description/proof of that composition of affine maps is affine map
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 map from affine set spanned by possibly-non-affine-independent set of base points on real vectors space.
- The reader knows a definition of affine map from convex set spanned by possibly-non-affine-independent set of base points on real vectors space.
- The reader admits any affine map from the affine or convex set spanned by any possibly-non-affine-independent set of base points on any real vectors space is linear.
Target Context
- The reader will have a description and a proof of the proposition that the composition of any affine maps from any affine or convex sets spanned by any possibly non-affine-independent set of base points on any real vectors spaces is an affine map.
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:
\(V_1\): \(\in \{\text{ the real vectors spaces }\}\)
\(V_2\): \(\in \{\text{ the real vectors spaces }\}\)
\(V_3\): \(\in \{\text{ the real vectors spaces }\}\)
\(S_1\): \(\in \{\text{ the affine or convex sets spanned by any possibly non-affine-independent set of base points on } V_1\}\), \(\subseteq V_1\)
\(S_2\): \(\in \{\text{ the affine or convex sets spanned by any possibly non-affine-independent set of base points on } V_2\}\), \(\subseteq V_2\)
\(f_1\): \(S_1 \to V_2\), \(\in \{\text{ the affine maps }\}\), such that \(f_1 (S_1) \subseteq S_2\)
\(f_2\): \(S_2 \to V_3\), \(\in \{\text{ the affine maps }\}\)
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Statements:
\(f_2 \circ f_1: S_1 \to V_3 \in \{\text{ the affine maps }\}\).
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2: Natural Language Description
For any real vectors spaces, \(V_1, V_2, V_3\), the affine or convex set spanned by any possibly non-affine-independent set of base points on \(V_1\), \(S_1 \subseteq V_1\), the affine or convex set spanned by any possibly non-affine-independent set of base points on \(V_2\), \(S_2 \subseteq V_2\), any affine map, \(f_1: S_1 \to V_2\), such that \(f_1 (S_1) \subseteq S_2\), and any affine map, \(f_2: S_2 \to V_3\), \(f_2 \circ f_1: S_1 \to V_3\) is an affine map.
3: Proof
Let \(b'_0, b'_1, ..., b'_k \in V_1\) be the affine-independent subset of the base points on \(V_1\) with which (the affine-independent subset) \(f_1\) is defined.
Let \(\sum_{j = 0 \sim k} t'^j b'_j \in S_1\) be any point. \(f_2 \circ f_1\) is \(\sum_{j = 0 \sim k} t'^j b'_j \mapsto f_1 (\sum_{j = 0 \sim k} t'^j b'_j) = \sum_{j = 0 \sim k} t'^j f_1 (b'_j) \mapsto f_2 (\sum_{j = 0 \sim k} t'^j f_1 (b'_j)) = \sum_{j = 0 \sim k} t'^j f_2 \circ f_1 (b'_j)\), because \(f_2\) is linear, by the proposition that any affine map from the affine or convex set spanned by any possibly non-affine-independent base points is linear. That exactly shows that \(f_2 \circ f_1\) is an affine map, according to the definition of affine map.
4: Note
Even when \(S_1\) or \(S_2\) is a convex (instead of affine) set, as any affine map from it is defined as the domain restriction of an affine map from the affine set spanned by the set of the base points, the logic works without any modification: although \(\sum_{j = 0 \sim k} t'^j b'_j\) is not with the conditions, \(0 \le t'^j\), the affine map is defined based on the expression.
