Homogeneous System of Linear Equations

In the homogeneous system of linear equations, the constant term in every equation is equal to 0. i.e., no equation in such systems has a constant term in it. A homogeneous linear system may have one or infinitely many solutions. But it has at least one solution always.

Let us learn how to find solve the homogeneous system of linear equations and let us see what is meant by trivial and nontrivial solutions.

A homogeneous system of linear equations is a linear system of equations in which there are no constant terms. i.e., a homogeneous linear system is of the form:

a₁₁ x₁ + a₁₂ x₂ + ... + a₁ₙ xₙ = 0

a₂₁ x₁ + a₂₂ x₂ + ... + a₂ₙ xₙ = 0

....

aₘ₁ x₁ + aₘ₂ x₂ + ... + aₘₙ xₙ = 0

This is a system in 'n' unknowns (x₁, x₂, ..., xₙ), and in each equation, the constant term is 0. When we solve these systems using matrices (by writing augmented matrix), there is no change in the last column (that is made up of zeros) though when row operations are applied. Thus, when solving a homogeneous system of linear equations, we often ignore the column of zeros in the augmented matrix and we only write the coefficient matrix. Here are some examples:

• 2x - 3y = 0
  x - y = 0
  is a homogeneous system in two variables.
• x + y + z = 0
  y - z = 0
  x + 2y = 0
  is a homogeneous system in three variables

A homogeneous system may have two types of solutions: trivial solutions and nontrivial solutions. Since there is no constant term present in the homogeneous systems, (x₁, x₂, ..., xₙ) = (0, 0, ..., 0) is obviously a solution to the system and is called the trivial solution (the most obvious solution). For example, the system formed by three equations x + y + z = 0, y - z = 0, and x + 2y = 0 has the trivial solution (x, y, z) = (0, 0, 0). But it may (or may not) have other solutions than the trivial solutions that are called nontrivial solutions. We can find them using the matrix method and applying row operations.

The above system can be written as:

x + y + z = 0
0x + y - z = 0
x + 2y + 0z = 0

Let us take the coefficient matrix of the above system and apply row operations in order to convert it into an upper diagonal matrix.

\(\begin{equation}
\left[\begin{array}{ccc}
1 & 1 & 1 \\
0 & 1 & -1 \\
1 & 2 & 0
\end{array}\right]
\end{equation}\)

Now, we apply R₃ → R₃ - R₁:

\(\begin{equation}
\left[\begin{array}{ccc}
1 & 1 & 1 \\
0 & 1 & -1 \\
0 & 1 & -1
\end{array}\right]
\end{equation}\)

Now, we apply R₃ → R₃ - R₂:

\(\begin{equation}
\left[\begin{array}{ccc}
1 & 1 & 1 \\
0 & 1 & -1 \\
0 & 0 & 0
\end{array}\right]
\end{equation}\)

We couldn't convert it into the upper diagonal matrix as we ended up with a row of zeros in the matrix. It means that the system has nontrivial solutions also. Let us see how to find them. Just expand the first two rows of the above matrix as equations.

x + y + z = 0 ... (1)

y - z = 0 ... (2)

We have two equations in three variables. Hence we should assume one of the variables to be a parameter (say t which is a real number). Let z = t. Substituting this in (2):

y - t = 0

y = t

Substituting in (1):

x + t + t = 0
x + 2t = 0
x = -2t

Thus, the solution is (x, y, z) = (-2t, t, t) which represents an infinite number of nontrivial solutions as 't' can be replaced with one of the real numbers (which is an infinite set). For example, some nontrivial solutions of the above homogeneous system can be:

When t = 1: (x, y, z) = (-2, 1, 1)
When t = 2: (x, y, z) = (-4, 2, 2)
When t = 0.5: (x, y, z) = (-1, 0.5, 0.5), etc

We can find whether a homogeneous linear system has a unique solution (trivial) or an infinite number of solutions (nontrivial) by using the determinant of the coefficient matrix. If A is the coefficient matrix of the system then:

• The system has a unique solution (trivial) of det (A) ≠ 0
• The system has an infinite number of solutions (nontrivial) of det (A) = 0

For example, let us find the determinant of the coefficient matrix of the same system that we have solved in the last section.

\(\begin{equation}
\left|\begin{array}{ccc}
1 & 1 & 1 \\
0 & 1 & -1 \\
1 & 2 & 0
\end{array}\right|
\end{equation}\)
= 1 (0 + 2) - 1 (0 + 1) + 1 (0 - 1)
= 2 - 1 - 1
= 0

Since the determinant is 0, the system had an infinite number of solutions (which we found already in the last section).

Here are the properties of a homogeneous system of linear equations:

• It always has at least one solution that is called a trivial solution where the value of each variable is 0.
• If a and b are two solutions of a homogeneous system, then their sum a + b is also a solution.
• If a is a solution, then ka is also a solution, where k is a scalar.
• A zero vector is always a solution of the homogeneous system.

☛ Related Topics:

• Solutions of Linear Equation
• Solving Linear Equations
• Applications of Linear Equations
• Multi Step Equations

What is a Homogeneous Linear Equation Example?

A homogeneous linear equation is a linear equation in which the constant term is 0. Examples: 3x - 2y + z = 0, x - y = 0, 3x + 2y - z + w = 0, etc.

What are Trivial and Nontrivial Solutions of a Homogeneous System of Linear Equations?

A vector formed by all zeros (zero vector) is always a solution of any homogeneous linear system and it is called a trivial solution. Any other solution than the trivial solution (if any) is called a nontrivial solution.

How do You Know if a System of Equations is Homogeneous?

A system has two or more equations in it. If each equation in it has its constant term to be zero, then the system is said to be homogeneous.

What is the Solution of Homogeneous System of Linear Equations?

A zero vector is always a solution to any homogeneous system of linear equations. For example, (x, y) = (0, 0) is a solution of the homogeneous system x + y = 0, 2x - y = 0. Sometimes, a homogeneous system has non-zero vectors also to be solutions, To find them, we have to use the matrices and the elementary row operations.

How Many Solutions Does a Homogeneous System of Linear Equations Has?

If A is the coefficient matrix formed by the given homogeneous system of linear equations then it has:

• only one solution (which is trivial (or) formed by zeros) if det (A) ≠ 0
• infinite number of solutions (which are nontrivial) if det (A) = 0