definition of affine map from affine set spanned by possibly-non-affine-independent set of base points on real vectors space
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
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.
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.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V_1\): \(\in \{\text{ the real vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V_1\), \(\in \{\text{ the possibly-non-affine-independent sets of base points on } V_1\}\)
\( S\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\)
\(*f\): \(: S \to V_2\)
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Conditions:
For an affine-independent subset of the base points, \(\{p'_0, ..., p'_k\} \subseteq \{p_0, ..., p_n\}\), that spans \(S\), \(f: \sum_{j = 0 \sim k} t'^j p'_j \mapsto \sum_{j = 0 \sim k} t'^j f (p'_j)\), where each \(f (p'_j)\) 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, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V_1 \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\), any map, \(f: S \to V_2\), such that for an affine-independent subset of the base points, \(\{p'_0, ..., p'_k\} \subseteq \{p_0, ..., p_n\}\), that spans \(S\), \(f: \sum_{j = 0 \sim k} t'^j p'_j \mapsto \sum_{j = 0 \sim k} t'^j f (p'_j)\), where each \(f (p'_j)\) can be chosen arbitrarily
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, \((t'^1, ..., t'^k)\), 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 \(p_2 = p_0 + 2 (p_1 - p_0)\), \(f (p_2) = - f (p_0) + 2 f (p_1)\)) 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.
