1002: Vectors Subspace Generated by Subset of Vectors Space

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definition of vectors subspace 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 vectors subspace 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:
$F$: $\in \{\text{ the fields }\}$
$V^{\prime }$: $\in \{\text{ the vectors spaces over }F\}$
$S$: $\subseteq V^{\prime }$
$\ast (S)$: $\in \{\text{ the vectors subspaces of }{V}^{\prime }\}$
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Conditions:
$S\subseteq (S)$
$\wedge$
$\mathrm{\neg }\mathrm{\exists }V\in \{\text{ the vectors subspaces of }{V}^{\prime }\}(S\subseteq V\subset (S))$
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In other words, $(S)$ is the smallest vectors subspace that contains $S$.

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2: Note

$(S)$ is indeed well-defined (uniquely exists), because it is the intersection of all the vectors subspaces that contain $S$: there is at least 1 vectors subspace that contains $S$, $V^{\prime }$; the intersection is indeed a smallest vectors subspace that contains $S$, because it contains $S$, the intersection of any vectors subspaces is a vectors subspace, by the proposition that for any vectors space and its any set of vectors subspaces, the intersection of the set is a vectors subspace, and it is a smallest, because it is contained in any vectors subspace that contains $S$; the intersection is indeed the unique smallest vectors subspace that contains $S$, because if there was another smallest vectors subspace that contained $S$, it would be a constituent of the intersection, and the intersection would be smaller than or equal to that another smallest vectors subspace, and if the intersection was smaller than that another smallest vectors subspace, that another smallest vectors subspace was not in fact smallest, and if the intersection was equal to that another smallest vectors subspace, that another smallest vectors subspace 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 vectors subspace 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 vectors subspace 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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