definition of affine-independent set of 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 linearly independent subset of module.
Target Context
- The reader will have a definition of affine-independent set of 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:
\( V\): \(\in \text{ the real vectors spaces }\)
\(*\{p_0, ..., p_n\}\): \(\subseteq V\)
//
Conditions:
\(\forall j \in \{0, ..., n\} (\{p_0 - p_j, ..., \widehat{p_j - p_j}, ..., p_n - p_j\} \text{ is linearly independent })\), where the hat mark denotes that the component is missing.
//
2: Natural Language Description
For any real vectors space, \(V\), any set of points, \(\{p_0, ..., p_n\} \subseteq V\), such that for each \(p_j\), \(\{p_0 - p_j, ..., \widehat{p_j - p_j}, ..., p_n - p_j\}\) is linearly independent, where the hat mark denotes that the component is missing
3: Note
In fact, if the set is linearly independent for a \(p_j\), inevitably, the set is linearly independent for each \(p_j\), by the proposition that for any finite set of points on any real vectors space, if for one of the points, the set of the subtractions of the point from the other points is linearly independent, it is so for each point.
