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Projecting vectors from R3 onto a subspace

  1. Nov 26, 2005 #1
    I want to project a vector from R3 onto a subspace.
    I'll let the bases for the subspace be [a,b,c]T
    (my T's mean transpose)
    ---
    I have the defintion for vector projection
    p = (<u,v>/<v,v>)*v
    ---
    I know v will be the [a,b,c]T vector but what is u?
    The only thing I could think of is let it be the triplet [x,y,z]T which could be any vector in R3.
    ---
    Using this I get
    p = [(a/(a^2 + b^2 + c^2))(ax + by + cz)]
    [(b/(a^2 + b^2 + c^2))(ax + by + cz)]
    [(c/(a^2 + b^2 + c^2))(ax + by + cz)]
    ---
    I'm not very confident with this solution, I was hoping someone could tell me if this is correct or show me where I've gone wrong.
     
  2. jcsd
  3. Nov 26, 2005 #2

    matt grime

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    Science Advisor
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    u is presumably the vector you want to project onto the subspace.
     
  4. Nov 26, 2005 #3
    I was just working an a problem that askes me to find the projection matrix P that projects vectors in R3 onto the orthoginal compliment of a two dimensional subsapce of R3 spanned by
    x1 = [1,0,2]T x2 = [0,1,-2]
    ---
    I've found that the bases of the orthoginal compliment is [-2,2,1]T
    ---
    Using the defintion of P that a posted above I've found
    P = [(-2/9)(-2x +2y +z)]
    [ (2/9)(-2x +2y +z) ]
    [ (1/9)(-2x +2y +z) ]

    where, like you said, u = (x,y,z) = the vector I want to project onto the subspace. Does this seem like a correct solution?
     
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