(Solved): A singular value decomposition of \( A \) is as follows: \[ A=U \Sigma V^{T}=\left[\begin{array}{c ...

A singular value decomposition of \( A \) is as follows: \[ A=U \Sigma V^{T}=\left[\begin{array}{cccc} -0.5 & 0.5 & -0.5 & 0.5 \\ 0.5 & 0.5 & -0.5 & -0.5 \\ 0.5 & 0.5 & 0.5 & 0.5 \\ -0.5 & 0.5 & 0.5 & -0.5 \end{array}\right]\left[\begin{array}{cc} 10 & 0 \\ 0 & 10 \\ 0 & 0 \\ 0 & 0 \end{array}\right]\left[\begin{array}{cc} 0.6 & 0.8 \\ 0.8 & -0.6 \end{array}\right] \] Find the least-squares solution of the linear system \( A \mathbf{x}=\mathbf{b} \), where \( \mathbf{b}=\left[\begin{array}{c}3 \\ -1 \\ -1 \\ 5\end{array}\right] \) \[ \begin{array}{l} \hat{x}_{1}=-\frac{29}{25}, \\ \hat{x}_{2}=-\frac{22}{25} \end{array} \]