567: For Affine Simplex and Ascending Sequence of Faces, Set of Barycenters of Faces Is Affine-Independent

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description/proof of that for affine simplex and ascending sequence of faces, set of barycenters of faces is affine-independent

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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: Proof

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Starting Context

• The reader knows a definition of affine simplex.
• The reader knows a definition of ascending sequence of faces of affine simplex.
• The reader knows a definition of barycenter of affine simplex.
• The reader admits the proposition that for any linearly independent finite subset of any module, the induced subset of the module with some linear combinations is linearly independent.

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Target Context

• The reader will have a description and a proof of the proposition that for any affine simplex and its any ascending sequence of faces, the set of the barycenters of the faces is affine-independent.

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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:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$[p_0,...,p_n]$: $=\text{ the affine simplex }$
$(p_0^{\prime },...,p_n^{\prime })$: $\in \text{ the sequences whose elements are }\{p_0,...,p_n\}$
$S$: $=([p_0^{\prime }],[p_0^{\prime },p_1^{\prime }],...,[p_0^{\prime },...,p_n^{\prime }])$
$S^{\prime }$: $=\{bary([p_0^{\prime }]),bary([p_0^{\prime },p_1^{\prime }]),...,bary([p_0^{\prime },...,p_n^{\prime }])\}$
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Statements:
$S^{\prime }\in \{\text{ the affine-independent sets of points on }V\}$.
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2: Natural Language Description

For any real vectors space, $V$, any affine-independent set of base points on $V$, $\{p_0,...,p_n\}$, the affine simplex, $[p_0,...,p_n]$, and any ascending sequence of faces of $[p_0,...,p_n]$, $S=([p_0^{\prime }],[p_0^{\prime },p_1^{\prime }],...,[p_0^{\prime },...,p_n^{\prime }])$, the set of the barycenters of the faces, $S^{\prime }=\{bary([p_0^{\prime }]),bary([p_0^{\prime },p_1^{\prime }]),...,bary([p_0^{\prime },...,p_n^{\prime }])\}$, is affine-independent. $$

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3: Proof

$bary([p_0^{\prime },...,p_m^{\prime }])=1/(m+1)(p_0^{\prime }+...+p_m^{\prime })$. Especially, $bary([p_0^{\prime }])=p_0^{\prime }$.

$bary([p_0^{\prime },...,p_m^{\prime }])-bary([p_0^{\prime }])=1/(m+1)(p_0^{\prime }+...+p_m^{\prime })-p_0^{\prime }=1/(m+1)(p_0^{\prime }+(p_1^{\prime }-p_0^{\prime })+p_0^{\prime }+...+(p_m^{\prime }-p_0^{\prime })+p_0^{\prime })-p_0^{\prime }=1/(m+1)((p_1^{\prime }-p_0^{\prime })+...+(p_m^{\prime }-p_0^{\prime })+(m+1)p_0^{\prime })-p_0^{\prime }=1/(m+1)((p_1^{\prime }-p_0^{\prime })+...+(p_m^{\prime }-p_0^{\prime }))+p_0^{\prime }-p_0^{\prime }=1/(m+1)((p_1^{\prime }-p_0^{\prime })+...+(p_m^{\prime }-p_0^{\prime }))$.

$\{bary([p_0^{\prime },p_1^{\prime }])-bary([p_0^{\prime }]),...,bary([p_0^{\prime },...,p_n^{\prime }])-bary([p_0^{\prime }])\}=\{1/2((p_1^{\prime }-p_0^{\prime })),1/3((p_1^{\prime }-p_0^{\prime })+(p_2^{\prime }-p_0^{\prime })),..,1/(n+1)((p_1^{\prime }-p_0^{\prime })+...+(p_n^{\prime }-p_0^{\prime }))\}$.

As $\{p_1^{\prime }-p_0^{\prime },p_2^{\prime }-p_0^{\prime },...,p_n^{\prime }-p_0^{\prime }\}$ is linearly independent, $\{1/2((p_1^{\prime }-p_0^{\prime })),1/3((p_1^{\prime }-p_0^{\prime })+(p_2^{\prime }-p_0^{\prime })),...,1/(n+1)((p_1^{\prime }-p_0^{\prime })+...+(p_n^{\prime }-p_0^{\prime }))\}$ is linearly independent, by the proposition that for any linearly independent finite subset of any module, the induced subset of the module with some linear combinations is linearly independent.

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References

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