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Question about projections and subspaces

  1. Aug 19, 2011 #1
    If we want to caculate the projection of a single vector, v=(1,2) (which is an element of an R2 vector space called V) onto the subspace of V (which we call W), do we use

    projection of v onto W = <v,w1>w1 + <v,w2>w2 + ........ <v,wn>wn

    However, if the individual values of v are not known (that is v=(x,y) ), do we calculate the matrix of projection ?

    that is, we do A(A[itex]^{T}[/itex]A)[itex]^{-1}[/itex]A[itex]^{T}[/itex]

    If we have to determine the matrix of projection, is it because we don't know what x and y is, so the safe bet is to determine the matrix of projection ?
     
  2. jcsd
  3. Aug 19, 2011 #2

    HallsofIvy

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    I really have no idea exactly what are asking. If you have a subspace, W, of an inner product space, V, and [itex]\{w_1, w2, ..., v_n\}[/itex] is an orthonormal basis for W, then, yes, the projection of v, a vector in v, onto W is [itex]<v, w_1>w_1+ <v, w_2>w_2+ ...+ <v, w_n>wn[/itex]. If you v is a "general" vector, then, yes, there exist a matrix such that multiplying any vector v by that matrix giives the projection. That is obviously true because projection is a linear operator. But I have no idea what you mean by "[itex]A(A^TA)^{-1}A^T[/itex]" because I do not know what "A" is. Where did you get the matrix A?
     
  4. Aug 19, 2011 #3
    sorry A is the matrix that is formed from the basis vectors in subspace W

    where W = { basis 1, basis 2 }
     
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