description/proof of that affine set spanned by non-affine-independent set of base points on real vectors space is affine set spanned by affine-independent subset of base points
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
- 4: Note
Starting Context
Target Context
- The reader will have a description and a proof of 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.
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:
\(V\): \(\in \text{ the real vectors spaces }\)
\(\{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the non-affine-independent sets of base points on } V\}\)
\(S\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\)
//
Statements:
\(\exists J \subset \{0, ..., n\}\)
(
\(\{p_j \vert j \in J\} \in \{\text{ the affine-independent sets of base points on } V\}\)
\(\land\)
\(S = \{\sum_{j \in J} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\)
)
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2: Natural Language Description
For any real vectors space, \(V\), and any non-affine-independent set of base points, \(p_0, ..., p_n \in V\), the affine set spanned by the set of the base points, \(S := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1\}\), is the affine set spanned by an affine-independent subset of the base points, \(\{p_j \vert j \in J\}\) where \(J \subset \{0, ..., n\}\).
3: Proof
There is a base point, \(p_k\), such that \(p_k - p_0 = \sum_{j \neq 0, k} t'^j (p_j - p_0)\) where \(t'^j \in \mathbb{R}\).
Any point on \(S\) is \(\sum_{j = 0 \sim n} t^j p_j = t^k p_k + \sum_{j \neq k} t^j p_j = t^k (p_0 + \sum_{j \neq 0, k} t'^j (p_j - p_0)) + \sum_{j \neq k} t^j p_j = (t^k - \sum_{j \neq 0, k} t^k t'^j + t^0) p_0 + \sum_{j \neq 0, k} (t^k t'^j + t^j) p_j\), which is a linear combination of the set of the base points except \(p_k\), which we will call the set of the reduced base points. The sum of the coefficients is \(t^k - \sum_{j \neq 0, k} t^k t'^j + t^0 + \sum_{j \neq 0, k} (t^k t'^j + t^j) = \sum t^j = 1\). So, it is an affine combination of the set of the reduced base points.
On the other hand, any affine combination of the set of the reduced base points, \(\sum_{j \neq k} t''^j p_j\), where \(\sum_{j \neq k } t''^j = 1\), is in \(S\), because it is just a special case of \(t''^k = 0\).
So, the affine set spanned by the set of the reduced base points, \(\{\sum_{j \neq k} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j \neq k} t^j = 1\}\), is nothing but \(S\).
If the set of the reduced base points is not affine independent, let us repeat the process, and eventually, the set of the reduced base points become affine independent, and \(S\) is the affine set spanned by the affine-independent subset of the base points.
Let us prove that \(S\) is indeed an affine subset of \(V\).
Let \(\sum_{j = 0 \sim n} t^j_1 p_j, \sum_{j = 0 \sim n} t^j_2 p_j \in S\) be any points. S's being affine is about that \(\sum_{j = 0 \sim n} t^j_1 p_j + t (\sum_{j = 0 \sim n} t^j_2 p_j - \sum_{j = 0 \sim n} t^j_1 p_j)\) is on \(S\) whenever \(t \in \mathbb{R}\).
\(\sum_{j = 0 \sim n} t^j_1 p_j + t (\sum_{j = 0 \sim n} t^j_2 p_j - \sum_{j = 0 \sim n} t^j_1 p_j) = \sum_{j = 0 \sim n} (t^j_1 (1 - t) + t t^j_2) p_j\). \(\sum_{j = 0 \sim n} (t^j_1 (1 - t) + t t^j_2) = \sum_{j = 0 \sim n} (t^j_1 (1 - t)) + \sum_{j = 0 \sim n} (t t^j_2) = (1 - t) \sum_{j = 0 \sim n} t^j_1 + t \sum_{j = 0 \sim n} t^j_2 = 1 - t + t = 1\).
So, \(\sum_{j = 0 \sim n} t^j_1 p_j + t (\sum_{j = 0 \sim n} t^j_2 p_j - \sum_{j = 0 \sim n} t^j_1 p_j) \in S\) whenever \(t \in \mathbb{R}\).
4: Note
The convex set spanned by a non-affine-independent set of base points is not necessarily any affine simplex spanned by an affine-independent subset of the base points, as is proved in 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.
