538: Convex Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

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definition of convex set spanned by possibly-non-affine-independent set of base points on real 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: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of %field name% vectors space.
• The reader knows a definition of affine-independent set of points on real vectors space.
• The reader knows a definition of convex combination of possibly-non-affine-independent set of base points on real vectors space.

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Target Context

• The reader will have a definition of convex set spanned by possibly-non-affine-independent set of base points on real 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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the possibly-non-affine-independent sets of base points on }V\}$
$\ast S$: $=\{\sum _{j=0\sim n}{t}^j{p}_j\in V|t^j\in \mathbb{R},\sum _{j=0\sim n}{t}^j=1\wedge 0\le t^j\}$
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Conditions:
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2: Natural Language Description

For any real vectors space, $V$, and any possibly-non-affine-independent set of base points, $p_0,...,p_n\in V$, the set, $S:=\{\sum _{j=0\sim n}{t}^j{p}_j\in V|t^j\in \mathbb{R},\sum _{j=0\sim n}{t}^j=1\wedge 0\le t^j\}$, which is the set of all the convex combinations of the set of the base points

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3: Note

$S$ is not necessarily any affine simplex spanned by an affine-independent subset of the base points, by the proposition that the convex set spanned by a non-affine-independent set of base points on a real vectors space is not necessarily any affine simplex spanned by an affine-independent subset of the base points.

But $S$ is a convex set anyway, as is proved in the proposition that the convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is convex.

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References

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