# System of homogeneous linear equations

Homogeneous linear equations form a specific type of system where all constant terms on the right-hand side of the equations are zero

These equations are of the form:

${a}_{11}{x}_{1}+{a}_{12}{x}_{2}+\cdots +{a}_{1n}{x}_{n}=0$

${a}_{21}{x}_{1}+{a}_{22}{x}_{2}+\cdots +{a}_{2n}{x}_{n}=0$

${a}_{m1}{x}_{1}+{a}_{m2}{x}_{2}+\cdots +{a}_{mn}{x}_{n}=0$

### Characteristics of Homogeneous Equations:

1. Zero Constant Terms:

• All equations have zero on the right-hand side, resulting in the absence of explicit constant values.
2. Homogeneity and Linearity:

• The term "homogeneous" refers to the property that the equations are balanced around zero, implying that the solutions form a subspace centered at the origin.
• The equations are linear because they don't contain higher powers of the variables or products between variables.

### Properties and Solutions:

1. Trivial Solution:

• The system always has at least one solution, known as the trivial solution, where all variables equal zero (${x}_{1}={x}_{2}=\cdots ={x}_{n}=0$).
2. Non-Trivial Solutions:

• If a system of homogeneous equations has non-trivial solutions (i.e., solutions where not all variables are zero), it implies that the equations are linearly dependent.
• Linearly dependent equations provide a set of values for variables that satisfy the equations without all variables being zero.
3. Determining Solutions:

• To determine non-trivial solutions, examine the determinant of the coefficient matrix ($\mathrm{\mid }A\mathrm{\mid }$) and apply row operations to derive the reduced echelon form.
• The number of free variables (variables that can take any value) indicates the dimension of the solution space.

### Example:

Consider a system of homogeneous equations:

$x+2y-z=0$

$2x-y+z=0$

$3x+y+2z=0$

Representing this system in matrix form:

$A=\left[\begin{array}{ccc}1& 2& -1\\ 2& -1& 1\\ 3& 1& 2\end{array}\right]$

1. Determine the number of solutions and whether there are non-trivial solutions based on the determinant and the reduced echelon form.
2. Identify the dimension of the solution space and the number of free variables, if any.