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 [1].
For a set , we shall denote this vector space by .
One application of this construction is given in [2],
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.,
(1) |
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
The mapping is universal in the following sense. If is an arbitrary mapping from to a vector space , then there exists a unique mapping such that the below diagram commutes:
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.
References
-
1
W. Greub,
Linear Algebra
, Springer-Verlag, Fourth edition, 1975.
-
2
I. Madsen, J. Tornehave,
From Calculus to Cohomology
, Cambridge University press, 1997.
Title | free vector space over a set |
---|---|
Canonical name | FreeVectorSpaceOverASet |
Date of creation | 2013-03-22 13:34:34 |
Last modified on | 2013-03-22 13:34:34 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 8 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 15-00 |
Synonym | vector space generated by a set |
Related topic | TensorProductBasis |