definition of sub-'vectors space' generated by subset of vectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of sub-'vectors space' generated by subset 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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V'\): \(\in \{\text{ the vectors spaces over } F\}\)
\( S\): \(\subseteq V'\)
\(*(S)\): \(\in \{\text{ the sub-'vectors space's of } V'\}\)
//
Conditions:
\(S \subseteq (S)\)
\(\land\)
\(\lnot \exists V \in \{\text{ the sub-'vectors space's of } V'\} (S \subseteq V \subset (S))\)
//
In other words, \((S)\) is the smallest sub-'vectors space' that contains \(S\).
2: Note
\((S)\) is indeed well-defined (uniquely exists), because it is the intersection of all the sub-'vectors space's that contain \(S\): there is at least 1 sub-'vectors space' that contains \(S\), \(V'\); the intersection is indeed a smallest sub-'vectors space' that contains \(S\), because it contains \(S\), the intersection of any sub-'vectors space's is a sub-'vectors space', by the proposition that for any vectors space and its any set of sub-'vectors space's, the intersection of the set is a sub-'vectors space', and it is a smallest, because it is contained in any sub-'vectors space' that contains \(S\); the intersection is indeed the unique smallest sub-'vectors space' that contains \(S\), because if there was another smallest sub-'vectors space' that contained \(S\), it would be a constituent of the intersection, and the intersection would be smaller than or equal to that another smallest sub-'vectors space', and if the intersection was smaller than that another smallest sub-'vectors space', that another smallest sub-'vectors space' was not in fact smallest, and if the intersection was equal to that another smallest sub-'vectors space', that another smallest sub-'vectors space' was not in fact "another".
When \(S \neq \emptyset\), \((S)\) is the set of all the linear combinations of the elements of \(S\), which can be another definition of sub-'vectors space' generated by subset of vectors space.
That is because the set contains \(S\); each element of the set is contained in \((S)\), because \((S)\) is a vectors space; the set indeed constitutes a vectors space, because it is closed under linear combinations (any linear combination of any linear combinations of the elements of \(S\) is a linear combination of the elements of \(S\)), \(0\) is in it as a linear combination, the inverse element of each element of the set is contained in the set as a linear combination, and the other requirements like commutativity, associativity, distributability, e.t.c. hold because they hold in ambient \(V'\), and so, the set is a sub-'vectors space' that contains \(S\) that (the set) is smaller than or equal to \((S)\), but the set cannot be smaller because \((S)\) is the smallest, so, the set is equal to \((S)\).
When \(S = \emptyset\), \((S) = \{0\}\).
