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Mathematical Expression Editor
Homogeneous Linear Systems
A system of linear equations is called homogeneous if the system can be written in
the form
A homogeneous linear system is always consistent because is a solution. This
solution is called the trivial solution. Geometrically, a homogeneous system can be
interpreted as a collection of lines or planes (or hyperplanes) passing through the
origin. Thus, they will always have the origin in common, but may have other points
in common as well.
If is the coefficient matrix for a homogeneous system, then the system can be
written as a matrix equation . The augmented matrix that represents the system
looks like this
As we perform elementary row operations, the entries to the right of the vertical bar
remain . It is customary to omit writing them down and apply elementary row
operations to the coefficient matrix only.
Solve the given homogeneous system and
interpret your solution geometrically.
We start by rewriting this system as a matrix equation
We will proceed to find the reduced row-echelon form of the matrix as usual, but will
omit writing the zeros to the right of the vertical bar.
and are leading variables, and is a free variable. We let and solve for and .
Each of the equations in the original system represents a plane through the origin in .
The system has infinitely many solutions. Geometrically, we can interpret these
solutions as points lying on the line shared by the three planes. The above solution is
a parametric representation of this line. Use the GeoGebra demo below to
take a better look at the system. (RIGHT-CLICK and DRAG to rotate the
image.)
General and Particular Solutions
Given any linear system , the system is called the associated homogeneous
system.
It turns out that there is a relationship between solutions of and solutions of the
associated homogeneous system.
Let
Consider the matrix equation . Row reduction produces the following.
We conclude that .
Let’s take a more careful look at .
We now see that the solution vector is made up of two distinct parts:
one specific vector
infinitely many scalar multiples of .
The vector is an example of a particular solution. This particular “particular
solution” corresponds to . We can find any number of particular solutions by letting
assume different values. For example, the particular solution that corresponds to is
. Let be any particular solution of . It turns out that all vectors of the
form
are solutions of . We can verify this as follows
This shows that the specific vector is not very special, as any solution of can be
used in its place.
The vector , however, is special. Note that
So and all of its scalar multiples are solutions to the associated homogeneous
system.
In Exploration init:generalplusparticular we found that the general solution of the equation has the
form:
It turns out that the general solution of any linear system can be written in this
format. Theorem th:homogeneous formalizes this result.
Suppose is a particular solution of .
(a)
If is a solution of the associated homogeneous system, then is a solution
of .
(b)
If is a solution of , then there exists a solution of the associated
homogeneous system, , such that .
Let
If possible, find a solution of and express it as a sum of a particular solution and the
general solution of the associated homogeneous system. ()
First, we obtain the
reduced row-echelon form
So
In this case
Practice Problems
Problems prob:hompluspart1-prob:hompluspart2 For each matrix and vector below, find a solution to and express your
solution as a sum of a particular solution and a general solution to the associated
homogeneous system.
Prove that a consistent system has infinitely many solutions if and only if the
associated homogeneous system has infinitely many solutions.
Problems prob:homexample1-prob:homexample2 If possible, construct an example of each of the following. If not possible,
explain why.
An inconsistent system with an associated homogeneous system that has infinitely
many solutions.
An inconsistent system with an associated homogeneous system that has a unique
(trivial) solution.
Prove that a linear combination of any number of solutions of a homogeneous
equation is a solution of the same equation.