Page 693 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

the equations of which are called the normal equations.

In the exercises it will be shown that the column vectors of M are linearly independent if and only if the n data points do not
lie on a vertical line in the -plane. In this case it follows from Theorem 6.4.4 that the least squares solution is unique and is
given by

In summary, we have the following theorem.
THEOREM 9.3.1

Least Squares Solution be a set of two or more data points, not all lying on a vertical line, and let
Let , ,…,

Then there is a unique least squares straight line fit
to the data points. Moreover,

is given by the formula

which expresses the fact that is the unique solution of the normal equations (6)
(7)

EXAMPLE 1 Least Squares Line: Using Formula 6
Find the least squares straight line fit to the four points (0, 1), (1, 3), (2, 4), and (3, 4). (See Figure 9.3.3.)