In mathematics, a linear map, linear transformation, or linear operator (in some contexts also called linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with that of linear map, while in analytic geometry it is less strict.
In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or in the language of category theory a morphism in KVect, the category of vector spaces over a given field K.
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Definition and first consequences
Let V and W be vector spaces over the same field K. A function f : V → W is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:
This is equivalent to requiring that for any vectors x_{1}, ..., x_{m} ∈ V and scalars a_{1}, ..., a_{m} ∈ K, the following equality holds:
It immediately follows from the definition that f(0) = 0.
Occasionally, V and 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 and W are considered as spaces over the field K as above, we talk about Klinear maps. For example, the conjugation of complex numbers is an Rlinear map C → C, but it is not Clinear.
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