description/proof of that subset of affine-independent set of points on real vectors space is affine-independent
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: Proof
Starting Context
- The reader knows a definition of affine-independent set of points on real vectors space.
Target Context
- The reader will have a description and a proof of the proposition that any subset of any affine-independent set of points on any real vectors space is affine-independent.
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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(\{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of points on } V\}\)
\(\{p'_0, ..., p'_m\}\): \(\subseteq \{p_0, ..., p_n\}\)
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Statements:
\(\{p'_0, ..., p'_m\} \in \{\text{ the affine-independent sets of points on } V\}\).
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2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of points on, \(V\), \(\{p_0, ..., p_n\}\), any subset, \(\{p'_0, ..., p'_m\} \subseteq \{p_0, ..., p_n\}\), is affine-independent. \(\)
3: Proof
\(p'_0 = p_k\) for a \(k \in \{0, ..., n\}\).
\(\{p_0 - p_k, ..., \widehat{p_k - p_k}, ..., p_n - p_k\}\) is linearly independent, where the hat mark denotes that the element is missing.
\(\{p'_1 - p'_0, ..., p'_m - p'_0\}\) is a subset of \(\{p_0 - p_k, ..., \widehat{p_k - p_k}, ..., p_n - p_k\}\), and so, is linearly independent, so, \(\{p'_0, ..., p'_m\}\) is affine-independent.
