2024-07-14

674: Dimension of Vectors Space

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definition of dimension of vectors space

Topics


About: vectors space

The table of contents of this article


Starting Context



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 in fact, they have the same cardinality, by the proposition that for any vectors space, the bases have the same cardinality.


References


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