% This is lecture no. 1 of MH1200/: Linear Algebra I.

\chapter{Linear Equations}

\begin{definition}
	Linear Equation.
	A linear equation in $n$ variables \newline $x_1, x_2, \ldots, x_n$ is an equation of the form:
	\[
	a_1x_1 + a_2x_2 + \ldots + a_nx_n = b
	.\] where $a_1,a_2,\ldots,a_n$ and $b$ are constant real numbers.
\end{definition}

\section{Solutions of linear equations}

Consider the equation:
\[
x + 3y = 7
.\]
The assignment
\begin{align*}
	x & = 1\\
	y & = 2
\end{align*}
is a solution to this equation, because substituting in these values the equation becomes true.

\begin{definition}
	$\mathbb{R}^n$ set of all possible $n$-tuples.
	Let $n$ be an integer $n > 0$.
	The set $\mathbb{R}^n$ is defined to be the set of all possible  $n$-tuples of real numbers.
\end{definition}

The explanation for the notation $\mathbb{R}^n$ is that we are taking the `cartesian product' of $n$ copies of the set $\mathbb{R}^n$:
\[
	\mathbb{R}^n = \underbrace{\mathbb{R}\times\ldots\times\mathbb{R}}_{n}
.\] 
\begin{definition}
	Solution set of a linear equation.
	For a linear equation of the form:
	\[
	c_1 x_1 + c_2 x_2 + \ldots + c_n x_n = d
	.\] 
	A solution set $\{x_1 = a_1 ,\ldots, x_n = a_n\}$ corresponds to the element of $\mathbb{R}^n$ you get by listing the values in the fixed order: $(a_1,\ldots,a_n)\in \mathbb{R}^n$.
\end{definition}

\bigbreak

\begin{definition}
	Parameterisation.
	Consider some linear equation in $n$ variales, such as:
	\[
	a_1 x_1 + a_2 x_2 + \ldots + a_n x_n = b
	.\]
	A \textbf{parameterisation} for the set of solutions of this equation is a collection of $n$ functions:
	\[
		f_1 (s_1 , \ldots,s_k),\ldots,f_n(s_1,\ldots,s_k)
	.\]
	of $k$ variables $s_1,\ldots,s_k$ such that whenever we substitute any choice of real numbers into the variable $s_i$ then we get a solution of the equation:
	\[
	\begin{array}{c}
		x_1=f_1(s_1,\ldots,s_k)\\
		\vdots\\
		x_n=f_n(s_1,\ldots,s_k)\\
	\end{array}
	.\]
	Moreover, we insist that every solution of the equation must arise from some choice of real number for $s_1,\ldots,s_k$.
	These new variables $s_1,\ldots,s_k$ are called free \textbf{free parameters}.
\end{definition}

\begin{exmp}
	\label{parametriceqn}
	Consider the equation:
	\[
		5x_1+2x_2 = 10
	.\] Note that whatever value we set for $x_2$, we can always solve the resulting equation for  $x_1$. So we can set $x_2$ to a new free parameter $t$, $x_2=t$, which can take any value, and then solve for $x_1$ in terms of $t$.
	\[
	5x_1+ 2t = 10 \Rightarrow x_1 = 2-\frac{2}{5}t
	.\] 
	So for any choice $t$, the formulas:
	\begin{align*}
		x_1 & = 2 - \frac{2}{5}t\\
		x_2 & = t
	\end{align*}
	give a solution to the original equation.
	An equivalent , short way of describing the resulting set of solutions is:
	\[
		S = \left\{\left(2 - \frac{2}{5}t,t\right)\in\mathbb{R}^2;t\in\mathbb{R}\right\} \subset \mathbb{R}^2
	.\] 
\end{exmp}
