544: Standard Simplex

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definition of standard simplex

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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

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Starting Context

• The reader knows a definition of Euclidean vectors space.
• The reader knows a definition of affine simplex.

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Target Context

• The reader will have a definition of standard simplex.

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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:
${\mathbb{R}}^{n+1}$: as the Euclidean vectors space
$\{e_1,...,e_{n+1}\}$: $\subseteq {\mathbb{R}}^{n+1}$, where $e_j$ is the unit vector for the $j$-th component of ${\mathbb{R}}^{n+1}$
$\ast {\mathrm{\Delta }}^n$: $=[e_1,...,e_{n+1}]$, which is the affine simplex with $V={\mathbb{R}}^{n+1}$ and $\{p_0,...,p_n\}=\{e_1,...,e_{n+1}\}$
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Conditions:
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2: Natural Language Description

For the Euclidean vectors space, ${\mathbb{R}}^{n+1}$, and the affine-independent set of base points, $\{e_1,...,e_{n+1}\}\subseteq {\mathbb{R}}^{n+1}$, where $e_j$ is the unit vector for the $j$-th component of ${\mathbb{R}}^{n+1}$, the affine simplex, $[e_1,...,e_{n+1}]$, with $V={\mathbb{R}}^{n+1}$ and $\{p_0,...,p_n\}=\{e_1,...,e_{n+1}\}$, denoted as ${\mathrm{\Delta }}^n$

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References

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