Page 263 - A Course in Linear Algebra with Applications

7.4: The Method of Least Squares 247

Here
1
A = and B =

so A has has rank 3. Since the augmented matrix has rank
4, the linear system is inconsistent. We know from 7.4.3 that
there is a unique least squares solution in this case. To find
it, first compute

3 0 1\ x / 11 1—3
T
A 1 A = 0 3 1 I and (A A)~ l = — | 1 1 1 - 3
1 1 4 - 3 - 3 9

Hence the least squares solution is

1
T i
A\-\1
L
X = {A Ay A B AT; =
that is, xi = 8/5, x 2 = 3/5, £3 = 6/5.

Example 7.4.2
A certain experiment yields the following data:

X - l 0 l 2
y 0 l 3 9

It is suspected that y is a quadratic function of x. Use the
Method of Least Squares to find the quadratic function that
best fits the data.
2
Suppose that the function is y = a + bx + ex . We need
to find a least squares solution of the linear system

a ~ b + c = 0
a = 1
a + b + c = 3
a + 26 + Ac = 9