# 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 $m \times n$ matrices over $K$. This vector space will be written as $K^{(m \times n)}$. 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 ${\operatorname{Hom}}_K(U, V)$. Once bases have been chosen for $U$ and $V$, we may identify ${\operatorname{Hom}}_K(U, V)$ with $K^{(m \times n)}$. Thus, $K^{(m \times n)}$ 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 $K^{(m \times n)}$. 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 $K^{(m \times n)}$.

## 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 $m \times n$-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 $K^{(m \times n)}$ (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 $K^{(n)}$.