = 0 → x is null space of cols (def) and rows (alixi=0) of A
    2) n = r + ν in either case
    3) row rank = column rank

    The intent is to convey the point of the proof as logically, intuitively, and conciseley as possible, to be easily recalled. Individual steps can then be filled in. Shortcuts should be obvious. It is assumed one has seen standard text-book proofs. Note convention sum over repeated indices.
    Last edited by Hartlw; October 6th 2012 at 07:57 AM.
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  2. #2
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    Re: Matrix Rank

    Typo in I),1): It should be Az=0, not Ax=0.
    Style in II),1): use z instead of x for consistency of notation between I and II
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