definition of standard simplex
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
Starting Context
- The reader knows a definition of Euclidean vectors space.
- The reader knows a definition of affine simplex.
Target Context
- The reader will have a definition of standard simplex.
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:
\( \mathbb{R}^{n + 1}\): as the Euclidean vectors space
\( \{e_1, ..., e_{n + 1}\}\): \(\subseteq \mathbb{R}^{n + 1}\), where \(e_j\) is the unit vector for the \(j\)-th component of \(\mathbb{R}^{n + 1}\)
\(*\Delta^n\): \(= [e_1, ..., e_{n + 1}]\), which is the affine simplex with \(V = \mathbb{R}^{n + 1}\) and \(\{p_0, ..., p_n\} = \{e_1, ..., e_{n + 1}\}\)
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Conditions:
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2: Natural Language Description
For the Euclidean vectors space, \(\mathbb{R}^{n + 1}\), and the affine-independent set of base points, \(\{e_1, ..., e_{n + 1}\} \subseteq \mathbb{R}^{n + 1}\), where \(e_j\) is the unit vector for the \(j\)-th component of \(\mathbb{R}^{n + 1}\), the affine simplex, \([e_1, ..., e_{n + 1}]\), with \(V = \mathbb{R}^{n + 1}\) and \(\{p_0, ..., p_n\} = \{e_1, ..., e_{n + 1}\}\), denoted as \(\Delta^n\)
