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Lecture IV:
The Canonical Form and Null Spaces
I.
To demonstrate the meaning of null space matrices, I first want to discuss the use of
canonical forms.
A canonical form is a solution to an underidentified system of
equations.
For example in a crop selection model we may have a land and a labor
constraint
xxxxx
xx
x
12345
12
34
5
100
2
3
4
250
+
+
+
+
=
++
=
A.
The matrix operations used to reduce this matrix to a row reduced form are
10
21
111111
0
0
231412
5
0
11
01
1 1
1
1
1
100
01 12 1 5
0
10 2
1 2 5
0
01 1 2
15
0
−
−
−−
−
1.
This representation is a canonical form representing a solution to the
system of equations.
Specifically,
x
1
=50,
x
2
=50,
x
3
=0,
x
4
=0,
x
5
=0
solves the system of equations.
2.
In addition, the solution says something about maintaining feasibility
in the nonbasic variables.
Taking the columns from the nonbasic, or
zerolevel, variables we can form a homogeneous set of equations
(
Ax
=0):
22
0
20
5
5
x
x
−
+
=
−+
−=
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 Fall '08
 Moss

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