Introduction

In this chapter we will discuss vector spaces and their linear transformations. Let K be a field. In Magma, the standard K-vector space is taken to be the set of n-tuples over the field K, which we shall write as K⁽ⁿ⁾.

A rectangular matrix over a field K is considered to be an element of the vector space consisting of all m x n matrices over K. This vector space will be written as K^{(m x n)}. Let U and V be K-vector spaces of dimensions m and n, respectively. The set of all linear transformations with domain U and codomain V will be denoted by Hom_K(U, V). Once bases have been chosen for U and V, we may identify Hom_K(U, V) with K^{(m x n)}. Thus, K^{(m x n)} is first of all a vector space and all the normal vector space operations apply. However, since it is also a set of mappings, some additional operations arising from this characterization apply.

We shall use the term vector space or K-vector space (if we wish to emphasize the coefficient field) to refer to both the space K⁽ⁿ⁾ and the space K^{(m x n)}. If we wish to differentiate between the two, we shall use the term tuple space when referring to K⁽ⁿ⁾ and the term matrix space referring to K^{(m x n)}.

Vector Space Categories

The family of all finite dimensional vector spaces over a given field K forms a category, while the set of all finite dimensional vector spaces forms a family of categories indexed by the field K. In this family of categories, objects are vector spaces and the morphisms are linear transformations. The (indexed family of) categories consisting of vector spaces of n-tuples has the name ModTupFld, while the (indexed family of) categories consisting of vector spaces of m x n-matrices has the name ModMatFld.

The Construction of a Vector Space

Every vector space V defined over a field K is created either as a subspace of the row space K⁽ⁿ⁾ (tuple spaces) or as a subspace of K^{(m x n)} (matrix modules). Thus, the construction of a general vector space is a two step process:

(i)

The appropriate row space K⁽ⁿ⁾, is constructed;

(ii)

The required vector space V is then defined as a subspace or quotient space of K⁽ⁿ⁾.