Real Linear Equations
Why
Linear equations are ubiquitous.
Definition
Given $a \in \R ^n$ and $y \in \R $, suppose
we want to find $x \in \R ^n$ satisfying
\[
a_1x_1 + a_2x_2 + \cdots + a_nx_n = y.
\]
The source of the terminology “linear” is by viewing the left hand side as a function. Define $f: \R ^n \to \R $ by $f(x) = \sum_{i} a_ix_i$. We want to find $x \in \R ^n$ to satisfy $f(x) = b$. Notice that $f$ is a linear real function.2
Moreover, to each linear function $f: \R ^d \to \R $ there exists a vector $a \in \R ^d$ so that $f(x) = \sum_{i} a_ix_i$. For this reason, if we are given several linear function $f_1, \dots , f_m$, then we can think of these as several vectors $a^1, \dots , a^n$. If we are also given $b_i \in \R $ for each $i = 1, \dots , m$, then we have the vector $b \in \R ^m$
We can define the two-dimensional array $A \in
\R ^{m \times n}$ by $A_{ij} = a^{i}_j$.
For this reason, a linear
system of equations is a pair $(A, b)$.
A solution of a linear system of equations is
a vector $x \in \R ^n$ satisfying the equations
\[
\begin{aligned}
A_{11}x_1 & + & A_{12}x_2 & + & \cdots \, & + & A_{1n}x_n
& = \, & b_1 \\
A_{21}x_1 & + & A_{22}x_2 & + & \cdots & + & A_{2n}x_n &
= & b_2 \\
\vdots & & \vdots & & & & \vdots & & \vdots \\
A_{m1}x_1 & + & A_{m2}x_2 & + & \cdots & + & A_{mn}x_n &
= & b_n \\
\end{aligned}
\]