Vector spaces

We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field ; the scalars to be used are to be elements of .

Definition 1. A vector space is a set of elements called vectors satisfying the following axioms.

(A) To every pair, and , of vectors in there corresponds a vector , called the sum of and , in such a way that

addition is commutative, ,
addition is associative, ,
there exists in a unique vector (called the origin ) such that for every vector , and
to every vector in there corresponds a unique vector such that .

(B) To every pair, and , where is a scalar and is a vector in , there corresponds a vector in , called the product of and , in such a way that

multiplication by scalars is associative, , and
for every vector .

(C)

Multiplication by scalars is distributive with respect to vector addition, , and
multiplication by vectors is distributive with respect to scalar addition, .

These axioms are not claimed to be logically independent; they are merely a convenient characterization of the objects we wish to study. The relation between a vector space and the underlying field is usually described by saying that is a vector space over . If is the field of real numbers, is called a real vector space ; similarly if is or if is , we speak of rational vector spaces or complex vector spaces .

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