- Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
- Homogeneity of degree 1: f(αx) = αf(x) for all α. It turns out that homogeneity follows from the additivity property in all cases where α is rational. (proof) In that case, provided that the function is continuous, it becomes useless to establish the condition of homogeneity as an additional axiom.
In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.
The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator], and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up.
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations.
1. Heinrich Wölfflin, Principles of Art History: the Problem of the Development of Style in Later Art, M. D. Hottinger (trans.), Mineola, N.Y.: Dover (1950): pp. 18-72.