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Introduced least-squares problems, gave example of polynomial fitting, gave normal equations, and derived them from the condition that the L2 error be minimized.
Discussed solution of normal equations. Discussed condition number of solving normal equations directly, and noted that it squares the condition number—not a good idea if we can avoid it.
Introduced the alternative of QR factorization (finding an orthonormal basis for the column space of the matrix). Explained why, if we can do it accurately, this will give a good way to solve least-squares problems.
Gave the simple, but unstable, construction of the Gram-Schmidt algorithm, to find a QR factorization.