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LINEAR ALGEBRA: Linear Mapping

  1. Oct 9, 2006 #1
    I have the linear mapping Pw(x).
    How can I prove that:
    [itex]||P_w(x)||^2 = \Sigma (<x, x_i>)^2[/itex]
    Where the sum is from i = 1 to k

    x is any vector which is an element of [itex]R^n[/itex]

    I have tried expanding [itex]||P_w(x)||^2[/itex] but it doesn't seem to give me the right side of the equation.
    Is there any other method you can suggest?
    Thank you.
     
  2. jcsd
  3. Oct 9, 2006 #2

    matt grime

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    You can start by defining what P_w is. And the x_i. Since the right hand side is independent of w and the left hand side is not, then it cannot be correct as written. (The dual observation also holds: the LHS is independent of the x_i and the RHS is not.)
     
  4. Oct 9, 2006 #3
    S = (x1,....,xk) is an orthonormal set in Rn.
    W = span{S}
     
  5. Oct 9, 2006 #4

    matt grime

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    Well, what is P_W(x) written with respect to the basis x_1,..,x_n?
     
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