Linear map: Difference between revisions

Revision as of 09:00, 9 December 2013

Revision as of 09:27, 9 December 2013

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== Examples ==

* The [[Identity function|identityzero map]] and [[zero map]]is arealways linear.

* The [[Identity function|identity map]] <math>x\mapstoof cx</math>,any wherevector ''c''space is a constant, is linear. operator

* Any [[homothecy]] centered in the origin of a vector space, <math>v\mapsto cv</math> where ''c'' is a scalar, is a linear operator.

* For real numbers, the map <math>x\mapsto x^2</math> is not linear.

* For real numbers, the map <math>x\mapsto x+1</math> is not linear (but is an [[affine transformation]]; <math>y=x+1</math> is a [[linear equation]], as used in [[analytic geometry]].)

* If ''A'' is a real ''m'' &times; ''n'' [[matrix (mathematics)|matrix]], then ''A'' defines a linear map from '''R'''^''n'' to '''R'''^''m'' by sending the [[column vector]] '''x''' ∈ '''R'''^''n'' to the column vector ''A'''''x''' ∈ '''R'''^''m''. Conversely, any linear map between [[finite-dimensional]] vector spaces can be represented in this manner; see the following section.

* The (definite) [[integralDerivative|Differentiation]] isdefines a linear map from the space of all real-valueddifferentiable integrablefunctions to the space of all functions. It also defines a linear operator on somethe space of all [[intervalsmooth (mathematics)|intervalfunction]] to '''R'''s.

* The (indefinitedefinite) [[integral]] (orover some [[antiderivativeinterval (mathematics)|interval]]) ''I'' is not considered a linear transformation,map asfrom the usespace of aall constantreal-valued ofintegrable integrationfunctions results in an infinite number ofon outputs''I'' perto input.'''R'''

* The (indefinite) [[integral]] (or [[antiderivative]]) with a fixed starting point defines a linear map from the space of all real-valued integrable functions on '''R''' to the space of all real-valued functions on '''R'''. Without fixed starting point it does not define a mapping at all, as the presence of a constant of integration in the result means it produces an infinite number of outputs for a single input.

* [[Derivative|Differentiation]] is a linear map from the space of all differentiable functions to the space of all functions.

* If ''V'' and ''W'' are finite-dimensional vector spaces over a field ''F'', then functions that send linear maps ''f'' : ''V'' → ''W'' to dim_''F''(''W'') × dim_''F''(''V'') matrices in the way described in the sequel are themselves linear maps (indeed linear isomorphisms).

* The [[expected value]] of a [[Random variable#Definition|random variable]] (which is in fact a function, and as such member of a vector space) is linear, as for random variables ''X'' and ''Y'' we have E[''X'' + ''Y''] = E[''X''] + E[''Y''] and E[''aX''] = ''a''E[''X''], but the [[variance]] of a random variable is not linear, as it violates the second condition, homogeneity of degree 1: V[''aX''] = ''a''²V[''X''].

== Matrices ==