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| In [[mathematics]], the term '''linear function''' refers to two different, although related, notions:<ref>"The term ''linear function'', which is not used here, means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1</ref>
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| * In [[calculus]] and related areas, a linear function is a [[polynomial function]] of degree zero or one, or is the zero polynomial.<ref>Stewart 2012, p. 23</ref>
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| * In [[linear algebra]] and [[functional analysis]], a linear function is a [[linear map]].<ref>Shores 2007, p. 71</ref>
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| == As a polynomial function ==
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| {{main|Linear function (calculus)}}
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| In calculus, [[analytic geometry]] and related areas, a linear function is a polynomial of degree one or less, including the [[zero polynomial]] (the latter not being considered to have degree zero).
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| When the function is of only one [[variable (mathematics)|variable]], it is of the form
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| :<math>f(x)=ax+b,</math>
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| where {{mvar|''a''}} and {{mvar|''b''}} are [[constant (mathematics)|constant]]s, often [[real number]]s. The [[graph of a function|graph]] of such a function of one variable is a nonvertical line.
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| For a function <math>f(x_1, \ldots, x_k)</math> of any finite number [[independent variable]]s, the general formula is
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| :<math>f(x_1, \ldots, x_k) = b + a_1 x_1 + \ldots + a_k x_k</math>,
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| and the graph is a [[hyperplane]] of dimension {{nowrap|''k''}}.
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| A [[constant function]] is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one independent variable, is a horizontal line.
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| In this context, the other meaning (a linear map) may be referred to as a [[homogeneous function|homogeneous]] linear function or a [[linear form]]. In the context of linear algebra, this meaning (polynomial functions of degree 0 or 1) is a special kind of [[affine map]].
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| == As a linear map ==
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| {{main|Linear map}}
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| In linear algebra, a linear function is a map ''f'' between two [[vector space]]s that preserves [[vector addition]] and [[scalar multiplication]]:
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| :<math>f(\mathbf{x} + \mathbf{y}) = f(\mathbf{x}) + f(\mathbf{y}) </math>
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| :<math>f(a\mathbf{x}) = af(\mathbf{x}). </math>
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| Here {{math|''a''}} denotes a constant belonging to some [[field (mathematics)|field]] {{math|''K''}} of [[Scalar (mathematics)|scalar]]s (for example, the [[real number]]s) and {{math|'''x'''}} and {{math|'''y'''}} are elements of a [[vector space]], which might be {{math|''K''}} itself.
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| Some authors use "linear function" only for linear maps that take values in the scalar field;<ref>Gelfand 1961</ref> these are also called [[linear functional]]s.
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| ==See also==
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| * [[Homogenous function]]
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| * [[Nonlinear system]]
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| * [[Piecewise linear function]]
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| * [[Linear interpolation]]
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| * [[Discontinuous linear map]]
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| == Notes ==
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| <references/>
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| == References ==
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| * Izrail Moiseevich Gelfand (1961), ''Lectures on Linear Algebra'', Interscience Publishers, Inc., New York. Reprinted by Dover, 1989. ISBN 0-486-66082-6
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| * Thomas S. Shores (2007), ''Applied Linear Algebra and Matrix Analysis'', Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6
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| *James Stewart (2012), ''Calculus: Early Transcendentals'', edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
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| * Leonid N. Vaserstein (2006), "Linear Programming", in Leslie Hogben, ed., ''Handbook of Linear Algebra'', Discrete Mathematics and Its Applications, Chapman and Hall/CRC, chap. 50. ISBN 1-584-88510-6
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| ==External links==
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| {{Polynomials}}
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| [[Category:Polynomials]]
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Nothing to write about me at all.
Great to be a member of this community.
I really wish I am useful in one way .