Global Positioning System Reference
In-Depth Information
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
cos
θ
01 0
sin
θ
0
−
sin
R
2
(
θ
)
=
(A.33)
θ
0
cos
θ
θ
θ
cos
sin
0
R
3
(
θ
)
=
−
sin
θ
cos
θ
0
(A.34)
0
0
1
de
scribe rotations by the angle
of a right-handed coordinate system around the first,
se
cond, and third axes, respectively. The rotation angle is positive for a counterclock-
wi
se rotation, as viewed from the positive end of the axis about which the rotation
tak
es place. The result of successive rotations depends on the specific sequence of
th
e individual rotations. An exception to this rule is differentially small rotations for
wh
ich the sequence of rotations does not matter.
θ
[34
Lin
—
6.6
——
Sho
PgE
A.
3 LINEAR ALGEBRA
Su
rveying computations rest heavily upon concepts from linear algebra. In general,
th
ere is a nonlinear mathematical relationship between the observations and other
qu
antities, such as coordinates, height, area, and volume. Seldom is there a natu-
ra
l linear relation between observations as there is in spirit leveling. Least-squares
ad
justment and statistical treatment require that nonlinear mathematical relations be
lin
earized; i.e., the nonlinear relationship is replaced by a linear relationship. Possible
er
rors caused by neglecting the nonlinear portion are eliminated through appropri-
ate
iteration. The result of linearization is a set of linear equations that is subject to
fu
rther analysis, thus the need to know the elements of linear algebra. The use of
lin
ear algebra in the derivation and analysis of surveying measurements fortunately
do
es not require the memorization of all possible proofs and theorems. Strang and
Bo
rre (1997) is a very useful reference that emphasizes linear algebra as it applies
to
geodesy and GPS. This appendix merely summarizes some elements from linear
alg
ebra for the sake of completeness.
[34
A.
3.1 Determinants and Matrix Inverse
Let the elements of a matrix
A
be denoted by
a
ij
, where the subscript
i
denotes the
row and
j
the column. A
u
×
u
square matrix
A
has a uniquely defined determinant,
denoted by
1 matrix equals
the matrix element. The determinant of
A
is expressed as a function of determinants
of submatrices of size
(u
|
A
|
, and said to be of order
u
. The determinant of a 1
×
2
)
, etc. until the size 2
or 1 is reached. The determinant is conveniently expressed in terms of minors and
cofactors.
−
1
)
×
(u
−
1
), (u
−
2
)
×
(u
−
