題組內容

(e) Consider the following linear equations characterized by \(\mathbf{A} \in \mathbb{R}^{m \times n}\):
\[\mathbf{A}\mathbf{x} = \mathbf{b}.\]
Then \(\widehat{\mathbf{x}} = \mathbf{A}^{\dagger}\mathbf{b} + \mathbf{v}\) is a solution to the above linear equations, where \(\mathbf{A}^{\dagger}\) is the pseudo-inverse of \(\mathbf{A}\) and \(\mathbf{v} \in \mathcal{N}(\mathbf{A})\) (null space of \(\mathbf{A}\)). If your answer is "False", please describe the physical meaning of \(\mathbf{A}\widehat{\mathbf{x}}\).