Linear equations

A Linear equation is first of all, an equation. The simplest example of a linear expression is an algebraic equation of the form y=3x, which if plotted as a graph gives a straight line (thus providing the terminology).

Such an equation has certain properties which are also present in more complicated equations, and which are often exploited in the solution of such equations. In mathematics, then, a linear function, f, (or equally often a linear operator, L) is defined in terms of these properties. Thus a function f from set A to set B is linear iff

• f(a x + b y) = a f(x) + b f(y)

where a and b are scalars.

A linear equation is an equation containing only linear functions.

Linear equations occur with great regularity in applied mathematics. Whilst they arise quite naturally when modelling many phenomena, they're particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.