1 Inhomogeneous linear systems.

Geometry at Lingotto.
LeLing3: Inhomogeneous linear systems.
Contents:
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• Inhomogeneous linear systems.
• Gauß-Jordan for inhomogeneous systems.
• The general solution.
• General solution as sum of the ‘homogeneous’ solution plus a particular solution.
• Geometric meaning of solutions.
Recommended exercises: GeoLing 3,5.
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1
Inhomogeneous linear systems.
A system of equations of the form:

a1 1 x 1 + a1 2 x 2 + · · · + a1 n x n = b 1




a2 1 x 1 + a2 2 x 2 + · · · + a2 n x n = b 2

a3 1 x 1 + a3 2 x 2 + · · · + a3 n x n = b 3
S=


.
.
.
................................



am 1 x 1 + am 2 x 2 + · · · + am n x n = b m
with at least one bj different from zero is called inhomogeneous or non-homogeneous
system of linear equations in n unknonws and m equations.
 
0
 0 
 
 0 
 
The zero column 0 =  ..  is never a solution of an inhomogeneous system.
 . 
 
 0 
0
Example 1.1. Two examples:
A=
x−y =1
x+y =0
B=
3x1 − x2 + x4 = 3
x5 − x6 = −2
The system is not hard to remember if one uses the augmented matrix :
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Geometry
1.1 The notion of solution.







Geometry at Lingotto.
···
···
···
..
.
a1 n
a2 n
a3 n
..
.
am 1 am 2 · · ·
am n
a1 1
a2 1
a3 1
..
.
a1 2
a2 2
a3 2
..
.
The matrix A is the coefficient matrix:

a1 1 a1 2
 a2 1 a2 2

 a3 1 a3 2

 ..
..
 .
.
am 1 am 2
···
···
···
..
.
a1 n
a2 n
a3 n
..
.
···
am n
b1
b2
b3
..
.







bm











and the last column of the augmented matrix is denoted with B = 


b1
b2
b3
..
.




.


bm
The symbol (A|B) is used to indicate the augmented matrix of an inhomogeneous
system having A as coefficient matrix and B as the above mentioned column.
Example 1.2. The augmented matrices of the previous examples are:
1 −1 1
3 −1 0 1 0 0 3
1 1 0
0 0 0 0 1 −1 −2
Note the relation between the unknown x1 and the first column of A, x2 and the
second column of A, etcetera, and at last, the relation between the equations of S and
the rows of A.
Remark 1.3. Be extremely carefulwhen writing
the matrix associated to a system.
3x + y = 0
3 1 0
The matrix of
is not
.
3y + x = 2
3 1 2
1.1
The notion of solution.
The most striking difference between homogeneous and inhomogeneous systems concerns
the existence of a solution (the zero column 0 is not a solution in the inhomogeneous
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Geometry
1.2 Equivalent systems and EROs.
Geometry at Lingotto.
case, by definition!!). That is to say, inhomogeneous system might admit no solution at
all, like in this example:
I=
x=0
.
x=1
It is a system
on 1 equation in 1 unknown that clearly has no solution. Its matrix
1 0
is:
1 1
Definition 1.4. A linear system is inconsistent if it has no solution. Otherwise the
system is called consistent.
Later we will explain how to decide whether a system is consistent or not.
As for the homogeneous systems, a solution to a system S is a column R = (ri )
whose elements satisfy each equation of the system S when substituted to the unknowns
xi . Solutions are thus written as columns.
1.2
Equivalent systems and EROs.
As for homogeneous systems, S and S 0 are said equivalent if they admit the same solutions.
The elementary row operations ERO1, ERO2, ERO3 can be used to generate
equivalent systems starting from a given system. Once again one works on the system’s
matrix.
The idea is to simplify the system using EROs.
1.3
Gauß-Jordan elimination.
The Gauß-Jordan method can be adapted to solve inhomogeneous systems as well. We
simply act on the augmented matrix as if dealing with a homogeneous system, using the
Gaußstep only on the coefficient matrix A.
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Geometry
1.3 Gauß-Jordan elimination.
Geometry at Lingotto.
1 −1 1
Example 1.5. We show how to solve the system with matrix
. We find
1 1 0
R2 /2
1 −1 1 R2 −R1 1 −1 1
1 −1 1
=⇒
=⇒
1 1 0
0 2 −1
0 1 −1/2
Here ends the Gauß step and begins the Jordan step. Then
1 −1 1
1 0 1/2
R1 +R2
=⇒
0 1 −1/2
0 1 −1/2
At the end x = 1/2 and y = −1/2, so the solution is the column
1/2
−1/2
.
The vertical line separating the matrix coefficient from the column B tells where to
end the Gauß step.
It may happen to have to solve many systems simultaneously. If so, we may just
juxtapose the columns B of the various systems after th vertical line and proceed as
before. For instance:
Example 1.6. The systems
3x1 − x2 + x4 + 3x5 = 1
x5 − x6 = −1
3x1 − x2 + x4 + 3x5 = 4
x5 − x6 = 7
have a common coefficient matrix and can therefore be solved at he same time by
considering:
3 −1 0 1 3 0 1 4
0 0 0 0 1 −1 −1 7
1 −1/3 0 1/3 1 0 1/3 4/3
With R1 /3 we get
0
0
0 0 1 −1 −1 7
and ends Gauß. By R1 − R2 we find
1 −1/3 0 1/3 0 1 4/3 −17/3
0
0
0 0 1 −1 −1
7
and finish the Jordan step. The equivalent systems obtained are:
x1 − 1/3x2 + 1/3x4 + x6 = 4/3
x5 − x6 = −1
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x1 − 1/3x2 + 1/3x4 + x6 = −17/3
x5 − x 6 = 7
Geometry
1.4 General solution
Geometry at Lingotto.
The solutions
come from assigningto x2 , x3 , x4 , x6 arbitrary values
finding x1 , x5
 x2 and
x4
x4
x2
4
−
−
x
+
−
− x6 − 17
6
3
3
3
3
3
3




