Results 1 to 6 of 6

Math Help - Subspaces and Dimension

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Subspaces and Dimension

    Let V be vector space with dim(V)=n. And let U,W be subspaces with n-1 dimension each,I need to prove that: U+W=V if and only if U!=W



    Thank you all!
    Last edited by Also sprach Zarathustra; December 2nd 2009 at 12:49 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    I'll assume you know how to prove that if U+W=V then U\neq W (it is trivial).

    Assume U,W are subspaces of V with dim V = n, ~ dimU = dim W = n-1 such that U \neq W.

    Let B = \{w_1,w_2,...,w_{n-1}\} be a basis for W.

    Let u \in U - W ~ \text{s.t.} ~ u \neq 0 (we know one such element exists otherwise U=W since 0\in U, ~ 0 \in W).
    Then u is not spanned by \{w_1,w_2,...,w_{n-1}\}, otherwise it would be an element of W (by definition). Then, B' = \{w_1,w_2,...,w_{n-1},u\} is an independent set and obviously dim B' = n and therefore it is a basis of V.

    This gives us that V \subset U+W.
    However, obviously there can be no element in U+W that is not in V since they are subspaces, therefore U+W \subset V \rightarrow U+W = V

    By the way, what uni are you studying in?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    Thanks a lot! But, your assumption is wrong, can you prove the first line please?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Well, assume U+W=V. Assume, by contradiction, that U = W. Then U+W = U = W = V

    What does this give us? (hint: look at the dimensions)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1
    That the dimension of U and W is not the dimension of V. Paradox?
    I am really bad at it! Thank you for your patients!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Aug 2009
    From
    Israel
    Posts
    976
    Yes, that is correct! since U+W=V, we must have that dim(U+W) = dim(V) but in this case dim(U+W)=n-1, dim(V)=n therefore we reach a contradiction, and then U \neq W.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Finding dimension of sum of subspaces
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 6th 2012, 10:36 AM
  2. dimension of subspaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 8th 2009, 11:07 AM
  3. Replies: 1
    Last Post: October 17th 2008, 11:35 AM
  4. Find a basis and the dimension of the following subspaces:
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 23rd 2008, 05:35 PM
  5. linear algebra...subspaces and dimension
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 13th 2008, 03:24 AM

Search Tags


/mathhelpforum @mathhelpforum