In the sense of "Linear Algebra" a function $f$ is said to be "linear" if it satisfies the following:

 An unavoidable consequence of these conditions is that $f(0)$ must be $0.$ To see this, consider the first condition with $x=k$ and $y=-k,$ or the second condition with $k=0.$

• $f(x+y)~=~f(x)+f(y)$
• $f(kx)~=~k.f(x)$
Linear functions don't need to be just from $\mathbb{R}$ to $\mathbb{R}$ or $\mathbb{C}$ to $\mathbb{C}$ but can be from any set to any set, provided it has suitable concepts of addition and multiplication - such a set forms a vector space. In this way we can then start to represent linear functions by matrices. This is why "Linear Algebra" is really the study of matrices, and matrix multiplication.

In geometry there is a different concept of "line" and "linear" which is discussed in Equation of a line.

MathematicsTaxonomy (none) (none)
EquationOfALine
LinearAlgebra
(none)

## You are here

LinearFunction
Function
Matrices
MatrixMultiplication
(none) (none) CoDomainOfAFunction
Commutative
DomainOfAFunction
ImageOfAFunction
MatrixTransformation
Polynomial
RangeOfAFunction
RealNumber
ScalarProduct
Vectors