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 the operations of addition and scalar multiplication.
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 u, v ∈ V and any scalar c ∈ K the following two conditions are satisfied:
f(u + v) = f(u) + f(v)
f(cu) = cf(u)