1002: Sub-'Vectors Space' Generated by Subset of Vectors Space

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definition of sub-'vectors space' generated by subset of vectors space

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of %field name% vectors space.

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Target Context

• The reader will have a definition of sub-'vectors space' generated by subset of vectors space.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$V^{\prime }$: $\in \{\text{ the vectors spaces over }F\}$
$S$: $\subseteq V^{\prime }$
$\ast (S)$: $\in \{\text{ the sub-'vectors space's of }{V}^{\prime }\}$
//

Conditions:
$S\subseteq (S)$
$\wedge$
$\mathrm{\neg }\mathrm{\exists }V\in \{\text{ the sub-'vectors space's of }{V}^{\prime }\}(S\subseteq V\subset (S))$
//

In other words, $(S)$ is the smallest sub-'vectors space' that contains $S$.

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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 $S$, $V^{\prime }$; 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 $S\ne \mathrm{\varnothing }$, $(S)$ 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 $V^{\prime }$, 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 $S=\mathrm{\varnothing }$, $(S)=\{0\}$.

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References

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