Results 1 to 2 of 2

Math Help - rank

  1. #1
    Super Member
    Joined
    Aug 2009
    Posts
    639

    rank

    im trying to understand the proof on rank (fg) = rank (f) if g is isomorphic.

    it states that since g is surjective, it means that im(f)= im (fg)..but i dont get how it means that im(f)= im (fg)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,967
    Thanks
    1635
    To prove that two sets are equal, prove that each is a subset of the other. That is, prove that "if x in A then it is in B" and vice versa. Here "A" and "B" are im(f) and im(fg).

    If vector w is in Im(f) then there exist vector v in V such that f(v)= w. Since g is isomorphic, there exist vector u in U such that g(u)= v. Then fg(u)= f(v)= w so w is in im(fg).

    Now do it the other way around: if vector w is in Im(fg) then there exist vector u in U such that fg(u)= w. Let v= g(u). Then f(v)= w so w is in im(f).
    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