1
Canonical Forms of Linear Systems
Lecture IV
I.
Systems of Equations with the Same Solution
A. Consistent equations and linear combinations
12
31
32
2:
2
7:
xx
xE
+=
+=
1. The ordered set x
1
=3, x
2
=2, x
3
=1 is said to be a solution of E
1
because
the values substituted into E
1
yield 2=2.
Thus, (3,2,1) is said to satisfy
equation E
1
.
In general, for an arbitrary mxn (m equations and n
unknowns) the solution (x
1
1
,x
2
1
, …x
n
1
) is a solution of the i
th
equation
if
1
11
1
1
22
i
i
i
n
ni
a
x
a
x
a
xb
+
+=
L
2. There are three possibilities for a given system.
a. The system possesses a unique solution.
b. The system possesses an infinite number of solutions.
c. The system possesses no solution.
3. A system that possesses solutions whether unique or not is called
consistent or solvable while a system containing no solution is
inconsistent or unsolvable.
4. The aggregate of solutions of a system is called the solution set.
If the
system is inconsistent its solution set is said to be empty (or a null set).
a. Given a system it is easy to construct new equations such that
any solution of the original equations also solves the new
equation.
( 29
( 29
123
122
3
27
2224
63
3
21
45
5
17
xxx

+
=
Notice that (3,2,1) solves the original system
3212
2(3
)217
That point also solves the new system
4(3
)
5(2
)
5(1
)
1
2
1
05
+=
 +=
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Professor Charles B. Moss
2
5. A scheme for generating new equations is then
( 29
( 29
( 29
1
1
1
1
1
2
211
2
2
1
1
2
2
222
1
1
22
1
1
1
1
1
1
2
2
11
2
1
1
2
2
2
1
1
1
1
nn
m
m
m
m
n
nm
mm
nnn
m
mn
ka
x a
x
a
xb
k a
x
a
x
a
d
x
d
x
d
xd
d
k
a
k
a
d
k
a
k
a
d
k
a
k
a
d
k
b
k
b
kb
+
+=
+
+
+
=++
L
L
M
L
L
L
L
M
L
L
a. An equation formed in this manner is called a linear
combination of the original equations.
The numbers
i
k
are
called multipliers or weights of the linear combination.
b. Writing these in detached form
[ ]
1
1
1
2
1
2
1
2
2
2
12
n
n
m
m
m
n
n
a
a
a
bk
a
a
a
a
a
a
d
ddd
L
L
M MO
M MM
L
L
6. Definition: If in a system of equations an equation is a linear
combination of the other equations, it is said to be dependent upon
them; the dependent equation is called redundant.
A vacuous
equation, i.e. an equation of the form
0
0
00
n
x
xx
+
L
is also called redundant when it occurs in a single equation system.
A
system containing no redundancy is called independent
( 29
( 29
123
1
23
1
5 304
3
5
1
5
20
3
6
1
6
xxx
x xx
x
++=
++=
+
=
+
Therefore, in the system of equations
1
1
34
3
6
1
6
x
+
+
the last equation is redundant
B. How systems are solved
1. The usual “elimination” procedure for finding a solution of a system of
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 Fall '08
 Moss
 Equations, Elementary algebra, Charles B. Moss, Professor Charles B., Forms Professor Charles

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