恒等写像

恒等写像 (こうとうしゃぞう British: identity mapping, identity function), 恒等作用素 (Sayo lie to ask in this way British: identity operator), 恒等変換 (こうとうへんかん British: identity transformation) in the mathematics are the mapping that just always pays values same as what I used as the argument. If say by words of set theory; 恒等写像 恒等関係 (asking mailing Kei British is identity relation.

Table of contents

Definition

If speak it closely, 恒等写像 f on M is the mapping that domain and 終域 are M together as a meeting in M; and for arbitrary former x of M

f(x) = x

I say a thing satisfying を [¹]. It is one representation from M which 恒等写像 on M lets x oneself support each former x of M, and is provided to M if I write it by words [²].

恒等写像 on M is often expressed in id_M or 1_M.

The opposite angle of function relations (English version) called 恒等関係 namely M gathers in 恒等写像 if I consider representation to be a binary relation (diagonal set)Δ= {(x, x) | I am given in x ∈ M} [³].

Property

f: When I assume M → N any representation,

${\displaystyle f\circ {\text{id}}_{M}=f={\text{id}}_{N}\circ f}$ 

But, it is managed (in "∘" composition of the representation). Particularly, id_M is an identity element in the semigroup (all conversion semigroup (English version) on M) T_M which the meeting that the whole representation (conversion on M) from M to M makes forms about composition (middle Tatemoto); therefore the T_M forms monoid.

Because the identity element of the monoid is only one, as another definition of 恒等写像 on M, I can establish it as an identity element of all conversion monoid. A concept of 恒等射 in the area idea can generalize such a definition. In this context, a self-model on M does not have to shoot it, and to be が representation.

with the structure in the meeting of relationships

• There is it by 1-double representation essentially when I think about 恒等写像 on the multiplication monoid that the whole positive integer makes and is of the perfection multiplication in the meaning of the function of the number idea again (English version) [⁴]
• 恒等写像 on the vector space is linear mapping [⁵]. 恒等写像 on the n-dimension linear space has n X n identity matrix I_n for an expression line, but this does not depend on basal how to get [⁶].
• 恒等写像 in the metric space is isometry mapping in a self-evident meaning. Any object which does not last as for any symmetricalness has a self-evident group only consisting of 恒等写像 as symmetry conversion group (English version) (a symmetry type is C₁); [⁷].
  • 恒等写像 id_X does not merely become with the equal spur conversion between two metric spaces (X, d₁), (X, d₂) about different distance d₁, d₂ on X when I think about 恒等写像 id_X on stand gathering X.
• When I regarded 恒等写像 I_X on stand gathering X as topological space (X,τ₁), (X,τ₂), a necessary and sufficient condition for I_X to become the consecutive representation is that τ ₁ is smaller than τ ₂.

Explanatory note

1. ^ (Knapp 2006)
2. ^ (1968, Matsusaka, p. 28)
3. ^ (Bourbaki 1984, p. 10)
4. ^ (Marshal, Odell & Starbird 2007)
5. ^ (Anton 2005)
6. ^ (Shores 2007)
7. ^ (Anderson 2005)

References

• Nichola Bourbaki "set theory abstract" Tokyo book 〈 mathematics basic principle (4) 〉, 1984. ISBN 978-4,489,001,048。
• Kazuo Matsuzaka "meeting, phase guide" Iwanami Shoten, 1968. ISBN 978-4,000,054,249。
• Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9 
• Marshall, D.; Odell, E.; Starbird, M. (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519. 
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International 
• Shores, T. S. (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 038-733-195-6. http://books.google.co.uk/books?id=8qwTb9P-iW8C&dq=Matrix+Analysis&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA. 
• James W. Anderson (2005), Hyperbolic Geometry, Springer, ISBN 1-85233-934-9 

Allied item

• Inclusion map

Outside link

• Weisstein, Eric W. "Identity Function". MathWorld (English). 

• identity map - PlanetMath.org(English)

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