definition of 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
- The reader knows a definition of %field name% vectors space.
- The reader knows a definition of affine-independent set of points on real vectors space.
- The reader knows a definition of convex combination of possibly-non-affine-independent set of base points on real vectors space.
Target Context
- The reader will have a definition of 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the possibly-non-affine-independent sets of base points on } V\}\)
\(*S\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
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Conditions:
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2: Natural Language Description
For any real vectors space, \(V\), and any possibly-non-affine-independent set of base points, \(p_0, ..., p_n \in V\), the set, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\), which is the set of all the convex combinations of the set of the base points
3: Note
\(S\) is not necessarily any affine simplex spanned by an affine-independent subset of the base points, by 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.
But \(S\) is a convex set anyway, as is proved in the proposition that the convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is convex.
