547: Affine Map from Affine Set Spanned by Possibly-Non-Affine-Independent Set of Base Points on Real Vectors Space

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definition of affine map from affine 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 affine set spanned by possibly-non-affine-independent set of base points on real vectors space.
• The reader knows a definition of affine-independent set of points on real vectors space.
• The reader admits the proposition that the affine set spanned by any non-affine-independent set of base points on any real vectors space is the affine set spanned by an affine-independent subset of the base points.

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Target Context

• The reader will have a definition of affine map from affine 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:
$V_1$: $\in \{\text{ the real vectors spaces }\}$
$V_2$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V_1$, $\in \{\text{ the possibly-non-affine-independent sets of base points on }{V}_1\}$
$S$: $=\{\sum _{j=0\sim n}{t}^j{p}_j\in V|t^j\in \mathbb{R},\sum _{j=0\sim n}{t}^j=1\}$
$\ast f$: $:S\to V_2$
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Conditions:
For an affine-independent subset of the base points, $\{p_0^{\prime },...,p_k^{\prime }\}\subseteq \{p_0,...,p_n\}$, that spans $S$, $f:\sum _{j=0\sim k}{t}^{\prime j}{p}_j^{\prime }\mapsto \sum _{j=0\sim k}{t}^{\prime j}f(p_j^{\prime })$, where each $f(p_j^{\prime })$ can be chosen arbitrarily.
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2: Natural Language Description

For any real vectors spaces, $V_1,V_2$, any possibly-non-affine-independent set of base points, $\{p_0,...,p_n\}\subseteq V_1$, and the affine set spanned by the set of the base points, $S:=\{\sum _{j=0\sim n}{t}^j{p}_j\in V_1|t^j\in \mathbb{R},\sum _{j=0\sim n}{t}^j=1\}$, any map, $f:S\to V_2$, such that for an affine-independent subset of the base points, $\{p_0^{\prime },...,p_k^{\prime }\}\subseteq \{p_0,...,p_n\}$, that spans $S$, $f:\sum _{j=0\sim k}{t}^{\prime j}{p}_j^{\prime }\mapsto \sum _{j=0\sim k}{t}^{\prime j}f(p_j^{\prime })$, where each $f(p_j^{\prime })$ can be chosen arbitrarily

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3: Note

Such an affine-independent subset of the base points always exists, by the proposition that the affine set spanned by any non-affine-independent set of base points on any real vectors space is the affine set spanned by an affine-independent subset of the base points.

$f$ is well-defined, because the coefficients, $(t^{\prime 1},...,t^{\prime k})$, are uniquely determined for each point on $S$ once the subset is determined.

$f$ cannot be defined based on the original base points like that, because $f(p_j)$ s cannot be chosen arbitrarily (for example, when $p_2=p_0+2(p_1-p_0)$, $f(p_2)=-f(p_0)+2f(p_1)$) and the coefficients are not uniquely determined.

But still, $f$ is linear with respect to the base points, by the proposition that any affine map from the affine or convex set spanned by any possibly-non-affine-independent base points is linear.

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References

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