free vector space over a set
In this entry we construct the free vector space over a set, or the vector space generated by a set . For a set , we shall denote this vector space by . One application of this construction is given in , where the free vector space is used to define the tensor product for modules.
To define the vector space , let us first define as a set. For a set and a field , we define
In other words, consists of functions that are non-zero only at finitely many points in . Here, we denote the identity element in by , and the zero element by . The vector space structure for is defined as follows. If and are functions in , then is the mapping . Similarly, if and , then is the mapping . It is not difficult to see that these operations are well defined, i.e., both and are again functions in .
0.0.1 Basis for
If , let us define the function by
These functions form a linearly independent basis for , i.e.,
Here, the space consists of all finite linear combinations of elements in . It is clear that any element in is a member in . Let us check the other direction. Suppose is a member in . Then, let be the distinct points in where is non-zero. We then have
To see that the set is linearly independent, we need to show that its any finite subset is linearly independent. Let be such a finite subset, and suppose for some . Since the points are pairwise distinct, it follows that for all . This shows that the set is linearly independent.
Let us define the mapping , . This mapping gives a bijection between and the basis vectors . We can thus identify these spaces. Then becomes a linearly independent basis for .
0.0.2 Universal property of
Proof. We define as the linear mapping that maps the basis elements of as . Then, by definition, is linear. For uniqueness, suppose that there are linear mappings such that . For all , we then have . Thus since both mappings are linear and the coincide on the basis elements.
|Title||free vector space over a set|
|Date of creation||2013-03-22 13:34:34|
|Last modified on||2013-03-22 13:34:34|
|Last modified by||mathcam (2727)|
|Synonym||vector space generated by a set|