We come now to the basic concept of this book. For the definition that follows we assume that we are given a particular field \(\mathbb{F}\) ; the scalars to be used are to be elements of \(\mathbb{F}\) .

Definition 1. A vector space is a set \(\mathcal{V}\) of elements called vectors satisfying the following axioms.

(A) To every pair, \(x\) and \(y\) , of vectors in \(\mathcal{V}\) there corresponds a vector \(x+y\) , called the sum of \(x\) and \(y\) , in such a way that

  1. addition is commutative, \(x+y=y+x\) ,
  2. addition is associative, \(x+(y+z)=(x+y)+z\) ,
  3. there exists in \(\mathcal{V}\) a unique vector \(0\) (called the origin ) such that \(x+0=x\) for every vector \(x\) , and
  4. to every vector \(x\) in \(\mathcal{V}\) there corresponds a unique vector \(-x\) such that \(x+(-x)=0\) .

(B) To every pair, \(\alpha\) and \(x\) , where \(\alpha\) is a scalar and \(x\) is a vector in \(\mathcal{V}\) , there corresponds a vector \(\alpha x\) in \(\mathcal{V}\) , called the product of \(\alpha\) and \(x\) , in such a way that

  1. multiplication by scalars is associative, \(\alpha(\beta x)=(\alpha \beta) x\) , and
  2. \(1 x=x\) for every vector \(x\) .

(C) 

  1. Multiplication by scalars is distributive with respect to vector addition, \(\alpha(x+y)=\alpha x+\alpha y\) , and
  2. multiplication by vectors is distributive with respect to scalar addition, \((\alpha+\beta) x=\alpha x+\beta x\) .

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 \(\mathcal{V}\) and the underlying field \(\mathbb{F}\) is usually described by saying that \(\mathcal{V}\) is a vector space over \(\mathbb{F}\) . If \(\mathbb{F}\) is the field \(\mathbb{R}\) of real numbers, \(\mathcal{V}\) is called a real vector space ; similarly if \(\mathbb{F}\) is \(\mathbb{Q}\) or if \(\mathbb{F}\) is \(\mathbb{C}\) , we speak of rational vector spaces or complex vector spaces .