Definition A.2.4.
The least squares solution to a system \(A\vec{x}=\vec{b}\) with no solution is:
\begin{equation*}
A^T\underbrace{
A\vec{x}^*
}_{
\mathord{\text{Proj}_{C(A)} \, \vec{b}}
}
=
A^T \vec{b}
\end{equation*}
Subsubsection Least Squares Solution
Remark A.2.5.
Importantly,
\begin{equation*}
A\vec{x}^*=\text{Proj}_{C(A)}\vec{b}
\end{equation*}
minimizes the \(L^2\) vector norm:
\begin{equation*}
\|A\vec{x}-\vec{b}\|^2\text{.}
\end{equation*}
Remark A.2.6.
By definition, \(A\vec{x}^*-\vec{b} \perp C(A)\text{.}\) Therefore,
\begin{equation*}
A\vec{x}^*-\vec{b} \in C(A) \implies A\vec{x}^*-\vec{b} \in N(A^T)\text{.}
\end{equation*}
