Homogeneous System of Linear Equations

A homogeneous system of linear equations is a set of linear equations where each equation is equal to zero on the RHS. For example, 3x - y = 0. If the right side of the equation has a non-zero number (x + 2y = 5) it would be a non-homogeneous system.
 

A homogeneous system of linear equations is a system where all equations are equal to zero on the right-hand side. It has special properties, which make it different from other types of systems.

• It always has at least one solution.
• A homogeneous system is always consistent, which means it always has at least one solution — the trivial solution (where all variables are zero).
• Homogeneous equations may have infinitely many solutions if certain conditions are met.
• The right-hand side is always zero in each equation of the system.
   

A homogeneous system of linear equations can be written in matrix form to make it simpler to solve the equation. In this form, the system is written as Ax=0, where all equations are equal to zero.
For example:
x - y = 0  (1)
2x + y = 0 (2)
Step 1:
We want to write this system as a matrix. Each equation gives the coefficients of variables x, and y.
Ax = 0

Let's write the coefficient matrix:
Coefficient matrix A:
Each variable has a coefficient in the matrix A
A = 21   1-1
Variable vector:
We have 2 variables = x and y

X = yx

Zero vector:

0 = 00

Matrix form of the system:

211-1  yx = 00

Step 2: 
Solve the system
From equation (1)

x - y = 0 
x = y
Substitute into equation (2)

2x + y = 0
2y + y = 0
3y = 0
y = 0

Now since x = y, we get 

x = 0

Let’s now confirm if det(A)  0. 
det(A) = 1·1 - (-1)·2 = 1 + 2 = 3  0. 
Since det(A)  0, we only have the trivial solution: x = 0, y=0

A homogeneous linear system has either a unique solution or an infinite number of solutions. To find out if it’s a trivial solution or infinitely many solutions, we take a look at the determinant of the coefficient matrix, which is denoted as A. 
The system has a unique solution if det(A) ≠ 0. 
The system has an infinite number of solutions if det(A) = 0. 

For example:
x + 2y + z = 0
2x + 3y + 4z = 0
x + y + z = 0

Step 1:
Coefficient Matrix: Write the coefficient matrix from the system.

1 2 1
2 3 4
1 1 1

A = 

Step 2: Calculate the determinant

Use the formula for a 3 × 3 determinant:
det(A) = 1 (3  1 - 4  1) - 2 (2  1 - 4  1) + 1 (2  1 - 3  1)
            = 1 (3 - 4) - 2 (2 - 4) + 1 (2 - 3)
            = 1 (-1) - 2 (-2) + 1 (-1) 
            = -1 + 4 -1 = 2

Step 3:
Interpret the result
So, det(A) = 2  0
This means the homogeneous system has only the trivial solution:
x = 0, y = 0, and z = 0.
 

A homogeneous system is a system of linear equations where all constant terms are equal to zero. It is usually written as Ax = 0, and is solved using methods like row reduction. Homogeneous systems always have a trivial solution. If the determinant ≠ 0, only the trivial solution exists. Here are the steps to solve a homogeneous system:

Make an augmented matrix using the coefficients from the system of linear equations.

Use row operations (like subtracting rows) to simplify the matrix.

Solve for variables, start from the bottom row, and go up.

Check if the system has only the trivial solution (where all variables are zero) or infinitely many solutions. 

For example:
x + y = 0
2x + 2y = 0

Step 1: Use 1st equation:
              x + y = 0
                    x = -y
Step 2: Substitute into 2nd equation:
              2(-y) + 2y = 0
Let y = t and x = -t

(x, y) = (-t, t)

In solving a system of linear equations, first find the values of the variables that make all the equations true at the same time. For example, solve the system:
2x + y = 8
3x -2y = -1

Step 1: Write it as augmented matrix

2   1    |  8
3  -2    | -1

Step 2: Replace the first element of the matrix with 1

To do that, we divide the first row by 2:

R1 = 12  R1     
1  0.5 |  4
3  -2   | -1

Step 3: Eliminate the 3 below the leading 1
Using row operation:

R2 = R2 - 3 × R1

R2 = 3,-2,-1 - 3  1, 0.5, 4 = 3-3,-2-1.5,-1-12  = 0,-3.5,-13 

So now, the new matrix is:
1    0.5  |  4
0   -3.5  | -13

Step 4: Make the second row’s middle into 1
Divide row 2 by -3.5

R2 = 1-3.5  R2 = 0,1,-13/-3.5 = 0,1,3.714...
Or as fraction: 133.5 = 13035 = 267

1   0.5  | 4
0      1  | 267

Step 5: make 0.5 into 0 
R1= R1 -0.5 × R2

1   0 | 157
0   1 | 267

x = 157  and y = 267
 

A homogeneous system of linear equations is a system where all the constant terms are zero in equations. It always has at least one solution — a trivial solution.

The formula for a homogeneous system of linear equations in a matrix is Ax = 0
Where A is the coefficient of the matrix
x is the column of variables 
0 is the zero vector
 

Homogeneous Systems of linear equations, where each of the equations is set to zero, are used to model real-world phenomena.

• Computer and graphics: In 3D graphics, homogeneous coordinates satisfy Ax = 0 for transformations like rotation.

• Chemistry: In chemistry, homogeneous reactions help in balancing chemical equations and studying reaction kinetics and energy.

• Engineering: In civil engineering, a homogeneous system helps determine, if there is a non-trivial solution, which is useful for analyzing forces and stability in structures.

• Economics: Economists use homogeneous systems of linear equations to model supply and demand across multiple markets, especially when the total excess demand is zero.

• Physics: Homogeneous systems help analyze wave functions, electrical circuits, and force balance.