Domain and Range

Recall that a function f is simply a rule for associating points in one set D —called the domain of f —to points in another set R —the range of f.

Linear Function

A function f that maps points in D to points in R is said to be a linear function whenever f satisfies the conditions that

Affine functions

However, f(x) = αx + β does not qualify for the title “linear function”—it is a linear function that has been translated by a constant β. Translations of linear functions are referred to as affine functions. 

General Form

it seems reasonable to suggest that a general linear function of the form

somehow represents a “linear” or “flat” surface passing through the origin 0 =(0, 0,..., 0) in Rn+1

.

Trace

Examples

differentiation and integration

trace function

matrix multiplication

linear system