Results 1 to 2 of 2

Math Help - Homogeneous equations

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

    Homogeneous equations

    Let W be sub-space in dimension m of F^n (m < n). Show that there exist m - n homogeneous equations in n variables, so that-

    W={(x_1,...x_n) | a_11*x_1+...+a_1n*x_n=0,...,
    a_n-m*x_1+...+a_n-mn*x_n=0}


    (x_1,...x_n) is column vector.
    Last edited by Also sprach Zarathustra; December 30th 2009 at 08:58 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by Also sprach Zarathustra View Post
    Let W be sub-space in dimension m of F^n (m < n). Show that there exist m - n homogeneous equations in n variables, so that-

    W={(x_1,...x_n) | a_11*x_1+...+a_1n*x_n=0,...,
    a_n-m*x_1+...+a_n-mn*x_n=0}


    (x_1,...x_n) is column vector.
    Pick a basis for W say \{w_1,...,w_m\} and extend to a basis for \mathbb{F} ^n say we add \{ v_{1},...,v_{n-m} \} and take T: \mathbb{F} ^n \rightarrow \mathbb{F} ^{n-m} (giving the later a basis \{ u_1,...,u_{n-m} \}) given by T(w_i)=0 and T(v_i)=u_i. We then have \ker (T)=W and so picking a suitable matrix to represent this we get our equations.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. homogeneous equations...
    Posted in the Differential Equations Forum
    Replies: 5
    Last Post: February 12th 2011, 03:06 PM
  2. homogeneous equations
    Posted in the Algebra Forum
    Replies: 0
    Last Post: March 28th 2010, 09:02 AM
  3. Homogeneous equations
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 11th 2010, 02:33 AM
  4. Homogeneous Differential Equations.
    Posted in the Differential Equations Forum
    Replies: 19
    Last Post: October 8th 2008, 02:31 PM
  5. Replies: 1
    Last Post: July 29th 2007, 03:37 PM

Search Tags


/mathhelpforum @mathhelpforum