System of Linear Equations

Core Idea

(Linear Equation)

An equation of the form $a_{1}x_{1}+ a_{2}x_{2}+\dots+a_{n}x_{n} = b$. Here, $a_i$ and $b$ are constants and $x \in \{1\dots n\}$ represents the variables. In a 2-D space, this equation would have 2 variables and all the solutions to the equation $a_{1}x+a_{2}y=b$ represent a straight line (see how this is similar to $mx+y=c$). In a 3-D space the set of solutions represents a plane. Generally, the set of solutions to such equation in an n-dimensional space represents a hyperplane in the $n$-dimensional space.

(System of Linear Equations)

A set of $m$ such Linear Equations would be called a system of linear equations. Note that the same set of $x_i$ would have to satisfy all the equations simultaneously. The solution could be (a) unique, (b) non-existent or (c) infinitely many.

Connections

• Forward Links: Row View of a System of Linear Equations, Column View of a System of Linear Equations, Matrix View of a System of Linear Equations, Gaussian Elimination
• Backward Links: Matrix, Vectors