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:

a11x1+a12x2++a1nxn=0

a21x1+a22x2++a2nxn=0

am1x1+am2x2++amnxn=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 (x1=x2==xn=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 (A) 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=[121211312]

  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.