Solution Preserving Operations

Earlier, while defining linear systems, we discussed the different possibilities of the solution of a linear system. In this chapter, we’ll describe how we can achieve those possibilities. Let’s start by learning the foundations before moving on to a systematic approach.

Augmented matrix

We’ve already learned two different representations of a system of linear equations. We started with a set of linear equations, that is,

$$a_{11}x_1 + a_{12}x_2+...+ a_{1n}x_n = y_1$$

$$a_{21}x_1 + a_{22}x_2+...+ a_{2n}x_n = y_2$$

$$\vdots$$

$$a_{m1}x_1 + a_{m2}x_2+...+ a_{mn}x_n = y_m$$

Later, we defined the matrix representation, $A\bold{x} = \bold{b}$, where $A$ is the coefficient matrix, $\bold{x}$ contains all the unknowns, and $\bold{b}$ is on the right-hand side of the equations.

$$\begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix} = \begin{bmatrix} y_1\\ y_2\\ \vdots\\ y_m \end{bmatrix}$$

Let’s look at yet another representation of the system of linear equations called the augmented matrix. It’s simply a combination of our system’s coefficient matrix, $A$, and the right-hand-side vector, $\bold b$, separated by a vertical line (the vertical line can be omitted if the context is clear).

$$\left(\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & y_1\\ a_{21} & a_{22} & \cdots & a_{2n} & y_2\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} & y_m \end{array}\right)$$

Each row in the augmented matrix represents an equation from the system of linear equations. In the augmented matrix, the variables aren’t included explicitly. However, each column on the left side of the vertical separating line represents the constraints on the corresponding variable in the equation, as depicted in the figure below.