MATH 17100
28446
11/11/2009
Varieties of Systems of Linear Equations
If in a
rowreduced form
of an augmented matrix, we encounter a row where all the elements
except the last one is 0, i.e.
0
0
0
,
0
0
0
0
0
a
a
,
in terms of the equation that row represents,
1
2
0
0
0
n
x
x
x
a
,
we have a contradictory statement . We say that the system of equations that gave rise to this
rowreduced echelon form is
inconsistent
. In other words, an inconsistent system has no
solution.
On the other hand, a
consistent
system has a solution. It may even have more than one solution.
If in
,
Ax
b
b
0
, we say that the system is a
homogeneous system
.
In a homogeneous system,
Ax
0
, the situation that gave rise to inconsistency can never occur
since the last column of the augmented matrix of the homogeneous system has all zero elements.
Hence a homogeneous system is always consistent. We know that
x
0
will always be a solution
to the homogeneous equation, in fact this solution is called the
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 Spring '10
 KITCHENS
 Linear Equations, Equations, Systems Of Linear Equations, augmented matrix, free variables, coefficient matrix, rowreduced echelon form

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