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description/proof of that affine subset of finite-dimensional real vectors space is spanned by finite affine-independent set of base points
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description/proof of that affine map from affine or convex set spanned by possibly-non-affine-independent set of base points on real vectors space is linear
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definition of face of orientated affine simplex
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About:
vectors space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of face of orientated affine 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( (p_0, ..., p_n)\): \(= \text{ the orientated affine simplex }\)
\(*face_j ((p_0, ..., p_n))\): \(= (-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\), where the hat mark denotes that the element is missing
//
Conditions:
//
\(face_j ((p_0, ..., p_n))\) is called \(j\)-th face of \((p_0, ..., p_n)\).
As \((p_0, ..., p_n) = (\sigma_0, ..., \sigma_n)\) for any even-parity permutation, \(\sigma\), of \((p_0, ..., p_n)\), the \(j\)-th face of \((p_0, ..., p_n)\) is not necessarily the \(j\)-th face of \((\sigma_0, ..., \sigma_n)\) (it cannot be when \(p_j \neq \sigma_j\)), but \(p_j = \sigma_k\) for a \(k\), and in fact, \(face_j ((p_0, ..., p_n)) = face_k ((\sigma_0, ..., \sigma_n))\) (proved in Note). So, although the term, "\(j\)-th face", requires the specification of the representation which \(j\) refers to, the face with the removed base point specified does not depend on the representation.
2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), and the orientated affine simplex, \((p_0, ..., p_n)\), the \(j\)-th face of \((p_0, ..., p_n)\), \(face_j ((p_0, ..., p_n))\) is \((-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\)
3: Note
Let us prove that when \((p_0, ..., p_n) = (\sigma_0, ..., \sigma_n)\), \(face_j ((p_0, ..., p_n)) = face_k ((\sigma_0, ..., \sigma_n))\) where \(p_j = \sigma_k\).
Let us suppose that \(j \le k\) without loss of generality.
\((p_0, ..., p_n)\) is permutated by the \(k - j\) switches such that \(p_j\) is at the \(k\)-th position: \(p_j\) and \(p_{j + 1}\) are switched; then, \(p_j\) and \(p_{j + 2}\) are switched; ...; then, \(p_j\) and \(p_{k}\) are switched. The result is permutated by some \(x\) switches to become \((\sigma_0, ..., \sigma_n)\): as \(p_j\) was already at the \(k\)-th position, it is permutating only the other points. \(k - j + x\) is even, \(2 m\), because \((p_0, ..., p_n)\) and \((\sigma_0, ..., \sigma_n)\) have the same parity. But \((p_0, ..., \hat{p_j}, ..., p_n)\) is permutated to become \((\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n)\) by the \(x\) switches, obviously, which means that \((\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^x (p_0, ..., \hat{p_j}, ..., p_n)\), which means that \((-1)^j (-1)^x (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (-1)^x (-1)^x (p_0, ..., \hat{p_j}, ..., p_n)\), which means that \((-1)^{j + x} (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\), but \(k + j + x = k - j + x + 2 j = 2 m + 2 j\), which means that when \(j + x\) is even, \(k\) is even; when \(j + x\) is odd, \(k\) is odd, which means that \((-1)^{j + x} = (-1)^k\), and so, \((-1)^k (\sigma_0, ..., \hat{\sigma_k}, ..., \sigma_n) = (-1)^j (p_0, ..., \hat{p_j}, ..., p_n)\).
We said that \(j \le k\) is without loss of generality, because otherwise, \((\sigma_0, ..., \sigma_n)\) can be permutated to \((p_0, ..., p_n)\) instead.
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definition of orientated affine simplex
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About:
vectors space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of orientated affine 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\(*(p_0, ..., p_n)\): \(= [p_0, ..., p_n]\) with the parity of the order of \((p_0, ..., p_n)\)
//
Conditions:
//
\(- (p_0, ..., p_n)\) is \([p_0, ..., p_n]\) with the opposite of the parity of the order of \((p_0, ..., p_n)\).
2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), the affine simplex, \([p_0, ..., p_n]\), with the parity of the order of \((p_0, ..., p_n)\)
\(- (p_0, ..., p_n)\) is \([p_0, ..., p_n]\) with the opposite of the parity of the order of \((p_0, ..., p_n)\).
