LINEAR ALGEBRA: Linear Mapping

[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.

 

[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.)

 

[piano.lisa]

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

 

[matt grime]

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