A homogenous system of linear equations is an important concept in algebraic mathematics. It plays an important role in various fields such as mathematics, engineering, computer science, and physics.

In this article we will learn about what is homogenous systems of linear equations, the matrix representation of a homogenous system, how we can determine the solution of a homogenous system, and why a homogenous system is always consistent.

A Homogeneous System of Linear Equations is Always Consistent.

Yes, a homogeneous system of linear equations is always consistent.

What is a Homogeneous System of Linear Equations?

A homogeneous linear equation system is one in which all the constant terms are equal to zero. We can represent it in the form of:

a₁₁x₁ + a₁₂x₂ + …………… + a_1nx_n = 0

a₂₁x₁ + a₂₂x₂ + …………… + a_2nx_n = 0

a₃₁x₁ + a₃₂x₂ + …………… + a_3nx_n = 0

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a_m1x₁ + a_m2x₂ + …………… + a_mnx_n = 0

where a_ij are the coefficients of the variables x₁, x₂, x₃, …….. x_n

Matrix Representation of Homogenous System

A homogeneous system of linear equations can be represented in matrix form as:

Ax = 0

where,

• A is Coefficient Matrix
• x is Column vector of Variables
• 0 is Zero Vector

Determining Solution of a Homogenous System

• If det(A) is not equal to 0 then the homogenous system has only one solution (unique) and it said to be trivial solution.
• If det(A) is equal to 0 then the homogenous system has infinitely many solutions and it said to be both trivial and non-trivial solution.

​Solution of Homogeneous System

Solution to a homogeneous system always includes the trivial solution, which is where all variables are zero (x = 0). This makes the system consistent because there is always at least one solution.

• Trivial Solution: We know that by definition, the trivial solution (x = 0) always satisfies the homogeneous system. No matter what is the value of the coefficients, when we substitute x = 0 into the system result in zero on both sides of each equation, fulfilling the requirements of the system.

• Non-Trivial Solutions: In addition to the trivial solution, a homogeneous system may have infinitely many non-trivial solutions. This occurs when the system has fewer equations than unknowns, which leads to free variables. These free variables can take on any value, which results in an infinite set of solutions.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations which have same set of variables. For example, a system of two linear equations in two variables can be represented as:

• a₁x + b₁y = c₁
• a₂x + b₂y = c₂

Solution to a System of Linear Equations

A solution to a system of linear equations is an ordered pair that satisfies all equations in the system simultaneously. Depending on the system, there can be three conditions:

• No Solution (inconsistent system or parallel lines)
• Exactly One Solution (consistent and independent system or intersecting lines)
• Infinitely Many Solutions (consistent and dependent system or coincident lines)

Practice Questions on Homogenous System of Linear Equation

Q1. Determine the solutions for the homogeneous system:

• 2x + 3y – z = 0
• 4x + 6y – 2z = 0
• x – y + 3z = 0

Q2. Find the solutions to the homogeneous system:

• x + y + z = 0
• 2x + 3y + z = 0

Q3. Solve the following system:

• 3x + y – 2z = 0
• 6x – 2y – 4z = 0

​Q4. Determine the solution set for:

• x – y + z = 0
• 2x + y – 3z = 0
• x + 2y + 2z = 0

Q5. Solve:

• x + y + 2z = 0
• 3x + 4y – z = 0

​Q6. Find the solution to the homogeneous system:

• 2x + 3y + 4z = 0
• x – y + z = 0

​Q7. Determine the solutions for the following system:

• x – 2y + z = 0
• 2x + y – 3z = 0

Q8. Solve:

• x + 2y + 3z = 0
• 4x + 5y + 6z = 0

Conclusion

In this article we have laernt why a homogeneous system of linear equations is always consistent because it inherently includes the trivial solution where all variables equal zero. This fundamental property of mathematics algebra is used in many areas of linear algebra and its applications, from theoretical constructs to practical problem-solving in science and engineering.

Read More:

• Homogeneous Linear Equations
• System of Linear Equations
• Linear Equations Formula

Frequently Asked Questions

What is a consistent system of linear equations?

Linear equations that has at least one solution are known as consistent systems of linear equation. There can be either a single unique solution or infinitely many solutions.

How can you determine if a system is consistent?

We can determine a given system is consistent by using various methods such as graphical methods (checking for intersections), algebraic methods (substitution and elimination), or matrix methods (Gaussian elimination).

What is the difference between independent and dependent systems?

An independent system has only one solution, while a dependent system has infinitely many solutions. Both dependent and independent system of equations are consistent systems, the only difference is number of solutions.

Can a system of linear equations have no solution?

Yes, a system with no solution is called inconsistent system. This occurs when the equations represent parallel lines that never intersect and does not have a single solution i.e. no solution.

What is Gaussian elimination?

Gaussian elimination is a method for solving systems of linear equations by transforming the system’s augmented matrix into an upper triangular form and then using back substitution to find the solutions.

What are the key characteristics of a consistent system?

Key characteristics of a consistent system include at least one solution and the possibility of the equations representing intersecting lines (independent or only one solution) or overlapping lines (dependent or infinitely many solutions).