Page 257 - A Course in Linear Algebra with Applications

7.4: The Method of Least Squares 241

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12. Let L be a linear operator on the Euclidean space R .
Prove that L is orthogonal if and only if L(X) — AX where
A is an orthogonal matrix.
13. Deduce from Exercise 12 and Example 7.3.7 that a lin-
ear operator on R 2 is orthogonal if and only if it is either a
rotation or a reflection.

7.4 The Method of Least Squares
A well known application of linear algebra is a method
of fitting a function to experimental data called the Method
of Least Squares. In order to illustrate the practical problem
involved, let us consider an experiment involving two measur-
able variables x and y where it is suspected that y is, approx-
imately at least, a linear function of x.
Assume that we have some supporting data in the form
of observed values of the variables and x and y, which can be
thought of as a set of points in the xy-plane

( a i , 6 i ) , . . . , ( a m , 6 m ) .

This means that when x = a*, it was observed that y = b{.
Now if there really were a linear relation, and if the data were
free from errors, all of these points would lie on a straight line,
whose equation could then be determined, and the linear rela-
tion would be known. But in practice it is highly unlikely that
this will be the case. What is needed is a way of finding the
straight line which "bests fits" the given data. The equation
of this best-fitting line will furnish a linear relation which is
an approximation to y.