System of Homogeneous Linear Equations

Homogeneous Linear Equations:-

a₁ x + b₁ y = 0
a₂ x + b₂ y = 0
is called a system of homogeneous linear equations in x and y.

Then the above system can be written as AX = 0

Definitions

Coefficient Matrix :-

In the system AX = 0, A is called as coefficient matrix.
solution :-
If x= x₁, y = y₁ satisfy the given system AX = 0 then

Trivial Solution :-
For the system AX = 0, X is clearly a solution. Null matrix 0 is called zero solution or trivial solution of AX = 0.

–> If the coefficient matrix A is non-singular matrix then the system AX = 0 has only zero solution.

–> If system AX = 0 has atleast 1 non-zero solution, then coefficient matrix A is singular. This can also be written as follows:
If there exists (x, y) not all zeros, such that
a₁ x + b₁ y = 0
a₂ x + b₂ y = 0
then x, y can be eliminated from the above equations.

–> If X₁ and X₂ are any two solutions of the system AX = 0, then for any two scalars a₁ and a₂, a₁X₁ + a₂X₂ is also a solution.

–> If AX = 0 has atleast 1 non-zero solution then the system has infinite number of solutions.

Homogeneous Linear Equations in x, y, z

a₁ x + b₁ y + c₁ z = 0
a₂ x + b₂ y + c₂ z = 0
a₃ x + b₃ y + c₃ z = 0 is called a system of homogeneous linear equations in x, y, z.

Then the above system can be written as AX = 0.

Remarks :-

1. If A is non-singular then 0 solution is the only solution of AX = 0.
2. If AX = 0 has atleast one non-zero solutions then A is singular. In other words if there exists x, y, z not all zeros such that a₁ x + b₁ y + c₁ z = 0
   a₂ x + b₂ y + c₂ z = 0 then x, y, z can be eliminated. The eliminant is the |A| = 0.