|
|
|
|
free vector space over a set
|
(Definition)
|
|
|
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 $\sK$ , we define \begin{eqnarray*} C(X) &=& \{ f:X\to \sK\,\, |\,\, f^{-1}(\sK\backslash\{0\}) \, \mbox{is finite} \}. \end{eqnarray*}In other words, $C(X)$ consists of functions $f:X\to \sK$ that are non-zero only at finitely many points in $X$ . Here, we denote the identity element in $\sK$ 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 $\alpha \in \sK$ , 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 $C(X)$ .
If $a\in X$ , let us define the function $\Delta_a \in C(X)$ by \begin{eqnarray*} \Delta_a(x)&=& \left\{ \begin {array}{ll} 1 & \mbox{when} \, x=a, \\ 0 & \mbox{otherwise.} \\ \end{array}. These functions form a linearly independent basis for $C(X)$ , i.e., \begin{eqnarray} \label{basiseq} C(X) &=& \lsp\{ \Delta_a\}_{a\in X}. \end{eqnarray}Here, the space $\lsp\{ \Delta_a\}_{a\in X}$ consists of all finite linear
combinations of elements in $\{ \Delta_a\}_{a\in X}$ . It is clear that any element in $\lsp\{ \Delta_a\}_{a\in X}$ is a member in $C(X)$ . Let us check the other direction. Suppose $f$ is a member in $C(X)$ . Then, let $\xi_1, \ldots, \xi_N$ be the distinct points in $X$ where $f$ is non-zero. We then have \begin{eqnarray*} f&=&\sum_{i=1}^Nf(\xi_i) \Delta_{\xi_i}, \end{eqnarray*}and we have established equality in equation ![[*]](http://images.planetmath.org:8080/cache/objects/4196/js//usr/share/latex2html/icons/crossref.png "[*]") .
To see that the set $\{ \Delta_a\}_{a\in X}$ is linearly independent, we need to show that its any finite subset is linearly independent. Let $\{ \Delta_{\xi_1}, \ldots, \Delta_{\xi_N} \}$ be such a finite subset, and suppose $\sum_{i=1}^N \alpha_i \Delta_{\xi_i }=0$ for some $\alpha_i \in \sK$ . Since the points $\xi_i$ are pairwise distinct, it follows that $\alpha_i=0$ for all $i$ . This shows that the set $\{ \Delta_a\}_{a\in X}$ is linearly independent.
Let us define the mapping $\iota:X\to C(X)$ , $x\mapsto \Delta_x$ . This mapping gives a bijection between $X$ and the basis vectors $\{ \Delta_a\}_{a\in X}$ . We can thus identify these spaces. Then $X$ becomes a linearly independent basis for $C(X)$ .
The mapping $\iota:X\to C(X)$ is universal in the following sense. If $\phi$ is an arbitrary mapping from $X$ to a vector space $V$ , then there exists a unique mapping $\bar{\phi}$ such that the below diagram commutes:
Proof. We define $\bar{\phi}$ as the linear mapping that maps the basis elements of $C(X)$ as $\bar{\phi}(\Delta_x) = \phi(x)$ . Then, by definition, $\bar{\phi}$ is linear. For uniqueness, suppose that there are linear mappings $\bar{\phi},\bar{\sigma}:C(X)\to V$ such that $\phi=\bar{\phi}\circ \iota =\bar{\sigma}\circ \iota$ . For all $x\in X$ , we then have $\bar{\phi}(\Delta_x)=\bar{\sigma}(\Delta_x)$ . Thus $\bar{\phi}=\bar{\sigma}$ since both mappings are linear and the coincide on the basis elements.
- 1
- W. Greub, Linear Algebra, Springer-Verlag, Fourth edition, 1975.
- 2
- I. Madsen, J. Tornehave, From Calculus to Cohomology, Cambridge University press, 1997.
|
"free vector space over a set" is owned by mathcam. [ owner history (1) ]
|
|
(view preamble | get metadata)
Cross-references: linear mapping, proof, diagram, universal, vectors, bijection, subset, equation, equality, clear, linear combinations, finite, basis, linearly independent, well defined, operations, mapping, structure, zero element, identity element, points, functions, field, modules, tensor product, application, vector space
There is 1 reference to this entry.
This is version 5 of free vector space over a set, born on 2003-04-20, modified 2003-07-07.
Object id is 4196, canonical name is FreeVectorSpaceOverASet.
Accessed 8116 times total.
Classification:
| AMS MSC: | 15-00 (Linear and multilinear algebra; matrix theory :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![$\displaystyle \xymatrix{ X \ar[r]^\phi\ar[d]_\iota & V \ C(X) \ar[ur]_{\bar{\phi}} & } $](http://images.planetmath.org:8080/cache/objects/4196/js/img1.png "$\displaystyle \xymatrix{ X \ar[r]^\phi\ar[d]_\iota & V \ C(X) \ar[ur]_{\bar{\phi}} & } $")