definition of affine map from convex 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
Target Context
- The reader will have a definition of affine map from convex 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:
\( V_1\): \(\in \{\text{ the real vectors spaces }\}\)
\( V_2\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V_1\), \(\in \{\text{ the possibly-non-affine-independent sets of base points on } V_1\}\)
\( S\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
\(*f\): \(: S \to V_2\)
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Conditions:
\(f\) is the domain restriction of any affine map from the affine set spanned by the set of the base points.
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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 convex set spanned by the set of the base points, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\), any map, \(f: S \to V_2\), that is the domain restriction of any affine map from the affine set spanned by the set of the base points
3: Note
\(f\) cannot be defined as an affine map from the affine simplex spanned by an affine-independent subset of the base points, because generally, the affine simplex does not cover \(S\) (see 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).
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.
