Introduction

Let us look at some examples of linear systems.

1. Suppose $a, b \in {\mathbb{R}}.$ Consider the system $a x = b.$
   1. If $a \neq 0$ then the system has a UNIQUE SOLUTION $x = \frac{b}{a}.$
   2. If $a = 0$ and
      1. $b \neq 0$ then the system has NO SOLUTION.
      2. $b = 0$ then the system has INFINITE NUMBER OF SOLUTIONS, namely all $x \in {\mathbb{R}}.$
2. We now consider a system with $2$ equations in $2$ unknowns.
   Consider the equation $a x + b y = c.$ If one of the coefficients, $a$ or $b$ is non-zero, then this linear equation represents a line in ${\mathbb{R}}^2.$ Thus for the system
   $$a_1 x + b_1 y = c_1 \; {\mbox{ and }} \; a_2 x + b_2 y = c_2,$$
   the set of solutions is given by the points of intersection of the two lines. There are three cases to be considered. Each case is illustrated by an example.
   1. UNIQUE SOLUTION
      $x+ 2 y = 1$ and $x + 3 y = 1.$ The unique solution is $(x, y)^t = (1, 0)^t.$
      Observe that in this case, $a_1 b_2 - a_2 b_1 \neq 0.$
   2. INFINITE NUMBER OF SOLUTIONS
      $x+ 2 y = 1$ and $2x + 4 y = 2.$ The set of solutions is $(x, y)^t = (1 - 2y, y)^t= (1, 0)^t + y (-2, 1)^t$ with $y$ arbitrary. In other words, both the equations represent the same line.
      Observe that in this case, $a_1 b_2 - a_2 b_1 = 0,\; a_1 c_2 - a_2 c_1 = 0$ and $b_1 c_2 - b_2 c_1 = 0.$
   3. NO SOLUTION
      $x+ 2 y = 1$ and $2x + 4 y = 3.$ The equations represent a pair of parallel lines and hence there is no point of intersection.
      Observe that in this case, $a_1 b_2 - a_2 b_1 = 0$ but $a_1 c_2 - a_2 c_1 \neq 0.$
3. As a last example, consider $3$ equations in $3$ unknowns.
   A linear equation $a x + b y + c z = d$ represent a plane in ${\mathbb{R}}^3$ provided $(a, b, c) \neq (0, 0, 0).$ As in the case of $2$ equations in $2$ unknowns, we have to look at the points of intersection of the given three planes. Here again, we have three cases. The three cases are illustrated by examples.
   1. UNIQUE SOLUTION
      Consider the system $x + y + z = 3, \;\; x + 4 y + 2 z = 7$ and $4 x + 10 y - z = 13.$ The unique solution to this system is $(x, y, z)^t = (1, 1, 1)^t;$ i.e. THE THREE PLANES INTERSECT AT A POINT.
   2. INFINITE NUMBER OF SOLUTIONS
      Consider the system $x + y + z = 3, \;\; x + 2 y + 2 z = 5$ and $3 x + 4 y + 4 z = 11.$ The set of solutions to this system is $(x, y, z)^t = (1, 2-z, z)^t = (1, 2, 0)^t + z (0, -1, 1)^t,$ with $z$ arbitrary: THE THREE PLANES INTERSECT ON A LINE.
   3. NO SOLUTION
      The system $x + y + z = 3, \;\; x + 2 y + 2 z = 5$ and $3 x + 4 y + 4 z = 13$ has no solution. In this case, we get three parallel lines as intersections of the above planes taken two at a time.

      The readers are advised to supply the proof.

DEFINITION 2.1.1 (Linear System)   A linear system of $m$ equations in $n$ unknowns $x_1, x_2, \ldots, x_n$ is a set of equations of the form

$$a_{11} x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1$$

$$a_{21} x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2$$

$$\vdots \hspace{1.5in} \vdots$$ (2.1.1)

$$a_{m1} x_1 + a_{m2}x_2 + \cdots + a_{mn}x_n = b_m$$

where for $1 \leq i \leq n,$ and $1 \leq j \leq m; \; a_{ij}, b_i \in {\mathbb{R}}.$ Linear System (2.1.1) is called HOMOGENEOUS if $b_1 = 0 = b_2= \cdots = b_m$ and NON-HOMOGENEOUS otherwise.

We rewrite the above equations in the form $A {\mathbf x}= {\mathbf b},$ where
$A = \begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdo... ...}, \; \; {\mathbf x}=\begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix},$ and ${\mathbf b}= \begin{bmatrix}b_1\\ b_2 \\ \vdots \\ b_m \end{bmatrix}$

The matrix $A$ is called the COEFFICIENT matrix and the block matrix $\left[ A \; \; {\mathbf b}\right],$ is the AUGMENTED matrix of the linear system (2.1.1).

Remark 2.1.2   Observe that the $i^{\mbox{th}}$ row of the augmented matrix $[A \;\; {\mathbf b}]$ represents the $i^{\mbox{th}}$ equation and the $j^{\mbox{th}}$ column of the coefficient matrix $A$ corresponds to coefficients of the $j^{\mbox{th}}$ variable $x_j.$ That is, for $1 \leq i \leq m$ and $1 \leq j \leq n,$ the entry $a_{ij}$ of the coefficient matrix $A$ corresponds to the $i^{\mbox{th}}$ equation and $j^{\mbox{th}}$ variable $x_j..$

For a system of linear equations $A {\mathbf x}= {\mathbf b},$ the system $A {\mathbf x}= {\mathbf 0}$ is called the ASSOCIATED HOMOGENEOUS SYSTEM.

DEFINITION 2.1.3 (Solution of a Linear System)   A solution of the linear system $A {\mathbf x}= {\mathbf b}$ is a column vector ${\mathbf y}$ with entries $y_1, y_2, \ldots, y_n$ such that the linear system (2.1.1) is satisfied by substituting $y_i$ in place of $x_i.$

That is, if ${\mathbf y}^t = [ y_1, y_2, \ldots, y_n ]$ then $A {\mathbf y}= {\mathbf b}$ holds.

Note: The zero $n$ -tuple ${\mathbf x}={\mathbf 0}$ is always a solution of the system $A {\mathbf x}= {\mathbf 0},$ and is called the TRIVIAL solution. A non-zero $n$ -tuple ${\mathbf x},$ if it satisfies $A {\mathbf x}= {\mathbf 0},$ is called a NON-TRIVIAL solution.