and called the sum of these two vectors. The second operation, called scalar multiplication，assigns to any scalar in and any vector in another vector in , which is denoted . For having a vector space, the eight following axioms must be satisfied for every , and in , and and in . When the scalar field is the real numbers the vector space is called a real vector space. When the scalar field is the complex numbers, the vector space is called a complex vector space. These two cases are the most common ones, but vector spaces with scalars in an arbitrary field are also commonly considered. Such a vector space is called an -vector space or a vector space over . An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication), say that this operation defines a ring homomorphism from the field into the endomorphism ring of this group. Subtraction of two vectors can be defined as Direct consequences of the axioms include that, for every and one has implies or Related concepts and properties Linear combination Given a set of elements of a -vector space , a linear combination of elements of is an element of of the form where and The scalars are called the coefficients of the linear combination. Linear independence The elements of a subset of a -vector space are said to be linearly independent if no element of can be written as a linear combination of the other elements of . Equivalently, they are linearly independent if two linear combinations of element of define the same element of if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero. Linear subspace A linear subspace or vector subspace of a vector space is a non-empty subset of that is closed under vector addition and scalar multiplication; that is, the sum of two elements of and the product of an element of by a scalar belong to . This implies that every linear combination of elements of belongs to . A linear subspace is a vector space