LECTURE 6:
HOMOGENEOUS SYSTEMS
Monday, September 12
In this lecture we’re going to consider a very special type of linear
system called a homogeneous system. Let’s start by defining what
such a system is.
Definition 0.1. A linear linear equation in n variables x1 , x2 , . . . , xn
is called homogeneous if it is of the form
a1 x1 ` a2 x2 ` ¨ ¨ ¨ ` an xn “ 0.
A linear system is called homogenous if all equations in the system are
homogenous equations.
From the above definition we see that a general homogenous linear
system with m equations and n variables will be of the form
a1,1 x1
a2,1 x1
..
.
` a1,2 x2
` a2,2 x2
..
.
` ¨¨¨
` ¨¨¨
..
.
am,1 x1 ` am,2 x2 ` ¨ ¨ ¨
` a1,n xn
` a2,n xn
..
.
“ 0
“ 0
(1)
` am,n xn “ 0
There are several reasons why homogeneous linear systems are
special, and one of these reasons has to do with the number of
solutions. Recall that a general linear system has either no solutions,
a unique solution, or infinitely many solutions. However, if the
system is homogeneous, there will always be at least one solution.
A homogenous will always have the trivial solution, which is
when all variables are equal to 0 (i.e. x1 “ x2 “ ¨ ¨ ¨ “ xn “ 0).
This means that a homogenous linear system will have either one
solution (the trivial solution) or infinitely many solutions (one of
which will be the trivial solution). A natural question to ask is how might we determine the number of solutions to a homogeneous
system? It turns out that the shape of the system (the number of
variables and equations) will go a long way towards answering this
question.
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1
more variables than equations
Since a homogenous linear system will always have the trivial solution, if we can find just one non-trivial solution to the system, it
would imply that the system must have infinitely many solutions.
So how can we determine if there is a non-trivial solution? Well,
suppose the homogeneous linear system has more variables than
equations, that is, suppose n ą m in (1). If we put such a system
into reduced echelon form, the resulting system will again be homogenous, so we know it has at least one solution. Furthermore,
the number of leading variables would be at most m, the number of
equations. This is because we can have at most one leading variable
per equation, and there are m equations. Now, since we are assuming that the number of variables n is greater than the number of
equations m, it must be the case that there is at least one free variable.
The fact that there is a solution and at least one free variable means
that the system must have infinitely many solutions. Let’s record
this.
Theorem 1.1. Every homogeneous linear system with more variables
than equations has infinitely many solutions.
Let’s take a look at a example.
Example 1. The following linear system has more variables than
equations:
2x1 ` 7x2 ` 4x3
“ 0
x1 ` 3x2 ` 2x3 ` x4 “ 0
2x1 ` 6x2 ` 5x3 ` 4x4 “ 0
If we transform this system into reduced echelon form, we know
that the maximum number of leading terms that it could have is
3, so there must be at least one free variable. Putting this linear
system in reduced echelon form, we get
x1
x2
x3
` 3x4 “ 0
´ 2x4 “ 0
` 2x4 “ 0.
From this we see that there are infinitely many solutions, all of
the form
x1 “ ´ 3x4
x2 “ 2x4
x3 “ ´ 2x4
x4 “ ”free”.
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Let’s consider what’s going on from a geometric perspective. Suppose we are considering a homogeneous linear system in three
variables x, y, and z. Then each equation in this system defines a
plane that passes through the origin p0, 0, 0q. For example, here is
the plane defined by x ` y ` z “ 0:
If we have fewer equations than variables, it means that we have at
most two planes. If there is one plane, then every point on it is a
solution to the system. If we have two planes, then every point in
their intersection is a solution to the linear system. Since we know
both planes pass through the origin, we know that they intersect (this
is why there is always a solution to a homogenous linear system)
and since the intersection of two planes is a line, we know that there
will be infinitely many solutions. For example, here are the planes
defined by x ` y ` z “ 0 and x ` 2y ´ z “ 0:
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Now, it is NOT necessarily true that a non-homogenous linear system
with more variables than equations has infinitely many solutions.
So what’s the difference? Well, for a general linear system, you are
not guaranteed that the system has a solution. Whereas, it is quite
possible that a non-homogenous linear system with more variables
than equations is inconsistent. Here’s a simple example:
Example 2. The following linear system has more variables than
equations, but is inconsistent.
x ` y ` z “ 0
x ` y ` z “ 1
Next, let’s consider the case in which a homogenous linear system
has the same number of variables and equations.
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same number of variables as equations
Suppose we have a homogeneous linear system with the same number of variables as equations, that is, suppose n “ m in (1). In this
case, rather than determine if the system will have infinitely many
solutions, we’ll determine when it will have just the trivial solution.
Suppose our linear system has just the trivial solution. This means
that when we put the system into reduced echelon form, it must
look like:
x1
“ 0
“ 0
..
.
x2
...
xn “ 0
Or equivalently, the augmented coefficient matrix of the reduced
linear system is the n ˆ n identity matrix:
»
fi
1 0 ¨¨¨ 0
—0 1 ¨ ¨ ¨ 0ffi
—
ffi
In “ — .. .. . . .. ffi .
–. .
. . fl
0 0 ¨¨¨
Let’s record this.
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Theorem 2.1. Let C be the coefficient matrix (not the augmented
coefficient matrix) of a homogenous linear system with n variables and
n equations. Then this system has only the trivial solution if and only
if C is row equivalent to In .
Note this theorem implies that if the coefficient matrix is not row
equivalent to the identity matrix, then it has infinitely many solutions.
Let’s look at a simple example to get a feel for this theorem.
Example 3. Consider the homogenous linear system:
x ` 2y “ 0
3x ` 7y “ 0
The coefficient matrix of this system is
„

1 2
.
3 7
Putting this matrix into reduced echelon form, we get
„

„

„

p´3qR1 `R2
p´2qR2 `R1
1 2
1 2
1 0
−−−−−−−Ñ
−−−−−−−Ñ
.
3 7
0 1
0 1
Since the coefficient matrix of the above system is row equivalent
to the identity matrix I2 , we know that the above system has only
the trivial solution.
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