Graphics Programs Reference
In-Depth Information
A
11
x
1
+
A
12
x
2
+···+
A
1
n
x
n
=
b
1
A
21
x
1
+
A
22
x
2
+···+
A
2
n
x
n
=
b
2
A
31
x
1
+
A
32
x
2
+···+
A
3
n
x
n
=
b
3
(2.1)
.
A
n
1
x
1
+
A
n
2
x
2
+···+
A
nn
x
n
=
b
n
where the coefficients
A
i j
and the constants
b
j
areknown, and
x
i
represent the un-
knowns. Inmatrix notation the equations are writtenas
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
A
11
A
12
···
A
1
n
x
1
x
2
.
x
n
b
1
b
2
.
b
n
A
21
A
22
···
A
2
n
=
(2.2)
.
.
.
.
.
.
A
n
1
A
n
2
···
A
nn
or,simply
Ax
=
b
(2.3)
A particularlyuseful representation of the equationsfor computational purposes
is the
augmented coefficient matrix
,obtained by adjoining the constant vector
b
to
the coefficient matrix
A
in the following fashion:
⎡
⎣
⎤
⎦
A
11
A
12
···
A
1
n
b
1
A
b
A
21
A
22
···
A
2
n
b
2
=
(2.4)
.
.
.
.
.
.
.
···
A
n
1
A
n
2
A
nn
b
n
Uniqueness of Solution
A system of
n
linear equations in
n
unknownshas a unique solution, provided that
the determinant of the coefficient matrix is
nonsingular
, i.e., if
0. The rows and
columnsofanonsingular matrix are
linearly independent
in the sense that no row (or
column) is a linear combination of otherrows(or columns).
If the coefficient matrix is
singular
, the equations may have an infinite number of
solutions, orno solutions at all, depending on the constant vector. As an illustration,
take the equations
|
A
| =
2
x
+
y
=
3
4
x
+
2
y
=
6
Since the second equation can beobtainedbymultiplying the first equationbytwo,
any combination of
x
and
y
thatsatisfies the first equationis also a solution of the
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