726: For Vectors Space and Linearly Independent Subset, Subset Can Be Expanded to Be Basis

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description/proof of that for vectors space and linearly independent subset, subset can be expanded to be basis

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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: Natural Language Description
• 3: Note
• 4: Proof

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Starting Context

• The reader knows a definition of basis of module.
• The reader admits the proposition that for any vectors space, any generator of the space, and any linearly independent subset contained in the generator, the generator can be reduced to be a basis with the linearly independent subset retained.

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Target Context

• The reader will have a description and a proof of the proposition that for any vectors space and any linearly independent subset, the subset can be expanded to be a basis.

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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:
$F$: $\in \{\text{ the fields }\}$
$V$: $\in \{\text{ the }F\text{ vectors spaces }\}$
$S$: $\in \{\text{ the linearly independent subsets of }V\}$
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Statements:
$\mathrm{\exists }B\subseteq V(S\subseteq B\wedge B\in \{\text{ the bases of }V\})$
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2: Natural Language Description

For any field, $F$, any $F$ vectors space, $V$, and any linearly independent subset, $S\subseteq V$, there is a basis, $B\subseteq V$, such that $S\subseteq B$.

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3: Note

$V$ does not need to be finite-dimensional.

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4: Proof

Whole Strategy: Step 1: apply the proposition that for any vectors space, any generator of the space, and any linearly independent subset contained in the generator, the generator can be reduced to be a basis with the linearly independent subset retained with $V$ as the generator.

Step 1:

$V$ is a generator of $V$.

By the proposition that for any vectors space, any generator of the space, and any linearly independent subset contained in the generator, the generator can be reduced to be a basis with the linearly independent subset retained with $V$ as the generator, there is a basis of $V$, $B$, such that $S\subseteq B\subseteq V$.

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References

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