quality:correction
Definition and first consequences

Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the same field K {\displaystyle K}. A function f: V → W {\displaystyle f:V\to W} is said to be a linear map if for any two vectors u, v ∈ V {\textstyle \mathbf {u},\mathbf {v} \in V} and any scalar c ∈ K {\displaystyle c\in K} the following two conditions are satisfied:

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By the associativity of the addition operation denoted as +, for any vectors u 1, …, u n ∈ V {\textstyle \mathbf {u} _{1},\ldots,\mathbf {u} _{n}\in V} and scalars c 1, …, c n ∈ K, {\textstyle c_{1},\ldots,c_{n}\in K,} the following equality holds: f (c 1 u 1 + ⋯ + c n u n) = c 1 f (u 1) + ⋯ + c n f (u n). {\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).} Thus a linear map is one which preserves linear combinations.

Denoting the zero elements of the vector spaces V {\displaystyle V} and W {\displaystyle W} by 0 V {\textstyle \mathbf {0} _{V}} and 0 W {\textstyle \mathbf {0} _{W}} respectively, it follows that f (0 V) = 0 W. {\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.} Let c = 0 {\displaystyle c=0} and v ∈ V {\textstyle \mathbf {v} \in V} in the equation for homogeneity of degree 1: f (0 V) = f (0 v) = 0 f (v) = 0 W. {\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v})=0f(\mathbf {v})=\mathbf {0} _{W}.}

A linear map V → K {\displaystyle V\to K} with K {\displaystyle K} viewed as a one-dimensional vector space over itself is called a linear functional.

These statements generalize to any left-module R M {\textstyle {}_{R}M} over a ring R {\displaystyle R} without modification, and to any right-module upon reversing of the scalar multiplication.