Results 1 to 13 of 13

Math Help - direct sum

  1. #1
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Lightbulb direct sum

    show that if f is in Hom(U,U) such that f2=f then V= ker f + ker(f - idv). also let B0 be an ordered basis For ker f and B1 be ordered basis for ker(f-idv). if B is an ordered basis and B=B0 + B1 for V. what is the matrix representation of f given B,B?
    Last edited by sipnayan; December 16th 2013 at 01:08 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: direct sum

    Where are you having trouble?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    in everything. i dont understand sir
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: direct sum

    Are you looking for a place to start? Let's start with the first statement. How do you show that two sets are equal?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    subsets of each other
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    i dont know what it means if it is in kerf and ker (f-idv)
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: direct sum

    Do you know what the kernel of a homomorphism is? If v is in the kernel of f, then f(v) = 0. If v is in the kernel of (f-id_V), then (f-id_V)(v)=0. So, the first thing you want to do is show that V=ker f (+) ker (f-id_V) is a subspace of U (where (+) is direct sum since I am having trouble with LaTeX).
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    its ok. how to show that it is direct sum?
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: direct sum

    Show that ker f intersect ker (f-id_V) is only the zero vector.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    and how should i do it?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Nov 2010
    Posts
    1,931
    Thanks
    782

    Re: direct sum

    I just told you. There is nothing magic about it. Suppose v is in ker f and v is in ker(f-id_V). Show that v is the zero vector. Since v is in ker f, you have f(v)=0. Since v is in ker(f-id_V), (f-id_V)(v) = 0. For the second one, you have (f-id_V)(v) = f(v) - id_V(v) = f(v)-v = 0. If f(v)-v=0 then f(v)=v. If f(v)=v and f(v)=0 then by transitivity, v=0.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    Newbie
    Joined
    Dec 2013
    From
    philippines
    Posts
    8

    Re: direct sum

    thanks a lot. how about the matrix representation?
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,401
    Thanks
    762

    Re: direct sum

    Let v be any element of V.

    Then if w = f(v), we have:

    f(w) = f(f(v)) = f2(v) = f(v) = w.

    Thus f(w) - w = 0, so:

    (f - idV)(w) = 0, that is: f(v) is in ker(f - idV).

    Write:

    v = f(v) + v - f(v) (which is always trivially true).

    We have, by the above, that f(v) is in ker(f - idV).

    Now let's compute f(v - f(v)):

    f(v - f(v)) = f(v) - f(f(v)) = f(v) - f2(v) = f(v) - f(v) = 0, so for ANY v, v - f(v) is in ker(f).

    So this tells us that we can write any element of V as the sum of something in ker(f) (namely: v - f(v)) and something in ker(f - idV) (namely: f(v)).

    Hence V = ker(f) + ker(f - idV).

    Now SlipEternal has already shown that the intersection of these two sets is {0}, hence the sum is direct.

    As for the matrix representation, we have:

    [f]B = [0 0...0 b1 b2....bk] (these are column vectors) where B1 = {b1,...,bk}

    Here is a simple example:

    Let V = R2, and let f(x,y) = (x,0).

    Then f2(x,y) = f(f(x,y)) = f(x,0) = (x,0), so we see that f2 = f.

    Clearly, a basis for ker(f) is {(0,1)}. let's find a basis for ker(f - idV).

    If f(x,y) = (x,y), then:

    (x,0) = (x,y), so y = 0. Thus a basis is: {(1,0)}. In the ordered basis B = {(0,1),(1,0)} (note this is not the USUAL ordered basis), we have

    [f]B =

    [0 1]
    [0 0].

    (Slight note: usually we write the ordered basis the "other way around" as B = B1 + B0, so the "0's come last").
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Direct Products/Direct Sums
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 15th 2012, 11:31 AM
  2. Direct Sum
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 17th 2011, 07:42 PM
  3. direct sum
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 16th 2010, 07:39 PM
  4. Direct Sum
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 6th 2009, 08:31 AM
  5. Help with a direct sum proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 4th 2009, 10:06 PM

Search Tags


/mathhelpforum @mathhelpforum