3: Note
For example, \((p_0, p_1, p_2) = (p_1, p_2, p_0) = (p_2, p_0, p_1)\) and \((p_0, p_2, p_1) = (p_1, p_0, p_2) = (p_2, p_1, p_0)\), but \((p_0, p_1, p_2) \neq (p_0, p_2, p_1)\), etc. and \((p_0, p_1, p_2) = - (p_0, p_2, p_1)\), etc..
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definition of standard simplex
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vectors space
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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 \(\{p_0, ..., p_n\} = \{e_1, ..., e_{n + 1}\}\)
//
Conditions:
//
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 \(\Delta^n\)
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description/proof of that convex set spanned by possibly-non-affine-independent set of base points on real vectors space is convex
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description/proof of that convex set spanned by non-affine-independent set of base points on real vectors space is not necessarily affine simplex spanned by affine-independent subset of base points
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definition of affine simplex
Topics
About:
vectors space
The table of contents of this article
Starting Context
Target Context
-
The reader will have a definition of affine 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\(*[p_0, ..., p_n]\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
//
Conditions:
//
\([p_0, p_1, ..., p_n]\) is called affine n-simplex.
2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), the set, \([p_0, ..., p_n] := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
\([p_0, ..., p_n]\) is called affine n-simplex.
3: Note
While it is usually defined with \(V = \mathbb{R}^d\), the Euclidean vectors space with the Euclidean topology, this definition made the more general definition with any general real vectors space. There is no critical difference between them, because with any \(d\)-dimensional \(V\) equipped with the canonical topology, \(V\) is homeomorphic to \(\mathbb{R}^d\), and the subspace, \([p_0, ..., p_n] \subseteq V\), is homeomorphic to the subspace, \([p_0, ..., p_n] \subseteq \mathbb{R}^d\).
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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
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description/proof of that determinant of square matrix whose last row is all 1 and whose each other row is all 0 except row number + 1 column 1 is -1 to power of dimension + 1
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About:
matrix
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Starting Context
Target Context
-
The reader will have a description and a proof of the proposition that the determinant of any square matrix whose last row is all \(1\) and whose each other row is all \(0\) except the row number \(+ 1\) column \(1\) is \(-1\) to the power of the dimension \(+ 1\).
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:
\(M\): \(\in \{\text{ the n x n matrices }\}\), the last row is all \(1\) and each other \(j\)-th row is \(0\) except the \(j + 1\)-th column \(1\)
//
Statements:
\(det M = (-1)^{n + 1}\).
//
2: Natural Language Description
For any \(\text{ n x n }\) matrix, \(M\), whose last row is all \(1\) and whose each other \(j\)-th row is \(0\) except the \(j + 1\)-th column \(1\), the determinant is \(det M = (-1)^{n + 1}\).
3: Proof
\(M = \begin{pmatrix} 0 & 1 & 0 & 0 & ... & 0 \\ 0 & 0 & 1 & 0 & ... & 0 \\ ... \\ 1 & 1 & 1 & 1 & ... & 1 \end{pmatrix}\).
Let us prove it inductively.
Let the determinant of the \(n\)-dimensional matrix be \(f (n)\).
When \(n = 1\), \(M = \begin{pmatrix} 1 \end{pmatrix}\), and \(f (1) = det M = 1 = (-1)^{1 + 1}\).
When \(n = 2\), \(M = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}\), and \(f (2) = det M = -1 = (-1)^{2 + 1}\).
Let us suppose that for \(n = m - 1\), \(det M = (-1)^{m - 1 + 1}\). For \(n = m\), let us expand \(det M\) by the Laplace expansion of determinant by the 1st row. The 1st row has only 1 nonzero column, the 2nd column, 1. The cofactor of the component is \((-1)^{1 + 2} f (m - 1)\), because \(M\) with the 1st row and the 2nd column removed is nothing but the matrix for the case, \(m - 1\). So, \(f (m) = -1 f (m - 1)\).
So, \(f (n) = (-1)^{n + 1}\).
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