1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
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: Rank/Image/Kernel proof

  1. Jun 17, 2011 #1
    1. The problem statement, all variables and given/known data

    Let V be a vector space. Show that for every three linear operators
    A,B,C: V -> V, we have
    rk(ABC) =< rk(B)

    2. Relevant equations

    V = rk(A) + dimKer(A)

    rk(A) = dimIm(A)

    3. The attempt at a solution

    Im(ABC) = {ABC(v) | vEV}
    = {AB(C(v)) | vEV}

    So Im(ABC) is a subset of Im(AB)

    So dimIm(ABC) =< dimIm(AB)

    So rk(ABC) =< rk(AB)

    Im(AB) = {A(B(w)) | wEV}

    Ker(B) subset of ker(AB) because if Bx=0, then ABx = A0 = 0

    By the dimension formula, this leads to rk(B) >= rk(AB)

    So putting the two results together we get rk(ABC) =< rk(B)

    Is this correct? Thanks!
  2. jcsd
  3. Jun 17, 2011 #2
    Hi Maybe_Memorie! :smile:

    Your proof seems correct! But it seems you'll need V to be finite dimensional for it to work.
  4. Jun 17, 2011 #3
    We've only dealt with finite dimensional vector spaces.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook