Page 264 - A Course in Linear Algebra with Applications

248 Chapter Seven: Orthogonality in Vector Spaces

Again the linear system is inconsistent. Here

/ I 4 X
1 \ 1
A = 0 0 and B =
1 3
1 V 1 1 W
2 4 /
and A has rank 3. We find that

T
A A =

and
12 -20
T
(AM) A\-l 36 -20
-20 20

The unique least squares solution is therefore

11
l T
T
X = (A A)- A B = — I 33
20
25
that is, a = 11/20, b = 33/20, c = 5/4. Hence the quadratic
function that best fits the data is
11 33 5 o
y
20 20 4

Least squares and QR-factorization
Consider once again the least squares problem for the
linear system AX = B where A is m x n with rank n; we
have seen that in this case there is a unique least squares so-
T 1 T
lution X — (A A)~ A B. This expression assumes a simpler
form when A is replaced by its QR-factorization. Let this be