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Projection matrix onto a subspace

  1. Oct 22, 2008 #1
    Alright so I am trying to find the projection matrix for the subspace spanned by the vectors

    [1] and [2]
    [-1] [0]
    [1] [1]

    I actually have the solution to the problem, it is ...

    P = [ 5 1 2 ]
    (1/6) [1 5 -2]
    [2 -2 2]

    Every formula I have found is not for two vectors but a vector and a matrix or something else. I found the formula P = A(ATA)-1AT but that makes no sense for what A could be since I'm given two vectors and not a matrix. I feel like I'm missing something simple.

    PLEASE help!!!!

  2. jcsd
  3. Oct 22, 2008 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    The vectors <1, -1, 1> and <2, 0, 1> are independent so there exist a third vector, independent of both and all 3 form a basis for R3. It should not be hard for you to find such a third vector. Then you want a matrix that maps <1, -1, 1> and <2, 0, 1> to themselves and the third vector to <0, 0, 0>. that gives you 9 equations to solve for the 9 numbers in the matrix. Actually they separate into 3 sets of 3 equations.
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