definition of dimension of vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of basis of module.
- The reader admits the proposition that any vectors space has a basis.
- The reader admits the proposition that the class of the ordinal numbers is well-ordered.
Target Context
- The reader will have a definition of dimension of vectors space.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( B'\): \(= \{\text{ the bases of } V\}\)
\(*min (\{\vert B \vert \vert B \in B'\})\): where \(\vert B \vert\) denote the cardinality of \(B\)
//
Conditions:
//
\(B' \neq \emptyset\), by the proposition that any vectors space has a basis.
\(min (\{\vert B \vert \vert B \in B'\})\) is well-defined even if there were some different cardinalities, by the proposition that the class of the ordinal numbers is well-ordered, while any cardinal number is an ordinal number.
2: Natural Language Description
For any field, \(F\), any \(F\) vectors space, \(V\), and the set of the bases of \(V\), \(B'\), \(min (\{\vert B \vert \vert B \in B'\})\), where \(\vert B \vert\) denote the cardinality of \(B\)
3: Note
This definition does not suppose that all the bases have the same cardinality, but at least when the dimension is finite, that has been proved to be true by the proposition that for any finite-dimensional vectors space, there is no basis that has more than the dimension number of elements.
