Math 6610 • Analysis of Numerical Methods I

Least Squares Problems. Let \(A \in \mathbb{R}^{m \times n}\) with \(m > n\) have rank \(n\) and \(b \in \mathbb{R}^m\).
The Least Squares problem is to find \(x \in \mathbb{R}^n\) which makes \(\|Ax - b\|_2\) as small as
possible.
\begin{enumerate}

\item
Show that \(x\) solves this Least Squares problem if and only if \(A^T Ax = A^T b\).


\item
What is meant by the QR factorization of matrix \(A\)?


\item
Show how to use the QR factorization of \(A\) to solve the Least Squares problem.

\end{enumerate}