Results 1 to 6 of 6

Math Help - Proving that the image is a subspace

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    19

    Proving that the image is a subspace

    If V and W are two vector spaces over the same field K, and T : V→W is a linear mapping, how do I prove that the image of the set T, given by
    Im(T) = (w belongs to W : there exists v belongs to V such that T(v)= w),
    Is a subspace of W




    Thank your for your time
    Last edited by MuhTheKuh; November 14th 2010 at 10:57 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by MuhTheKuh View Post
    If V and W are two vector spaces over the same field K, and T : V→W is a linear mapping, how do I prove that the image of the set T, given by
    Im(T) = (w [IMG]file:///C:/Users/Tristan/AppData/Local/Temp/moz-screenshot-1.png[/IMG]belongs to [IMG]file:///C:/Users/Tristan/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG][IMG]file:///C:/Users/Tristan/AppData/Local/Temp/moz-screenshot.png[/IMG] W : there exists v belongs to[IMG]file:///C:/Users/Tristan/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG] V such that T(v)= w),
    Is a subspace of W




    Thank your for your time
    as you can see, your images are not showing up...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    May 2010
    Posts
    19
    Thanks. I was just trying to get the "belongs to" sign in there, but when that did not work out I just left it and completely forgot about that one.
    Sorry.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by MuhTheKuh View Post
    Thanks. I was just trying to get the "belongs to" sign in there, but when that did not work out I just left it and completely forgot about that one.
    Sorry.
    So you're trying to prove that if V,W are two K-spaces and T:V\to W is a linear transformation then T\left(V\right) is a subspace of W, right? But this amounts to proving that x,y\in T\left(V\right)\text{ and }\alpha,\beta\in K\implies \alpha x+\beta y\in T\left(V\right). But, by definition x=T(x') and y=T(y') for some x',y'\in V so that you're trying to prove that \alpha T(x')+\beta T(y')\in T\left(V\right) but \alpha T(x')+\beta T(y')=\cdots?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    May 2010
    Posts
    19
    aT(x')+bT(y')= ax+by

    but is that enough to prove it?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by MuhTheKuh View Post
    aT(x')+bT(y')= ax+by

    but is that enough to prove it?
    No. Try using the linearity of T
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proving that something is a subspace
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 26th 2010, 12:37 AM
  2. Proving intersection is a subspace
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 9th 2010, 09:07 PM
  3. Proving this is a subspace
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: June 1st 2009, 07:47 PM
  4. Proving a set is a subspace
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 5th 2009, 12:25 PM
  5. Prove that the inverse image of V is a subspace in X.
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: June 22nd 2008, 02:25 AM

Search Tags


/mathhelpforum @mathhelpforum