Results 1 to 4 of 4

Math Help - direct sum

  1. #1
    Junior Member
    Joined
    Oct 2010
    Posts
    47

    direct sum

    Suppose x1,...., xn (where n>= 2) is a basis of a vector space V. Choose r in {1,....,n-1} and define

    M= Span{x1,...,xr},
    N= Span{xr+1,....xn}
    show that V = M (direct sum) N
    since need to show direct sum that I have to show v= M+N
    and the intersection of M, N is {0}.
    But I don't know how to show that the intersection of M, N is {0}.

    Plx help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by mathbeginner View Post
    since need to show direct sum that I have to show v= M+N
    and the intersection of M, N is {0}.
    But I don't know how to show that the intersection of M, N is {0}.

    Plx help
    It's patently true that M+N=V, right? Now, the reason why the intersection is trivial is because if v\in M\cap N then v=\alpha_1x_1+\cdots+\alpha_rx_r and v=\alpha_{r+1}x_{r+1}+\cdots+\alpha_nx_n and thus \alpha_1x_1+\cdots+\alpha_rx_r=\alpha_{r+1}x_{r+1}  +\cdots+\alpha_nx_n or equivalently \alpha_1x_1+\cdots+\alpha_rx_r+(-\alpha_{r+1})x_{r+1}+\cdots+(-\alpha_n)x_n=\bold{0}. But, what does that tell us?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2010
    Posts
    47
    Quote Originally Posted by Drexel28 View Post
    It's patently true that M+N=V, right? Now, the reason why the intersection is trivial is because if v\in M\cap N then v=\alpha_1x_1+\cdots+\alpha_rx_r and v=\alpha_{r+1}x_{r+1}+\cdots+\alpha_nx_n and thus \alpha_1x_1+\cdots+\alpha_rx_r=\alpha_{r+1}x_{r+1}  +\cdots+\alpha_nx_n or equivalently \alpha_1x_1+\cdots+\alpha_rx_r+(-\alpha_{r+1})x_{r+1}+\cdots+(-\alpha_n)x_n=\bold{0}. But, what does that tell us?
    [tex]\alpha_1=...=\alpha_n=0[tex] they are linearly independent, am I right?
    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 mathbeginner View Post
    [tex]\alpha_1=...=\alpha_n=0[tex] they are linearly independent, am I right?
    Oui oui!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Direct Sum
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 17th 2011, 07:42 PM
  2. Direct Sum
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 6th 2009, 08:31 AM
  3. direct products
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 4th 2009, 11:57 AM
  4. Help with a direct sum proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 4th 2009, 10:06 PM
  5. Direct product
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: January 7th 2009, 08:41 PM

Search Tags


/mathhelpforum @mathhelpforum