# Linear Algebra - (Linear system|System of Linear equations)

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

\begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 1 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& -2 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}

Each linear system corresponds to a linear system with zero right-hand sides:

• $a_1 . x = beta_1 \rightarrow a_1 . x = 0$
• $a_2 . x = beta_2 \rightarrow a_2 . x = 0$
• $a_n . x = beta_n \rightarrow a_n . x = 0$

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.

## 3 - Type

### 3.1 - Homogeneous

A system of homogeneous linear equations is called a homogeneous linear system.

\begin{alignat}{7} 3x &&\; + \;&& 2y &&\; - \;&& z &&\; = \;&& 0 & \\ 2x &&\; - \;&& 2y &&\; + \;&& 4z &&\; = \;&& 0 & \\ -x &&\; + \;&& \tfrac{1}{2} y &&\; - \;&& z &&\; = \;&& 0 & \end{alignat}

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

### 3.2 - Triangular

A triangular linear system as a triangular form

• $10 = 2x_1 + 3x_2 - 4x_3$
• $~3 = 1x_2 + 2x_3$
• $15 = 5x_3$

and can be expressed as a triangular matrix