In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an m × n {\displaystyle m\times n} matrix, which takes vectors in n {\displaystyle n} -dimensions into vectors in m {\displaystyle m} -dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars. When the two vector spaces are the same, a linear map is also called a linear transformation or linear endomorphism. A linear map is a homomorphism of vector spaces. Thus, a linear map T : V → W {\displaystyle T:V\to W} satisfies T ( a x + b y ) = a T x + b T y {\displaystyle T(ax+by)=aTx+bTy} , where a {\displaystyle a} and b {\displaystyle b} are scalars, and x {\displaystyle x} and y {\displaystyle y} are vectors (elements of the vector space V {\displaystyle V} ). A linear mapping always maps the origin of V {\displaystyle V} to the origin of W {\displaystyle W} , and linear subspaces of V {\displaystyle V} onto linear subspaces in W {\displaystyle W} (possibly of a lower dimension); for example, it maps a plane through the origin in V {\displaystyle V} to either a plane through the origin in W {\displaystyle W} , a line through the origin in W {\displaystyle W} , or just the origin in W {\displaystyle W} . Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.
Definition and first consequences Let V {\displaystyle V} and W {\displaystyle W} be vector spaces over the same field K {\displaystyle K} , such as the real or complex numbers. 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:
Additivity / operation of addition f ( u + v ) = f ( u ) + f ( v ) {\displaystyle f(\mathbf {u} +\mathbf {v} )=f(\mathbf {u} )+f(\mathbf {v} )}
Homogeneity of degree 1 / operation of scalar multiplication f ( c u ) = c f ( u ) {\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )}
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 sides of the above examples) or after (the left 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 {\displaystyle c_{1},\ldots ,c_{n}\in K} , the following equality holds:
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