Results 1 to 4 of 4

Math Help - Linear Algebra - Direct Sum Problem.

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    14

    Linear Algebra - Direct Sum Problem.

    I'm doing some pre-term reading and I've come across direct sums which I'm having a bit of trouble getting my head around.

    If V is a vector space and U is a proper subspace (non trivial), I'm told that there are infinitely many different subspaces W of V such that V=U(+)V
    [ I've used (+) as the sign for direct sum as i'm not sure how to get it on my computer].

    I don't really understand this, and would have no idea on how to prove it. The notes I'm reading hint at thinking about what happens with V is 2-dimensional. Any clues and help would be appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    769
    Quote Originally Posted by nlews View Post
    I'm doing some pre-term reading and I've come across direct sums which I'm having a bit of trouble getting my head around.

    If V is a vector space and U is a proper subspace (non trivial), I'm told that there are infinitely many different subspaces W of V such that V=U(+)V
    [ I've used (+) as the sign for direct sum as i'm not sure how to get it on my computer].

    I don't really understand this, and would have no idea on how to prove it. The notes I'm reading hint at thinking about what happens with V is 2-dimensional. Any clues and help would be appreciated!
    What is it you're expected to prove?

    To the moderators - through no fault of my own, my post was duplicated on this thread.
    Last edited by wonderboy1953; September 22nd 2010 at 09:23 AM. Reason: explanation
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    769

    Response

    Quote Originally Posted by nlews View Post
    I'm doing some pre-term reading and I've come across direct sums which I'm having a bit of trouble getting my head around.

    If V is a vector space and U is a proper subspace (non trivial), I'm told that there are infinitely many different subspaces W of V such that V=U(+)V
    [ I've used (+) as the sign for direct sum as i'm not sure how to get it on my computer].

    I don't really understand this, and would have no idea on how to prove it. The notes I'm reading hint at thinking about what happens with V is 2-dimensional. Any clues and help would be appreciated!
    What is it you're expected to prove?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,396
    Thanks
    1847
    V is a vector space, U is a non-trivial subspace ("non-trivial" meaning here: there exist a non-zero vector in U, there exist a vector, w, in V that is not in U). Choose a basis, B_U, for U. Then there exist a basis B_V for V which contains B_U. Then B_V- B_U is also a basis for a subspace, W of V and V is the direct sum of U and W. If u is any vector in U, then B_U+ u, by which I mean the set of vectors created by adding v to each vector in B_U, is a basis for a different subspace of V and V is the direct sum of U and this new subspace.

    Examples in R^3:
    Any non-trivial subspace of R^3 is either a line through the origin or a plane through the origin.

    1) Let U be a plane containing the origin. Then the subset of R^3, W, consisting of any single line through the origin that is not contained in U is a subspace of R^3 and R^3 is the direct sum of U and W.

    2) Let U be a line through the origin. Then the subset or R^3, W, consisting of any single plane through the origin that does not contain U is a supspace of R^3 and R^3 is the direct sum of U and W.

    Example in R^2:
    Any subspace of R^2 is a line through the origin.
    Let U be a line through the origin. Let W be any other line through the origin. Then V is the direct sum of U and W.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Direct Sum & Linear Mappings
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 20th 2010, 09:27 PM
  2. Writting Linear Functional as Direct Sum.
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 28th 2010, 02:57 AM
  3. Linear Algebra Problem
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 4th 2009, 03:36 PM
  4. linear algebra - direct sums
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 15th 2008, 10:00 PM
  5. Linear Algebra problem
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 13th 2007, 06:13 PM

Search Tags


/mathhelpforum @mathhelpforum