x
x
2
2








x3
x3




from the system 
for
the
first
system
and



x
x
4
4








x6 − 1
x6 + 7
x6
x6
for the second.
1.4
General solution

The solutions of
x2
3



3x1 − x2 + x4 + 3x5 = 1
are 

x5 − x6 = −1


is a sum of two








x2
3
−
x4
3
x2
x3
x4
x6
x6
− x6


 
 
 
+
 
 
 
4
3
0
0
0
−1
0
−
x4
3
− x6 +
x2
x3
x4
x6 − 1
x6

4
3




 . This column










where the former involves the parameters, while the latter does not depend on the
unknowns and is called particular solution.
To a non-homogeneous system S with matrix (A|B) one can associate a homogeneous system: just take A as matrix of a homogeneous system.
Theorem 1.7. Let S be an inhomogeneous system with matrix (A|B). Call X0 an
arbitrary solution of S . Then every solution of S can be expressed as:
X0 + Y
where Y is the solution of the associated homogeneous system.
So the knowledge of an inhomogeneous solution allows to find all solutions by solving
the associated homogeneous system.
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Geometry
1.4 General solution
Geometry at Lingotto.


1
Example 1.8. The column  2  solves the system
3
x+y+z =6
S=
2x − y − 3z = −9

  
1
x



2
y  with
The solutions of S can be written as:
+
3
z
x+y+z =0
2x − y − 3z = 0
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Geometry
1.5 Geometric interpretation.
1.5
Geometry at Lingotto.
Geometric interpretation.
A linear equation ax = b represents the point x =
b
a
of the real line, provided a 6= 0.
Example 1.9. The system 3x = 3 has x = 1 as solution, a point of the real line.
Similarly, a linear equation ax + by = c in two unknowns
x, y represents a straight line in the plane, if a 6= 0 or b 6= 0.
Example 1.10. The system 3x + 2y = 3 has as solution
a straight line in the plane.
Any equation of a system with two variables can be understood geometrically as a line in the plane. If these lines
intersect at a common point the system is consistent, otherwise it will be inconsistent.
y−x=1
x
1
The solution of
is
=
,
y + 2x = 4
y
2
hence the intersection point of the two lines.
Take the inconsistent system
y−x=1
. The lines
y − x = −1
are parallel, hence do not meet.
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Geometry
1.5 Geometric interpretation.
Geometry at Lingotto.
The equation x+y +3z = 0, with three variables, is a plane
in space.
The inconsistent system
z=1
has no solution:
z = −1
At last, here is the line solution of a consisten system, seen
as intersection of two planes in space:
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Geometry