Identity function: Difference between revisions
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*In an ''n''-dimensional [[vector space]] the identity function is represented by the [[identity matrix]] ''I''<sub>''n''</sub>, regardless of the [[Basis (linear algebra)|basis]].<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=038-733-195-6|series=Undergraduate Texts in Mathematics|url=http://books.google.co.uk/books?id=8qwTb9P-iW8C&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}</ref> |
*In an ''n''-dimensional [[vector space]] the identity function is represented by the [[identity matrix]] ''I''<sub>''n''</sub>, regardless of the [[Basis (linear algebra)|basis]].<ref>{{cite book|title=Applied Linear Algebra and Matrix Analysis|author=T. S. Shores|year=2007|publisher=Springer|isbn=038-733-195-6|series=Undergraduate Texts in Mathematics|url=http://books.google.co.uk/books?id=8qwTb9P-iW8C&printsec=frontcover&dq=Matrix+Analysis&hl=en&sa=X&ei=SCd1UryWD_LG7Aag_4HwBg&ved=0CGQQ6AEwCA#v=onepage&q=Matrix%20Analysis&f=false}}</ref> |
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*In a [[metric space]] the identity is trivially an [[isometry]]. An object without any [[symmetry]] has as [[symmetry group]] the trivial group only containing this isometry (symmetry type ''C<sub>1</sub>).<ref>{{aut|James W. Anderson}}, ''Hyperbolic Geometry'', Springer 2005, ISBN 1-85233-934-9</ref> |
*In a [[metric space]] the identity is trivially an [[isometry]]. An object without any [[symmetry]] has as [[symmetry group]] the trivial group only containing this isometry (symmetry type ''C<sub>1</sub>).<ref>{{aut|James W. Anderson}}, ''Hyperbolic Geometry'', Springer 2005, ISBN 1-85233-934-9</ref> |
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*In a [[topological space]], the identity function is always continuous. |
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==See also== |
==See also== |
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Revision as of 22:38, 4 August 2016
In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.
Definition
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies
- f(x) = x for all elements x in M.[1]
In other words, the function assigns to each element x of M the element x of M.
The identity function f on M is often denoted by idM.
In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of M.
Algebraic property
If f : M → N is any function, then we have f ∘ idM = f = idN ∘ f (where "∘" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.
Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.
Properties
- The identity function is a linear operator, when applied to vector spaces.[2]
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.[3]
- In an n-dimensional vector space the identity function is represented by the identity matrix In, regardless of the basis.[4]
- In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type C1).[5]
- In a topological space, the identity function is always continuous.
See also
References
- ^ Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9
- ^ Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
- ^ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
- ^ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 038-733-195-6.
- ^ James W. Anderson, Hyperbolic Geometry, Springer 2005, ISBN 1-85233-934-9