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 August 11th, 2012, 03:52 PM #1 Newbie   Joined: Jul 2012 Posts: 6 Thanks: 0 Best least-squares approximation Given the subspace $W= \{x=\begin{bmatrix} x_1\\x_2\\x_3\end{bmatrix}, x_1-2x_2+x_3=0\}" /> and the vector $v=\begin{bmatrix} 1\\2\\6\end{bmatrix}$ 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 $A^TAx=A^Tb$. If I understand it correctly, Ax=w* and b = v so it should be $A^Tw*=A^T\begin{bmatrix} 1\\2\\6\end{bmatrix}$ I get the feeling I'm missing something obvious and I'm using the wrong equation because $A^T$ can't be constructed without more information can it?
 August 12th, 2012, 02:26 PM #2 Newbie   Joined: Jul 2012 Posts: 6 Thanks: 0 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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