Linear Map - Definition and First Consequences

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 the same for any linear combination of vectors, i.e. that for any vectors x₁, ..., x_m ∈ V and scalars a₁, ..., a_m ∈ K, the following equality holds:

Denoting the zeros of the vector spaces by 0, it follows that f(0) = 0 because letting α = 0 in the equation for homogeneity of degree 1,

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 K-linear maps. For example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear.

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional.

These statements generalize to any left-module _RM over a ring R without modification.