**for**

**Background****
**We have seen how Gaussian elimination can
be used to obtain the reduced row echelon form of a matrix and the
solution of a linear system . Now
we consider the special case of solving a homogeneous linear

system where the column vector is .

(1)

which can be written in matrix form

.

At first glance it seems that the solution to (1) should be , and certainly this is the case. Since it was easy to come up with, , it is called the "trivial solution," and if , then it is the unique solution. What makes the solution to (1) interesting is the case when and if this happens there is an infinite number of solutions.

An important technique for solving a
homogeneous system of linear equations is
to form the augmented matrix and
reduce to
reduced row echelon form.

**Definition (Reduced Row Echelon
Form).** A matrix is said to be in row-reduced
echelon form provided that

**(i)** In
each row that does not consist of all zero elements, the first
non-zero element in this row is a 1. (called. a "leading
1).

**(ii)** In
each column that contains a leading 1 of some row, all other elements
of this column are zeros.

**(iii)** In
any two successive rows with non-zero elements, the leading 1 of the
the lower row occurs farther to the right than the leading 1 of the
higher row.

**(iv)** If
there are any rows contains only zero elements then they are grouped
together at the bottom.

**Definition
(Rank).** The number of nonzero rows in the
reduced row echelon form of a matrix is
called the rank of
and is denoted by .

**Lemma.** Consider
the m × n homogeneous
linear system , where
is the augmented matrix then .

**Theorem.** Consider
the n × n homogeneous
linear system .

**(i)** If then
the system has the unique solution .

**(ii)** If then
the system has an infinite number of solution.

When the matrix , is an n × n, square matrix, then the determinant can be used to express these ideas.

**Theorem.** Consider
the n × n homogeneous
linear system .

**(i)** If
then
the system has the unique solution .

**(ii)** If
then
the system has an infinite number of solution.

**Proof ****Homogeneous
Linear Systems** **Homogeneous
Linear Systems**

**Computer
Programs ****Homogeneous
Linear Systems** **Homogeneous
Linear Systems**

**Mathematica Subroutine (Complete
Gauss-Jordan Elimination).**

**Example 1.** Solve
the homogeneous linear system of equations

**Solution
1.**

**Example 2.** Solve
the homogeneous linear system of equations

**Solution
2.**

**Example 3.** Solve
the homogeneous linear system of equations

**Solution
3.**

**Free Variables**

When the linear system is underdetermined, we
needed to introduce free variables in the proper location. The
following subroutine will rearrange the equations and introduce free
variables in the location they are needed. Then all that
is needed to do is find the row reduced echelon form a second
time. This is done at the end of the next example.

**Mathematica Subroutine (Under Determined
Equations).**

**Example 4.** Solve
the linear system

**Solution
4.**

**Application to
Chemistry****
**Propane is used to reduce automobile
emissions. Consider the chemical equation which relates
how propane molecules () combine
with oxygen atoms ()
to form carbon dioxide ()
and water ():

When a chemist wants to "balance this equation," whole numbers must be found so that the number of atoms of carbon (

The next example shows how to solve this system.

**Example 5.** Balance
the propane-oxygen equation by solve the homogeneous linear
system

**Solution
5.**

**Research Experience for
Undergraduates**

**Homogeneous
Linear Systems** **Homogeneous
Linear Systems** Internet hyperlinks to web
sites and a bibliography of articles.

**Download this
Mathematica Notebook****
****Homogeneous
Linear Systems**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004