LINEAR ALGEBRA: Linear Mapping

1. Oct 9, 2006

piano.lisa

I have the linear mapping Pw(x).
How can I prove that:
$||P_w(x)||^2 = \Sigma (<x, x_i>)^2$
Where the sum is from i = 1 to k

x is any vector which is an element of $R^n$

I have tried expanding $||P_w(x)||^2$ 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. Oct 9, 2006

matt grime

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.)

3. Oct 9, 2006

piano.lisa

S = (x1,....,xk) is an orthonormal set in Rn.
W = span{S}

4. Oct 9, 2006

matt grime

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