Results 1 to 2 of 2

Math Help - rank

  1. #1
    Super Member
    Joined
    Aug 2009
    Posts
    639

    rank

    how do you prove that rank(A) = rank (A^t)?

    im trying to states that Ax=0, A^tAx=O then nullspace of A = nullspace of A^t.

    then rank(A) = rank (A^t)..but this wont work if A is not a sq matrix right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,154
    Thanks
    595
    if A is an mxn matrix, Aᵀ is an nxm matrix.

    AAᵀ is an mxm matrix, AᵀA is an nxn matrix, both these products are pefectly well-defined, but...

    by the rank-nullity theroem, dim(null(A)) = n - rank(A), whereas dim(null(Aᵀ)) = m - rank(Aᵀ),

    so unless m = n, the nullspaces will NOT have the same dimensions.

    what you want to do is prove that if rank(A) = k, that A has k linearly independent rows, as well as k linearly independent columns.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Row Rank = Column Rank?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 9th 2011, 12:10 AM
  2. Proof: rank(AB)+n >= rank(A)+rank(B)
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: September 9th 2010, 05:28 PM
  3. Replies: 3
    Last Post: August 20th 2010, 05:32 AM
  4. Row rank and column rank
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 13th 2010, 07:40 AM
  5. Short proof that rows-rank=column-rank?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 26th 2009, 10:02 AM

Search Tags


/mathhelpforum @mathhelpforum