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 $X$, we shall denote this vector space by $C(X)$. 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 $C(X)$, let us first define $C(X)$ as a set. For a set $X$ and a field $\mathbb{K}$, we define
$C(X)$ | $=$ | $\mathrm{\{}f:X\to \mathbb{K}|{f}^{-1}(\mathbb{K}\backslash \{0\})\text{is finite}\}.$ |
In other words, $C(X)$ consists of functions $f:X\to \mathbb{K}$ that are non-zero only at finitely many points in $X$. Here, we denote the identity element^{} in $\mathbb{K}$ by $1$, and the zero element^{} by $0$. The vector space structure for $C(X)$ is defined as follows. If $f$ and $g$ are functions in $C(X)$, then $f+g$ is the mapping $x\mapsto f(x)+g(x)$. Similarly, if $f\in C(X)$ and $\alpha \in \mathbb{K}$, then $\alpha f$ is the mapping $x\mapsto \alpha f(x)$. It is not difficult to see that these operations are well defined, i.e., both $f+g$ and $\alpha f$ are again functions in $C(X)$.
0.0.1 Basis for $C(X)$
If $a\in X$, let us define the function ${\mathrm{\Delta}}_{a}\in C(X)$ by
${\mathrm{\Delta}}_{a}(x)$ | $=$ | $\{\begin{array}{cc}1\hfill & \text{when}x=a,\hfill \\ 0\hfill & \text{otherwise.}\hfill \end{array}$ |
These functions form a linearly independent^{} basis for $C(X)$, i.e.,
$C(X)$ | $=$ | $span{\{{\mathrm{\Delta}}_{a}\}}_{a\in X}.$ | (1) |
Here, the space $span{\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$ consists of all finite linear combinations^{} of elements in ${\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$. It is clear that any element in $span{\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$ is a member in $C(X)$. Let us check the other direction. Suppose $f$ is a member in $C(X)$. Then, let ${\xi}_{1},\mathrm{\dots},\xc3\mathrm{\x82}\xc2\mathrm{}{\xi}_{N}$ be the distinct points in $X$ where $f$ is non-zero. We then have
$f$ | $=$ | $\sum _{i=1}^{N}}f({\xi}_{i}){\mathrm{\Delta}}_{{\xi}_{i}},$ |
To see that the set ${\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$ is linearly independent, we need to show that its any finite subset is linearly independent. Let $\{{\mathrm{\Delta}}_{{\xi}_{1}},\mathrm{\dots},{\mathrm{\Delta}}_{{\xi}_{N}}\}$ be such a finite subset, and suppose ${\sum}_{i=1}^{N}{\alpha}_{i}{\mathrm{\Delta}}_{{\xi}_{i}}=0$ for some ${\alpha}_{i}\in \mathbb{K}$. Since the points ${\xi}_{i}$ are pairwise distinct, it follows that ${\alpha}_{i}=0$ for all $i$. This shows that the set ${\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$ is linearly independent.
Let us define the mapping $\iota :X\to C(X)$, $x\mapsto {\mathrm{\Delta}}_{x}$. This mapping gives a bijection between $X$ and the basis vectors ${\{{\mathrm{\Delta}}_{a}\}}_{a\in X}$. We can thus identify these spaces. Then $X$ becomes a linearly independent basis for $C(X)$.
0.0.2 Universal property of $\iota :X\to C(X)$
The mapping $\iota :X\to C(X)$ is universal in the following sense. If $\varphi $ is an arbitrary mapping from $X$ to a vector space $V$, then there exists a unique mapping $\overline{\varphi}$ such that the below diagram commutes:
$$\text{xymatrix}X\text{ar}{[r]}^{\varphi}\text{ar}{[d]}_{\iota}\mathrm{\&}VC(X)\text{ar}{[ur]}_{\overline{\varphi}}\mathrm{\&}$$ |
Proof. We define $\overline{\varphi}$ as the linear mapping that maps the basis elements of $C(X)$ as $\overline{\varphi}({\mathrm{\Delta}}_{x})=\varphi (x)$. Then, by definition, $\overline{\varphi}$ is linear. For uniqueness, suppose that there are linear mappings $\overline{\varphi},\overline{\sigma}:C(X)\to V$ such that $\varphi =\overline{\varphi}\circ \iota =\overline{\sigma}\circ \iota $. For all $x\in X$, we then have $\overline{\varphi}({\mathrm{\Delta}}_{x})=\overline{\sigma}({\mathrm{\Delta}}_{x})$. Thus $\overline{\varphi}=\overline{\sigma}$ since both mappings are linear and the coincide on the basis elements.$\mathrm{\square}$
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 |