每日彙整: 2017-08-01

樹莓派、樹莓派之學習、樹莓派之教育

GoPiGo 小汽車︰格點圖像算術《投影幾何》【四‧平面國】《丙》

佛說四十二章經之

第十八章 念等本空

佛言。吾法念無念念，行無行行。言無言言。修無修修。會者近爾 。迷者遠乎。言語道斷。非物所拘。差之毫釐。失之須臾 。

 述說著什麼呢？『言無言言』而已矣夫？？

所謂『言語道斷』

是講用『語言文字』來『表述真理』是『可能的』嗎？如果將一個人的『整體認知』編碼成『概念符號系統』，那麼『認知』概念之『認知』怎能『不循環』指稱的呢？？無論訴 諸於『什麼』 ── 『理性』、『本能』、『人性』…『天性』 ── ，終究是將『此』歸之於『彼』而已，從『邏輯推演』上講，『套套邏輯』依然是『套套邏輯』，甚至這個『理則』是『經驗的』還是『自明的』，都尚且陷在『爭 論』之 泥沼裡的哩！更不要說讀過『荷蘭人之書』的人︰

也許二十世紀初，『量子力學』的『機率問題』的推波助瀾，更加深了所謂的『事件』之『觀察者』是『主觀』的或者是『客觀』的 ，之天下『大哉辯』！因而『機率』到底是『什麼』？就會是人們不得不『思考』與『面對』之問題的了！！

因此在一個『相容自洽』的『概念系統』中，『概念』 C_{\bigcirc} 如果不是『公設的』、『自明的』、『先驗的』…，『它』就應該能從那些『初始』的概念 C_{\Box}_1, \ C_{\Box}_2, \ C_{\Box}_3, \ \dots，演繹推理『得到』。『它』就得面對是否『☆』與『★』才是更『根本』的『原故』、『存在』、…… 『如來』？？

 

或將可守『念無念念』之法！終能了『非物所拘。差之毫釐。失之須臾 。』耶！！

【大同而與小同異，此之謂小同異；萬物畢同畢異，此之謂大同異。 】

若是我們『研究』自然中『發生』『事物』的『同異』，那麼我們將會『認為』『魚上陸地』與『鳥飛上天』是『變化大者』之『大』『變』的嗎？或者說『用進廢退』來說就是『相較之小』之『小』『變』的呢？也許無須『好辯哉』，也不必作『大哉辨』，『自然』自有其『道理』的啊！所以人才會問著『世間』果真有『道德律』的嗎？？

假使我們思考『如是』的『形式數學』方程式

如果 X \neq F ，這稱之為 F 之『定點』

F(F(F\cdots F(X))) = F^{x}(X) = X ；

要是用於 X = F ，此表達是 F 的『自身』

\forall x, \ F(F(F\cdots F(F))) = F^{x}(F) = F ；

這時 F 只能是『恆等函數』嗎？那能帶給我們什麼『啟示』呢？？

或許當真那位證明了『哥德爾定理』的人，善於引爆『邏輯炸彈』，於是乎︰

一九四九年，這位『庫爾特‧哥德爾』 Kurt Gödel 開啟一條『封閉類時曲線』 CTC closed timelike curve 的研究風潮，這是一種『特殊的世界線』，它『首尾相環』，於是乎，『未來』可以通向『過去』。

220px-Black_Swans
可否證性
Are all swans white?

200px-Leprechaun_ill_artlibre_jnl
空洞假設
If someone wants to believe in leprechauns, they can avoid ever being proven wrong by using ad hoc hypotheses (l.g. by adding “they are invisible”, then “their motives are complex”, and so on).

還是那不說『如之何』、『如之何』者，更加令人不知『如之何』的哩！！

─── 摘自《佛說四十二章經》

 

這『會者近爾。迷者遠乎。』，往往容易引發『學習者』之『邏輯炸彈』，通常恐怕含糊帶過的吧★

就像追問『點投派』者有疑焉︰

假使 P 是『點投派』一『投影函數』，滿足

P \left( \begin{array}{cc} x \\ y \end{array} \right) = \left( \begin{array}{cc} x/y \\ 1 \end{array} \right)

且  P^2 = P 後，

劍指

A := \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)

之罔然也︰

Basic definitions

Definition 1.1   The is the set of all lines in passing through the origin.

Note that if and are two nonzero vectors lying on the same line through the origin, then they are multiples of each other. This remark allows us to redefine as the quotient of by the equivalence relation (or ) if and are multiples of each other.

Exercise 1.6 (00)   Let be the slope of the line passing through the origin and let

be an invertible matrix. Verify that the slope of the line equals .

This exercise tells us that in suitable coordinates projective transformations have the form .

As explained in [2], projective geometry arose from the artists’ needs to represent the three-dimensional world on a two-dimensional canvas. An artist in Flatland just has to worry about representing a two-dimensional world on a one-dimensional canvas. The figure below shows a Flatland artist copying a one-dimensional image onto a canvas. Notice how distances are distorted.

This type of correspondence between the points in two lines is called a perspective.

─ 摘自《GoPiGo 小汽車︰格點圖像算術《投影幾何》【三下】》

 

因為依據『行列式性質』︰

Properties of the determinant

The determinant has many properties. Some basic properties of determinants are

  1.   \det(I_{n})=1 where In is the n × n identity matrix.
  2. \det(A^{\rm {T}})=\det(A).
  3. \det(A^{-1})={\frac {1}{\det(A)}}=\det(A)^{-1}.
  4. For square matrices A and B of equal size,
\det(AB)=\det(A)\det(B).
  1. \det(cA)=c^{n}\det(A) for an n × n matrix.
  2. If A is a triangular matrix, i.e. ai,j = 0 whenever i > j or, alternatively, whenever i < j, then its determinant equals the product of the diagonal entries:
\det(A)=a_{1,1}a_{2,2}\cdots a_{n,n}=\prod _{i=1}^{n}a_{i,i}.

遵循『逆矩陣規矩』︰

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that

  \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n}\

where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A and is called the inverse of A, denoted by A−1.

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that a square matrix randomly selected from a continuous uniform distribution on its entries will almost never be singular.

Non-square matrices (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such a matrix may have a left inverse or right inverse. If A is m-by-n and the rank of A is equal to n, then A has a left inverse: an n-by-m matrix B such that BA = In. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = Im.

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. However, in the case of the ring being commutative, the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than being nonzero. For a noncommutative ring, the usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated since a notion of rank does not exist over rings.

The set of n × n invertible matrices together with the operation of matrix multiplication form a group, the general linear group of degree n.

 

豈能不是

A^{-1} \cdot A^2 = A^{-1} \cdot A = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)

哩◎