II Find a least squars solution of \( A x_{i}=b \) for:
\[
A=\left[\begin{array}{llll}
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 \\
1 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 \\
1 & 0 & 0 & 1
\end{array}\right] \quad b=\left[\begin{array}{c}
-3 \\
-1 \\
0 \\
2 \\
5 \\
1
\end{array}\right]
\]

Answer

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Answer

\(x_i = (A^T A)^{-1}(A^T b) = \left[\begin{array}{c} \frac{-3}{2} \\ \frac{5}{2} \\ 1 \\ \frac{7}{2} \end{array}\right]\)

Steps

Step 1 ：\(A^T A = \left[\begin{array}{cccc} 6 & 2 & 2 & 2 \\ 2 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 2 & 0 & 0 & 2 \end{array}\right]\)

Step 2 ：\(A^T b = \left[\begin{array}{c} 4 \\ -4 \\ 2 \\ 6 \end{array}\right]\)

Step 3 ：\(x_i = (A^T A)^{-1}(A^T b) = \left[\begin{array}{c} \frac{-3}{2} \\ \frac{5}{2} \\ 1 \\ \frac{7}{2} \end{array}\right]\)