Results 1 to 2 of 2

Math Help - Orthogonal Prjection Question

  1. #1
    Member
    Joined
    Jan 2008
    Posts
    78

    Orthogonal Prjection Question

    Find the matrix for the orthogonal projection onto the subspace of R^3 defined
    by 3x_1 -x_2  + 3x_3
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by flaming View Post
    Find the matrix for the orthogonal projection onto the subspace of R^3 defined
    by 3x_1 -x_2  + 3x_3 \color{red}{} = 0.
    The projection P onto the 1-dimensional subspace spanned by the vector \mathbf{v} = (3,-1,3) is given by P\mathbf{x} = \langle\mathbf{x},\mathbf{v}\rangle/\|\mathbf{v}\|^2 = \tfrac1{19}(3x_1-x_2+x_3)\mathbf{v}, where x is the vector (x_1,x_2,x_3). In matrix notation, this looks like P\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix} = \frac1{19}\begin{bmatrix}9x_1-3x_2+9x_3 \\-3x_1+x_2-3x_3 \\9x_1-3x_2+9x_3 \end{bmatrix} = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmat  rix}. So the matrix of P is M_P = \frac1{19}\begin{bmatrix}9&-3&9 \\-3&1&-3 \\9&-3&9 \end{bmatrix}.

    The 2-dimensional subspace given by 3x_1 -x_2  + 3x_3 = 0 is the orthogonal complement of the above 1-dimensional subspace. So the projection onto it will have matrix I-M_P.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: August 15th 2011, 04:32 AM
  2. orthogonal matrices question
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: June 23rd 2011, 12:28 PM
  3. Orthogonal Complement Question
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: June 7th 2010, 03:36 AM
  4. Question about orthogonal trajectories
    Posted in the Calculus Forum
    Replies: 2
    Last Post: July 16th 2009, 09:58 AM
  5. orthogonal curves question
    Posted in the Calculus Forum
    Replies: 2
    Last Post: June 11th 2008, 09:00 AM

Search Tags


/mathhelpforum @mathhelpforum