definition of 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 %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 affine combination of possibly-non-affine-independent set of base points on real vectors space.
Target Context
- The reader will have a definition of 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:
\( V\): \(\in \text{ the real vectors spaces }\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the possibly-non-affine-independent sets of base points on } V\}\)
\(*S\): \(\{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\)
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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, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\), which is the set of all the affine combinations of the set of the base points
3: Note
\(S\) is the affine set spanned by an affine-independent subset of the base points, 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.
