# Find a least squares solution of Ax=b by constructing and solving the normal equations. A=[(3,1),(1,1),(1,4)], b[(1),(1),(1)] bar(x)=?

Find a least squares solution of Ax=b by constructing and solving the normal equations.
$A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$
$\stackrel{―}{x}=$?
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nitruraviX
Step 1
We have to find the least square solution of Ax = B, by constructing the normal equations, where
$A=\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right],b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$
The set of solutions of the non-empty solutions is given by
${A}^{T}Ax={A}^{T}$ b.To solve this normal equations, we first compute the relevant matrices.
${A}^{T}A=\left[\begin{array}{ccc}3& 1& 1\\ 1& 1& 4\end{array}\right]\left[\begin{array}{cc}3& 1\\ 1& 1\\ 1& 4\end{array}\right]=\left[\begin{array}{cc}11& 8\\ 8& 18\end{array}\right]$
${A}^{T}b=\left[\begin{array}{ccc}3& 1& 1\\ 1& 1& 4\end{array}\right]\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]=\left[\begin{array}{c}5\\ 6\end{array}\right]$
Step 2
Now, we need to solve $\left[\begin{array}{cc}11& 8\\ 8& 18\end{array}\right]x=\left[\begin{array}{c}5\\ 6\end{array}\right]$.
The augmented matrix is given by
$\left[\begin{array}{ccc}11& 8& 5\\ 8& 18& 6\end{array}\right]\to \left[\begin{array}{ccc}-3& 10& 1\\ 8& 18& 6\end{array}\right]$
$\to \left[\begin{array}{ccc}1& 10& -3\\ 6& 18& 8\end{array}\right]$
$\to \left[\begin{array}{ccc}1& 10& -3\\ 0& -42& -10\end{array}\right]$
$\to \left[\begin{array}{ccc}1& 10& -3\\ 0& 21& 5\end{array}\right]$
Step 3
From the final matrix, we get the following equations ${x}_{1}+10{x}_{2}=-3$
$21{x}_{2}=5$
$⇒{x}_{2}=\frac{5}{21},{x}_{1}=-\frac{113}{21}$
$⇒x=\left[\begin{array}{c}\frac{-113}{21}\\ \frac{5}{21}\end{array}\right]$
Jeffrey Jordon