2
f
8
(t)
=
cos (1.7  t)
we would like to make a curve fit of the data obtained as follows:
xt
c f t
kk
k
(29 = (29
=
∑
1
8
where c
k
, k = 1, .
..., 8, are the coefficients which must be determined.
If a merit
function is defined by
Ξ
2
1
8
1
21
2
=

(29
=
=
∑
∑
xc
f
t
ik
k
i
k
i
i
σ
where (t
i
, x
i
,
σ
i
), i = 1, .
.., 21, are the data set, then the coefficient c
k
, k = 1, .
.., 8,
are determined so as to minimize the merit function
Ξ
2
.
That is, the coefficients c
k
,
k = 1, .
.., 8, of the curve fit is the solution of the least squares problem .
If a matrix
A
= [a
i
j
] and a vector
b
= {b
i
} are defined by
a
ft
b
x
ij
ji
i
i
i
i
=
=
σσ
and
the merit function X
2
can be written as
Ξ
2
1
2
=

bA
c bA
c
T
.
(a)
Determine the coefficient matrix
A
and the vector
b
using the data given.
(b)
Obtain the range R(
A
) of the matrix
A
.
What is the dimension of R(
A
)?
(c)
Obtain the null space N(
A
).
What is the dimension of N(
A
(d)
Using the Householder transformation, modify the matrix
AA
T
to the form
of the matrix
T
*
such that its lower triangular portion becomes zero, that is,