Definition – Elementary row operation

Contents
1. Definitions
2. Theorems

Definition

Let us consider a system of linear equations. We define two types of elementary row operations.

Type 1: Where two rows are swapped, and the remaining rows are left unchanged.
Type 2: Where a multiple of row $i$ is added to any row $k$ .

Theorems

Theorem 1 – An elementary row operation produces a system of equations which is equivalent to the original system of equations.

Proof –

In the case of type 1, it is obvious that the solutions will remain the same as the solution is not dependent on the order of equations, but rather solely on the coefficients of the equations. For the case of an elementary row operation of type 2, we will prove that the original set of solutions is the same (be the set empty or infinite).

Firstly, let us consider a system of equations,

$\begin{array}{cccccc} \alpha_1 x_1 &+&...&+&\alpha_n x_n & = m_1 \\ \beta_1 x_1 &+&...& +&\beta_n x_n & = m_2 \\ &\vdots & & \vdots \\ \lambda_1 x_1 &+&...& +&\lambda_n x_n & = m_n \end{array}$

And let us apply an elementary row operation of type two to this set of equations. Let $c\in\mathbb{R}$ ,

$\begin{array}{ccccccccccccc} \alpha_1 x_1 &+&...&+&\alpha_n x_n &+& c(\beta_1 x_1 &+&...& +& \beta_n x_n) &=& m_1 + m_2 \\ \beta_1 x_1 &+&...& +&\beta_n x_n & & &&&&&=& m_2 \\ &\vdots & & \vdots &&&&&&&&\vdots\\ \lambda_1 x_1 &+&...& +&\lambda_n x_n & & & && & &=& m_n \end{array}$

Now, it is clear that if $s = (s_1, \dots, s_n)$ is a solution to the original equation, $s$ will also be a solution to the new set of equations. However, is the converse true?

We can prove this simply by showing that we can obtain the original set of equations. Thus, let us add $-c$ times row $k$ to row $1$ , we obtain the original system of equations. Thus, every solution to the original system is a solution to the new system, and vice versa. This proves the theorem. $\square$