Homogeneous Linear Equations

Linear algebra serves as the backbone for various mathematical concepts, from computer graphics to economic modeling. One fundamental aspect of linear algebra is solving systems of linear equations. Among these, homogeneous systems of linear equations hold particular significance due to their unique properties and applications across diverse fields. In this article, we will understand homogeneous systems, explore their characteristics, methods of solution, and real-world implications.

What is Homogeneous Linear Equation?

A system of linear equations is said to be homogeneous if all the constant terms on the right-hand side of the equations are zero. Mathematically, a homogeneous system of linear equations can be represented as:

\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = 0 \\ \vdots \\ a_{m1}x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = 0 \end{cases} 

where a_ij represents the coefficients of the variables x_i in the ith equation.

Key Characteristics

• Zero Solutions: Every homogeneous system has at least one solution, namely the trivial solution where all variables are zero.

• Homogeneity Preserved: If x is a solution to the system, then any scalar multiple of \(x\) is also a solution.

• Linear Independence: The solutions to a homogeneous system form a vector space, known as the null space or kernel of the associated matrix.

Example of Homogeneous System in two variable

4x - y = 0

3x - 2y = 0

Example of Homogeneous System in three variable

x + y - z = 0

x + y + z = 0

x - y + 2z = 0

Solving Homogeneous System of Linear Equations

A homogeneous system of linear equations may yield two types of solutions: trivial and nontrivial solutions. The trivial solution, (x₁, x₂, ..., xₙ) = (0, 0, ..., 0), is evident since there are no constant terms present in the system. However, there may exist nontrivial solutions beyond this obvious one. These solutions can be found using the matrix method and applying row operations.

Trivial Solution: When all variables in a system of equations are zero, it's termed a trivial solution. For example, in a system x + y = 0, the trivial solution is x = 0 and y = 0

Nontrivial Solution: Any solution to a system of equations where at least one variable is not zero. For instance, in the system x + y = 3, there's a nontrivial solution where x = 1 and y = 2.

Example: Consider the following homogeneous system:

1. x + y + z = 0

2. 0x + y - z = 0

3. x + 2y + 0z = 0

Matrix Representation:

The coefficient matrix of the system is:

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -1 \\ 1 & 2 & 0 \\ \end{bmatrix}

Row Operations:

1. Apply R₃ → R₃ - R₁:

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -1 \\ 0 & 1 & -1 \\ \end{bmatrix}

2. Apply R₃ → R₃ - R₂:

\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ \end{bmatrix}

Matrix couldn't be converted into upper triangular form, indicating nontrivial solutions.

Solution:

Expressing the first two rows as equations:

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

2. y - z = 0 ...(2)

Assuming one variable as a parameter (say t), let z = t. Substituting into (2):

y - t = 0

y = t

Substituting into (1):

x + t + t = 0

x + 2t = 0

x = -2t

Thus, the solution is (x, y, z) = (-2t, t, t), representing an infinite number of nontrivial solutions as t can be any real number.

Formula for Homogeneous System of Linear Equations

Determining whether a homogeneous linear system possesses a unique solution (trivial) or an infinite number of solutions (nontrivial) involves examining the determinant of the coefficient matrix. If A represents the coefficient matrix of the system, then:

The system has a unique solution (trivial) if \text{det}(A) \neq 0.

The system has an infinite number of solutions (nontrivial) if det(A)=0.

For example, let's calculate the determinant of the coefficient matrix of a system examined in the previous section:

\left| \begin{array}{ccc} 1 & 1 & 1 \\ 0 & 1 & -1 \\ 1 & 2 & 0 \end{array} \right|

= 1 · (0 + 2) - 1 · (0 + 1) + 1 · (0 - 1)

= 2 - 1 - 1

= 0

As the determinant equals 0, the system possesses an infinite number of solutions.

Properties of Homogeneous System

• Trivial Solution: Every homogeneous system has at least one solution, known as the trivial solution, where all variables equal 0.

• Closure under Addition: If a and b are two solutions of a homogeneous system, their sum a+b is also a solution.

• Scalar Multiplication: If a is a solution, then ??ka is also a solution, where k is any scalar.

• Zero Vector Solution: The zero vector is always a solution of any homogeneous system.

Real-world Applications of Homogeneous Linear Equations

Homogeneous systems of linear equations find applications in various fields, including:

• Physics: Modeling equilibrium conditions and linear transformations in physics phenomena.

• Engineering: Analyzing structural stability, electrical circuits, and control systems.

• Economics: Studying input-output models and equilibrium in economic systems.

• Computer Graphics: Determining transformations and transformations in 3D graphics rendering.

• Chemistry: Balancing chemical equations and studying reaction kinetics.

Homogeneous Linear System Solved Examples

Example 1:

2x + 3y - z = 0

4x - y + 2z = 0

-x + 5y - 3z = 0

Solution:

To solve this system, we can represent it in matrix form Ax = 0, where A is the coefficient matrix:

A = \begin{pmatrix} 2 & 3 & -1 \\ 4 & -1 & 2 \\    -1 & 5 & -3    \end{pmatrix}

To find the solutions, we need to find the null space (kernel) of matrix A. This can be done by row reducing A to its reduced row echelon form (RREF):

\text{RREF}(A) = \begin{pmatrix}    1 & 0 & 1 \\    0 & 1 & 1 \\    0 & 0 & 0    \end{pmatrix}

From the RREF, we can see that the system has infinitely many solutions. The general solution is given by:

x = -z

y = -z​

Where z can be any real number.

Example 2:

3x - 2y + z = 0

x + 4y - 2z = 0

2x - y + 5z = 0

Solution:

In matrix form:

   A = \begin{pmatrix}    3 & -2 & 1 \\    1 & 4 & -2 \\    2 & -1 & 5    \end{pmatrix}

After row reducing A to RREF, we get:

 \text{RREF}(A) = \begin{pmatrix}    1 & 0 & -\frac{7}{17} \\    0 & 1 & \frac{1}{17} \\    0 & 0 & 0    \end{pmatrix}

The system also has infinitely many solutions. The general solution is:

   x = 7/17 z

   y = 1/17 z

Where z can be any real number.

Homogeneous Linear System Practice Examples

1. Solve the following homogeneous linear system:

2x + 3y - z &= 0

4x - y + 2z &= 0

-x + 5y - 3z &= 0

2. Solve the following homogeneous linear system:

3x - 2y + z &= 0

x + 4y - 2z &= 0

2x - y + 5z &= 0

3. Solve the following homogeneous linear system with parameters:

2x + y + z &= 0

x - 2y + 3z &= 0

3x + 2y + 2z &= 0

Conclusion

Homogeneous systems of linear equations play a vital role in mathematical modeling and problem-solving across numerous disciplines. Understanding their properties, methods of solution, and real-world applications provides a solid foundation for tackling complex problems in science, engineering, economics, and beyond. As such, mastering the concepts surrounding homogeneous systems equips individuals with powerful analytical tools applicable in various domains of study and practice.

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Homogeneous Differential Equations