Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. The function f : R 2 → R 2 { extstyle f:mathbb {R} ^{2} o mathbb {R} ^{2}} with f ( x , y ) = ( 2 x , y ) { extstyle f(x,y)=(2x,y)} is a linear map. This function scales the x { extstyle x} component of a vector by the factor 2 { extstyle 2} .The function is additive: It doesn't matter whether first vectors are added and then mapped or whether they are mapped and finally added: f ( a + b ) = f ( a ) + f ( b ) { extstyle f(a+b)=f(a)+f(b)} The function is homogeneous: It doesn't matter whether a vector is first scaled and then mapped or first mapped and then scaled: f ( λ a ) = λ f ( a ) { extstyle f(lambda a)=lambda f(a)} In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. An important special case is when V = W, an endomorphism of V, sometimes the term linear operator refers to this case. In another convention, linear operator allows V and W to differ, while requiring them to be real vector spaces. Sometimes the term linear function has the same meaning as linear map, while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring. Let V { extstyle V} and W { extstyle W} be vector spaces over the same field K . { extstyle mathbf {K} .} A function f : V → W { extstyle f:V o W} is said to be a linear map if for any two vectors u , v ∈ V { extstyle mathbf {u} ,mathbf {v} in V} and any scalar c ∈ K { extstyle cin mathbf {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 or after the operations of addition and scalar multiplication. This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors u 1 , … , u n ∈ V { extstyle mathbf {u} _{1},ldots ,mathbf {u} _{n}in V} and scalars c 1 , … , c n ∈ K , { extstyle c_{1},ldots ,c_{n}in mathbf {K} ,} the following equality holds: Denoting the zero elements of the vector spaces V { extstyle V} and W { extstyle W} by 0 V { extstyle mathbf {0} _{V}} and 0 W { extstyle mathbf {0} _{W}} respectively, it follows that f ( 0 V ) = 0 W . { extstyle fleft(mathbf {0} _{V} ight)=mathbf {0} _{W}.} Let c = 0 { extstyle c=0} and v ∈ V { extstyle mathbf {v} in V} in the equation for homogeneity of degree 1: Occasionally, V { extstyle V} and W { extstyle W} can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of 'linear'. If V { extstyle V} and W { extstyle W} are considered as spaces over the field K { extstyle mathbf {K} } as above, we talk about K { extstyle mathbf {K} } -linear maps. For example, the conjugation of complex numbers is an R { extstyle mathbf {R} } -linear map C → C { extstyle mathbf {C} o mathbf {C} } , but it is not C { extstyle mathbf {C} } -linear.