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August 11th, 2012, 03:52 PM   #1
Joined: Jul 2012

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Best least-squares approximation

Given the subspace
=\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix}, x_1-2x_2+x_3=0\}" />
and the vector

Find a basis for the indicated subspace W. For the given vector v, solve the normal equations and determine the best approximation w*. Verify that v-w* is orthogonal to the basis vectors.

I know how to find the basis and check orthogonality, and I know how to solve for w* using an orthogonal basis and how to get an orthogonal basis, but I don't understand how the normal equation is used to solve for w*.

According to my textbook the normal equation is .
If I understand it correctly, Ax=w* and b = v so it should be

I get the feeling I'm missing something obvious and I'm using the wrong equation because can't be constructed without more information can it?
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August 12th, 2012, 02:26 PM   #2
Joined: Jul 2012

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Re: Best least-squares approximation

I figured out what I was supposed to do. It took me forever to think to use the subspace as a constraint for the system of equations to solve for w*.
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