如何『轉變』 Transform 想法呢？比方求 。假使先計算 再除以 ，恐怕使用『全球計算機網路』，還得計算很久哩！要是『轉換』觀點，借著『模算數』之『定理』︰

。

那麼知道

的人就可以利用

得知也！！

此所以天下『觀物』與『物觀』之觀點，尚得能有『形變』 Transformation 之『心』乎？？

在《一個奇想！！》一文中，提及『計算機種子』 Lick 一張標題為『Members and Affiliates of the Intergalactic Computer Network』給工作同仁的備忘錄︰
“imagined as an electronic commons open to all, ‘the main and essential medium of informational interaction for governments, institutions, corporations, and individuals.’”

許下了『銀河際網路』的願景，開啟了今天的『網際網路』！！

─── 摘自《【Sonic π】電聲學之電路學《四》之《 !!! 》》

 

如是『   』尺寸無涉？祇靠幾何概念！

※ 此處 都是本地座標系，皆如是賦值︰

。

 

又將通往哪裡矣！！

如果 將 投射到 ，且 將 投射到

 

 

此時若講

。

 

，當下是否可證『 f∘g 』 必滿足 組合耶◎

Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function which maps x in X to g(f(x)) in Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.^{[note 1]} The notation g ∘ f is read as “g circle f “, or “g round f “, or “g composed with f “, “g after f “, “g following f “, or “g of f“, or “g on f “. Intuitively, composing two functions is a chaining process in which the output of the inner function becomes the input of the outer function.

The composition of functions is a special case of the composition of relations, so all properties of the latter are true of composition of functions.^[1] The composition of functions has some additional properties.

 

Concrete example for the composition of two functions.

Composition monoids

Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these are often called transformations. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom) composition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup^[3] or symmetric semigroup^[4] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.^[5])

If the transformation are bijective (and thus invertible), then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. A fundamental result in group theory, Cayley’s theorem, essentially says that any group is in fact just a subgroup of a permutation group (up to isomorphism).^[6]

The set of all bijective functions f: X → X (called permutations) forms a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group.

In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a regular semigroup.^[7]

The similarity that transforms triangle EFA into triangle ATB is the composition of a homothety H  and a rotation R, of which the common centre is S.  For example, the image of A  under the rotation R is U,  which may be written  R (A) = U.  And  H(U) = B  means that the mapping H transforms U  into B.  Thus  H(R (A)) = (H ∘ R )(A) = B.

 

※ 參考

pi@raspberrypi:~ $ ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: from sympy import *

In [2]: init_printing()

In [3]: k1, k2 = symbols('k1, k2')

In [4]: f = Matrix(([1,0],[1 - 1/k1 , 1/k1]))

In [5]: f
Out[5]: 
⎡  1     0 ⎤
⎢          ⎥
⎢    1   1 ⎥
⎢1 - ──  ──⎥
⎣    k₁  k₁⎦

In [6]: g = Matrix(([1,0],[1 - 1/k2 , 1/k2]))

In [7]: g
Out[7]: 
⎡  1     0 ⎤
⎢          ⎥
⎢    1   1 ⎥
⎢1 - ──  ──⎥
⎣    k₂  k₂⎦

In [8]: g * f
Out[8]: 
⎡       1           0  ⎤
⎢                      ⎥
⎢        1             ⎥
⎢    1 - ──            ⎥
⎢        k₁   1     1  ⎥
⎢1 + ────── - ──  ─────⎥
⎣      k₂     k₂  k₁⋅k₂⎦

In [9]: (g * f)[1,0].simplify()
Out[9]: 
      1  
1 - ─────
    k₁⋅k₂

In [10]: f * g
Out[10]: 
⎡       1           0  ⎤
⎢                      ⎥
⎢        1             ⎥
⎢    1 - ──            ⎥
⎢        k₂   1     1  ⎥
⎢1 + ────── - ──  ─────⎥
⎣      k₁     k₁  k₁⋅k₂⎦

In [11]: (f * g)[1,0].simplify()
Out[11]: 
      1  
1 - ─────
    k₁⋅k₂

In [12]: 

 

Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.

Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

Binary lower unitriangular Toeplitz matrices, multiplied using F₂ operations
They form the Cayley table of Z₄ and correspond to powers of the 4-bit Gray code permutation.

Simultaneous triangularisability

A set of matrices are said to be simultaneously triangularisable if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix P. Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the  denoted Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel subalgebra.

The basic result is that (over an algebraically closed field), the commuting matrices  or more generally  are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.

The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert’s Nullstellensatz: commuting matrices form a commutative algebra  over  which can be interpreted as a variety in k-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables.

This is generalized by Lie’s theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.

More generally and precisely, a set of matrices  is simultaneously triangularisable if and only if the matrix is nilpotent for all polynomials p in k non-commuting variables, where  is the commutator; note that for commuting  the commutator vanishes so this holds. This was proven in (Drazin, Dungey & Gruenberg 1951); a brief proof is given in (Prasolov 1994, pp. 178–179). One direction is clear: if the matrices are simultaneously triangularisable, then is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any  or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.