Let \(\mathcal{V}\subseteq\mathbb{R}^n\) be a vector space and \(\mathcal{V}^\perp \perp \mathcal{V}\text{.}\) Then the normal equation of the linear system \(\vec{A}\vec{x}=\vec{b}\) is defined as:
\begin{equation*}
A^T A \vec{x} = A^T \vec{b}
\end{equation*}
where we assume \(A\in\mathbb{R}^{n\times k}\) and \(\vec{b}\in\mathbb{R}^n\text{.}\)
