System of homogeneous linear equations
Homogeneous linear equations form a specific type of system where all constant terms on the righthand 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:

Zero Constant Terms:
 All equations have zero on the righthand side, resulting in the absence of explicit constant values.

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:

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

NonTrivial Solutions:
 If a system of homogeneous equations has nontrivial 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.

Determining Solutions:
 To determine nontrivial 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:
$x+2yz=0$
$2xy+z=0$
$3x+y+2z=0$
Representing this system in matrix form:
$A=\left[\begin{array}{ccc}{\textstyle 1}& {\textstyle 2}& {\textstyle 1}\\ {\textstyle 2}& {\textstyle 1}& {\textstyle 1}\\ {\textstyle 3}& {\textstyle 1}& {\textstyle 2}\end{array}\right]$
 Determine the number of solutions and whether there are nontrivial solutions based on the determinant and the reduced echelon form.
 Identify the dimension of the solution space and the number of free variables, if any.