Results 1 to 3 of 3

Math Help - Linear Mapping Rank Proofs

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    41

    Linear Mapping Rank Proofs

    I have the following problem:
    Let U, V, W be finite dimensional vector spaces over R and let
    L : V -> U and M : U -> W be linear mappings

    a) Prove rank(M(L)) <= rank(M)
    b) Prove rank(M(L)) <= rank(L)
    c) Prove that if M is invertible, then rank(M(L)) = rank(L)

    Now...I know that rank(N) = dim(Range(N)) for some linear mapping N

    I can use this to prove part a, since range(M(L)) must be some subset of range(M) and therefore dim(range(M(L))) <= dim(range(M)) and rank(M(L)) <= rank(M).

    I don't know how I am supposed to prove b though, is there a way to show that dim(range(M(L))) <= dim(range(L))

    edit: I think I can use the rank nullity theorem to show that since:
    rank(M(L(V))) + nullity(M(L(V))) = dim (L(V))
    rank(M(L(V))) <= dim(L(V))
    and since L(V) is, by definition, the range of L:
    rank(M(L)) <= rank(L)
    I am however, still stuck on how to go about proving c...

    I also don't know how to go about proving c, is there some related property based on the invertibility of M?

    Any help would be greatly appreciated.
    Last edited by crymorenoobs; September 21st 2010 at 06:20 PM. Reason: Revised facts
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433
    What is the nullity of an invertible map?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2010
    Posts
    41
    Thanks, sorry I solved this but forgot to edit it afterwards. Since the nullity of an invertible map is 0, the rank nullity theorem presents a simple solution.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Mapping..
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 20th 2009, 10:55 AM
  2. linear mapping
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 25th 2009, 06:33 PM
  3. linear mapping
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2008, 04:57 PM
  4. linear mapping
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 18th 2008, 09:37 PM
  5. linear mapping
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 16th 2008, 09:12 PM

Search Tags


/mathhelpforum @mathhelpforum