SOLUTION TO ASSIGNMENT 8  MATH 251, WINTER 2007
In this assignment
F
=
R
or
C
.
1. Consider results of an experiment given by a series of points:
(
x
1
,y
1
)
,
(
x
2
,y
2
)
,...,
(
x
n
,y
n
)
,
where
x
1
< x
2
<
···
< x
n
and the
x
i
,y
i
are real numbers.
We assume that the actual law governing this data is linear. Namely, that there is an equation of the
form
f
A,B
(
x
) =
Ax
+
B
that ﬁts the data up to experimental errors. Therefore, we look for such an
equation
Ax
+
B
that ﬁts the data best. Our measure for that is “the method of list squares”. Namely,
given a line
Ax
+
B
, let
d
i
=

y
i

(
Ax
i
+
B
)

(the distance between the theoretical
y
and the observed
y
).
Then we seek to minimize
d
2
1
+
d
2
2
+
···
+
d
2
n
.
Let
T
:
R
2
→
R
n
,
be the map
T
(
A,B
) = (
f
A,B
(
x
1
)
,...,f
A,B
(
x
n
))
.
Prove that
T
is a linear map:
We can represent
T
by a matrix
(
A,B
)
7→
x
1
1
x
2
1
.
.
.
.
.
.
x
n
1
±
A
B
¶
.
So clearly it is linear.
and that the problem we seek to solve is to minimize
k
T
(
A,B
)

(
y
1
,...,y
n
)
k
2
.
Let us calculate:
k
T
(
A,B
)

(
y
1
,...,y
n
)
k
2
=
k
(
Ax
1
+
B,.
..,Ax
n
+
B
)

(
y
1
,...,y
n
)
k
2
=
∑
n
i
=1
(
Ax
i
+
B

y
i
)
2
=
∑
n
i
=1
d
2
i
.
Let
W
be the subspace of
R
n
which is the image of
T
. Prove that
W
is two dimensional and that
{
s
1
,s
2
}
is a basis for
W
, where
s
1
= (1
,
1
,...,
1)
,s
2
= (
x
1
,x
2
,...,x
n
).
W
is spanned by
T
(0
,
1)
and
T
(1
,
0)
, which are just
s
1
and
s
2
.
W
is two dimensional, unless
x
1
=
x
2
=
···
=
x
n
, which is not the case.
Assume for simplicity that
∑
n
i
=1
x
i
= 0 (this can always be achieved by shifting the data). Find an
orthonormal basis for
W
and use it to ﬁnd the vector in
W
closest to (
y
1
,...,y
n
).
1