Linear function - Wikipedia

In mathematics, the term linear function refers to two distinct but related notions:^[1]

In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one.^[2] For distinguishing such a linear function from the other concept, the term affine function is often used.^[3]
In linear algebra, mathematical analysis,^[4] and functional analysis, a linear function is a linear map.Cite error: A <ref> tag is missing the closing </ref> (see the help page). these are more commonly called linear forms.

The "linear functions" of calculus qualify as "linear maps" when (and only when) $f (0, ..., 0) = 0$ , or, equivalently, when the constant $b$ equals zero in the one-degree polynomial above. Geometrically, the graph of the function must pass through the origin.

Notes

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^ "The term linear function means a linear form in some textbooks and an affine function in others." Vaserstein 2006, p. 50-1
^ Stewart 2012, p. 23
^ A. Kurosh (1975). Higher Algebra. Mir Publishers. p. 214.
^ T. M. Apostol (1981). Mathematical Analysis. Addison-Wesley. p. 345.

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
Thomas S. Shores (2007), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer. ISBN 0-387-33195-6
James Stewart (2012), Calculus: Early Transcendentals, edition 7E, Brooks/Cole. ISBN 978-0-538-49790-9
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

v t e Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Graph of a function Linear function Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Order of approximation (ε, δ)-definition of limit
Differential calculus	Derivative Second derivative Partial derivative Differential Differential operator Mean value theorem Notation Leibniz's notation Newton's notation Rules of differentiation linearity Power Sum Chain L'Hôpital's Product General Leibniz's rule Quotient Other techniques Implicit differentiation Inverse functions and differentiation Logarithmic derivative Related rates Stationary points First derivative test Second derivative test Extreme value theorem Maximum and minimum Further applications Newton's method Taylor's theorem Differential equation Ordinary differential equation Partial differential equation Stochastic differential equation
Integral calculus	Antiderivative Arc length Riemann integral Basic properties Constant of integration Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Tangent half-angle substitution Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method Integral equation Integro-differential equation
Vector calculus	Derivatives Curl Directional derivative Divergence Gradient Laplacian Basic theorems Line integrals Green's Stokes' Gauss'
Multivariable calculus	Divergence theorem Geometric Hessian matrix Jacobian matrix and determinant Lagrange multiplier Line integral Matrix Multiple integral Partial derivative Surface integral Volume integral Advanced topics Differential forms Exterior derivative Generalized Stokes' theorem Tensor calculus
Sequences and series	Arithmetico-geometric sequence Types of series Alternating Binomial Fourier Geometric Harmonic Infinite Power Maclaurin Taylor Telescoping Tests of convergence Abel's Alternating series Cauchy condensation Direct comparison Dirichlet's Integral Limit comparison Ratio Root Term
Special functions and numbers	Natural logarithm Exponential function Stirling's approximation Bernoulli numbers e (mathematical constant)
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	List of limits List of derivatives List of integrals
Miscellaneous topics	Complex calculus Contour integral Differential geometry Manifold Curvature of curves of surfaces Tensor Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7 exceeds π Regiomontanus' angle maximization problem Steinmetz solid

See also

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References