Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Linear transformation + ranks

  1. Feb 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Let S(U)=V and T(V)=W be linear maps where U,V, W are vector spaces over the same field K. Prove :

    2. Relevant equations
    a) Rank (TS) <= Rank (T)
    b) Rank (TS) <= Rank (S)
    c) if U=V and S is nonsingular then Rank (TS) = Rank (T)
    d) if V=W and T is nonsingular then Rank (TS) = Rank (S)

    3. The attempt at a solution
    a) TS maps to W, so is T
    b) TS maps to W, but S to V, but how do I show the ranks for (a) and (b)?
    c) d) So inverse of S and T exists, and err...

    U,V,W are vector space over the SAME field, does that mean they have the same number of entries, say R2, R3, etc etc
     
  2. jcsd
  3. Feb 10, 2010 #2

    HallsofIvy

    User Avatar
    Science Advisor

     
  4. Feb 10, 2010 #3
    "TS maps to W, so is T"

    sorry, what I meant was T also maps to W

    for (c) and (d)
    what does the the nonsingularity of the matrix imply?
    how does that show that the ranks are both equal?
     
  5. Feb 10, 2010 #4
    I think the best way to show that the ranks are equal is by considering the surjectivity and injectivity of S and T respectively (since S and T are non-singular). From there, it should be pretty much straightforward.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook