Decide whether the linear system of equations \(A \vec{x} = \vec{b}\) has a solution. Then find a least squares solution. \[A = \begin{bmatrix} -2 & -1 \\ -4 & 1 \\ -3 & 4 \end{bmatrix}, \qquad \vec{b} = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix}\]
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Decide whether the linear system of equations \(A \vec{x} = \vec{b}\) has a solution. Then find a least squares solution. \[A = \begin{bmatrix} -2 & -1 \\ -4 & 1 \\ -3 & 4 \end{bmatrix}, \qquad \vec{b} = \begin{bmatrix} 3 \\ 2 \\ 2 \end{bmatrix}\]
\begin{solution} Solution: The system \(A \vec{x} = \vec{b}\) does not have solutions. The normal equations are: \[A^T A = \begin{bmatrix} 29 & -14 \\ -14 & 18 \end{bmatrix}\] \[A^T \vec{b} = \begin{bmatrix} -20 \\ 7 \end{bmatrix}\] The reduced row echelon form (RREF) of \(A^TA \vec{x} = A^T\vec{b}\) is \[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}\] \end{solution}