In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
<MATH> \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat} </MATH>
Each linear system corresponds to a linear system with zero right-hand sides:
If a linear system has a solution u1 then that solution is unique if the only solution to the corresponding homogeneous linear system is 0.
A system of Linear Algebra - Linear Equation is called a homogeneous linear system.
<MATH> \begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 0 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& 0 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat} </MATH>
The solution set of a homogeneous linear system is a vector space.
Lemma: Let u1 be a solution to a linear system. Then, for any other vector u2, u2 is also a solution if and only if u2 - u1 is a solution to the corresponding homogeneous linear system
A triangular linear system as a triangular form
and can be expressed as a triangular matrix