linguistic:translation
Terminology

The vector space V is called the representation space of φ and its dimension (if finite) is called the dimension of the representation (sometimes degree, as in). It is also common practice to refer to V itself as the representation when the homomorphism φ is clear from the context; otherwise the notation (V, φ) can be used to denote a representation.

When V is of finite dimension n, one can choose a basis for V to identify V with F, and hence recover a matrix representation with entries in the field F.

An effective or faithful representation is a representation (V, φ), for which the homomorphism φ is injective.