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Equation that does not involve powers or products of variables
In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0
Linear_equation
Differential equation that is linear with respect to the unknown function
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written
Linear_differential_equation
Several equations of degree 1 to be solved simultaneously
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example
System_of_linear_equations
Type of differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives
Partial_differential_equation
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
Mathematical formula expressing equality
(see System of polynomial equations). A system of linear equations (or linear system) is a collection of linear equations involving one or more variables
Equation
System where changes of output are not proportional to changes of input
regardless of whether known linear functions appear in the equations. In particular, a differential equation is linear if it is linear in terms of the unknown
Nonlinear_system
Polynomial equation of degree two
(If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by
Quadratic_equation
Differential equation containing derivatives with respect to only one variable
differential equations (SDEs) where the modeled process is random. A linear differential equation is a differential equation that is defined by a linear polynomial
Ordinary differential equation
Ordinary_differential_equation
Properties of mathematical relationships
Linear actuator Linear element Linear foot Linear system Linear programming Linear differential equation Bilinear Multilinear Linear motor Linear interpolation
Linearity
Branch of mathematics
Linear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b
Linear_algebra
Type of functional equation (mathematics)
linear differential equations (see Holonomic function). A non-linear differential equation is a differential equation that is not a linear equation in
Differential_equation
Nonlinear form of the Schrödinger equation
waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state
Nonlinear Schrödinger equation
Nonlinear_Schrödinger_equation
Type of ordinary differential equation
solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by
Homogeneous differential equation
Homogeneous_differential_equation
Least squares approximation of linear functions to data
include inverting the matrix of the normal equations and orthogonal decomposition methods. Consider the linear equation where A ∈ R m × n {\displaystyle A\in
Linear_least_squares
Polynomial function of degree at most one
are related to linear equations. A linear function is a polynomial function in which the variable x has degree at most one (a linear polynomial): f (
Linear_function_(calculus)
Mathematical relation defining a sequence
linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates of a variable—that is, in
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Type of differential equation
q_{0}(x)=0} the equation reduces to a Bernoulli equation, while if q 2 ( x ) = 0 {\displaystyle q_{2}(x)=0} the equation becomes a first order linear ordinary
Riccati_equation
Symbolic representation of a chemical reaction
Simple equations can be balanced by inspection, that is, by trial and error. Another technique involves solving a system of linear equations. Balanced
Chemical_equation
Straight figure with zero width and depth
linear equations. More precisely, every line L (including vertical lines) is the set of all points whose coordinates (x, y) satisfy a linear equation;
Line_(geometry)
Set of equations to be solved together
System of linear equations System of nonlinear equations System of bilinear equations System of polynomial equations System of differential equations System
System_of_equations
Basic concepts of algebra
associated plot of the equations. For other ways to solve this kind of equations, see below, System of linear equations. A quadratic equation is one which includes
Elementary_algebra
Statistical modeling method
2: Linear Regression, Linear Regression with Error Bars and Nonlinear Regression. National Physical Laboratory (1961). "Chapter 1: Linear Equations and
Linear_regression
Partial differential equations with random force terms and coefficients
include stochastic versions of famous linear equations, such as the wave equation and the Schrödinger equation. One difficulty is their lack of regularity
Stochastic partial differential equation
Stochastic_partial_differential_equation
Linear recurrence equation
P-recursive equation is a linear equation of sequences where the coefficient sequences can be represented as polynomials. P-recursive equations are linear recurrence
P-recursive_equation
algebra, linear equations and systems of linear equations over a field are widely studied. "Over a field" means that the coefficients of the equations and
Linear_equation_over_a_ring
Form of cryptanalysis
linear equations in conjunction with known plaintext-ciphertext pairs to derive key bits. For the purposes of linear cryptanalysis, a linear equation
Linear_cryptanalysis
Differential equation for the description of waves or standing wave
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves
Wave_equation
Branch of mathematics
methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them
Algebra
Algebraic structure in linear algebra
concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their dimension, which, roughly
Vector_space
In mathematics, vector subspace
by a homogeneous system of linear equations will yield a subspace. (The equation in example I was z = 0, and the equation in example II was x = y.) Again
Linear_subspace
Vectors mapped to 0 by a linear map
A\mathbf {x} =\mathbf {0} \right\}.} The matrix equation is equivalent to a homogeneous system of linear equations: A x = 0 ⇔ a 11 x 1 + a 12 x 2 + ⋯ + a 1 n
Kernel_(linear_algebra)
More equations than unknowns (mathematics)
solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others. The terminology
Overdetermined_system
Method for solving continuous operator problems (such as differential equations)
operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets
Galerkin_method
Class of ordinary differential equations
applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form d d x [ p ( x ) d y d x ] + q ( x ) y = − λ
Sturm–Liouville_theory
Dimension of the column space of a matrix
is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions
Rank_(linear_algebra)
Concepts from linear algebra
expressed in the form of an n × n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
Eigenvalues_and_eigenvectors
Quantum algorithm for solving systems of linear equations
obtaining certain limited information about the solution to a system of linear equations, introduced by Aram Harrow, Avinatan Hassidim, and Seth Lloyd. Specifically
HHL_algorithm
Algorithm to solve systems of equations
plane of a pinhole camera, and homographies. An ordinary system of linear equations x k = A y k {\displaystyle \mathbf {x} _{k}=\mathbf {A} \,\mathbf {y}
Direct_linear_transformation
Mathematical model of how solid objects deform
analysis. Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum
Linear_elasticity
Matrix consisting of linearly independent solutions to a linear differential equation
mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations x ˙ ( t ) = A ( t ) x ( t ) {\displaystyle {\dot {\mathbf
Fundamental matrix (linear differential equation)
Fundamental_matrix_(linear_differential_equation)
Equation from stability analysis
Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical
Lyapunov_equation
Finding values for variables that make an equation true
unknowns or auxiliary variables. This is always possible when all the equations are linear. Such infinite solution sets can naturally be interpreted as geometric
Equation_solving
Concept in differential equation mathematics
substantially different. More precisely, consider an ordinary linear differential equation of n-th order f ( n ) ( z ) + ∑ i = 0 n − 1 p i ( z ) f ( i )
Regular_singular_point
Type of ordinary differential equation
that n {\displaystyle n} not be 0 or 1 as they cause the equation to become linear. The equation was first discussed in a work of 1695 by Jacob Bernoulli
Bernoulli differential equation
Bernoulli_differential_equation
Polynomial equation, generally univariate
Sextic equation (degree = 6) Septic equation (degree = 7) System of linear equations System of polynomial equations Linear Diophantine equation Linear equation
Algebraic_equation
Algorithm for generating pseudo-randomized numbers
pseudo-randomized numbers calculated with a discontinuous piecewise linear equation. The method represents one of the oldest and best-known pseudorandom
Linear_congruential_generator
Equation to derive time of sunset and sunrise
The sunrise equation or sunset equation can be used to derive the time of sunrise or sunset for any solar declination and latitude in terms of local solar
Sunrise_equation
Operator equation in the style of Fredholm theory
into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is f ( t ) = ∫ a t K ( t , s ) x ( s ) d s {\displaystyle
Volterra_integral_equation
Partial differential equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas
Burgers'_equation
Mathematical function, in linear algebra
In mathematics, and more specifically in linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which
Linear_map
Whether or not there exists a set of values to satisfy a given system of equations
of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in
Consistent and inconsistent equations
Consistent_and_inconsistent_equations
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical
List_of_equations
Inequality which involves a linear function
equal to A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign. Two-dimensional linear inequalities
Linear_inequality
Equations of motion for viscous fluids
we arrive at the linear constitutive equation in the form usually employed in thermal hydraulics: Linear stress constitutive equation (expression used
Navier–Stokes_equations
Limiting set in dynamical systems
the basin of attraction. Similar features apply to linear differential equations. The scalar equation d x / d t = a x {\displaystyle dx/dt=ax} causes all
Attractor
Equation involving both integrals and derivatives of a function
integro-differential equation is an equation that involves both integrals and derivatives of a function. The general first-order, linear (only with respect
Integro-differential_equation
Computer vision algorithm
a homogeneous linear equation, where the solution is directly related to E {\displaystyle \mathbf {E} } , and then solves the equation, taking into account
Eight-point_algorithm
State of linear equations
differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if
Linear_stability
Study of geometry using a coordinate system
y = x is said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections
Analytic_geometry
Partial differential equation describing the evolution of temperature in a region
specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier
Heat_equation
Elliptic partial differential equation
identically, we obtain Laplace's equation. Poisson's equation is linear inhomogeneous partial differential equation, and invariant under rotations or
Poisson's_equation
Finding linear approximation of function at given point
systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete
Linearization
Shape formed from points common to other shapes
solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically
Intersection_(geometry)
Balancing. The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "[i]t is not certain just what the
History_of_algebra
Pattern defining an infinite sequence of numbers
be calculated by repeatedly applying the equation. In linear recurrences, the nth term is equated to a linear function of the k {\displaystyle k} previous
Recurrence_relation
Description of a quantum-mechanical system
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery
Schrödinger_equation
Array of numbers
used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m × n matrix, x designates
Matrix_(mathematics)
Concept in geometry including line and circle
}}-r^{2}\right).\end{aligned}}} This is a homogeneous bivariate linear polynomial equation in terms of the complex variable z {\displaystyle z} and its conjugate
Generalised_circle
is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector
Outline_of_linear_algebra
Vectors whose linear combinations are nonzero
\dots ,\mathbf {v} _{n}} is said to be linearly independent if it is not linearly dependent, that is, if the equation a 1 v 1 + a 2 v 2 + ⋯ + a n v n = 0
Linear_independence
Regularization technique for ill-posed problems
ordinary least squares linear regression.[clarification needed] However, if no x {\displaystyle \mathbf {x} } satisfies the equation or more than one x {\displaystyle
Ridge_regression
Equations describing classical electromagnetism
Gauss's laws. For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But
Maxwell's_equations
Equations with an unknown function under an integral sign
integral. Third kind: An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: g ( t
Integral_equation
Mathematical theorem
the number of linear equations associated with an integral. This term reflects the common practice of bracketing each linear equation. The complexity
Ramanujan's_master_theorem
Second-order partial differential equation
functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution.
Laplace's_equation
Improved reduction for specific matrices
solve cyclic block tri-diagonal and cyclic block penta-diagonal linear systems of equations". Applied Mathematics and Computation. 210 (2): 558–563. doi:10
Tridiagonal_matrix_algorithm
Linear optimal control technique
The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called
Linear–quadratic_regulator
Partial differential equation
mechanics and information theory, the Fokker–Planck equation is a partial differential equation that describes the time evolution of the probability
Fokker–Planck_equation
Relativistic wave equation in quantum mechanics
where the equation describes the dynamics of spin-0 fields. Mathematically, it is a linear second-order hyperbolic partial differential equation that is
Klein–Gordon_equation
Numerical method for differential equations
analysis, the local linearization (LL) method is a general strategy for designing numerical integrators for differential equations based on a local (piecewise)
Local_linearization_method
Matrix used in finite element analysis
elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to
Stiffness_matrix
Nonlinear equation which arises on linear optimal control problems
equation will be a matrix equation. The algebraic Riccati equation determines the solution of the infinite-horizon time-invariant Linear-Quadratic Regulator
Algebraic_Riccati_equation
Class of differential equations expressible in differential algebra
In mathematics, an algebraic differential equation is a differential equation that can be expressed by means of differential algebra. There are several
Algebraic differential equation
Algebraic_differential_equation
Type of statistical model
simultaneous equations at once, this often leads to a computationally costly non-linear optimization problem even for the simplest system of linear equations. This
Simultaneous_equations_model
Methods used to find numerical solutions of ordinary differential equations
assume the differential equation is either of the form or it has been locally linearized about a background state to produce a linear term − A y {\displaystyle
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
Sum of terms, each multiplied with a scalar
system of linear equations can easily be solved. First, the first equation simply says that a3 is 1. Knowing that, we can solve the second equation for a2
Linear_combination
Affine subspace of a Euclidean space
called a linear manifold or linear variety to distinguish it from other manifolds or varieties. A flat can be described by a system of linear equations. For
Flat_(geometry)
Type of partial differential equations
to one less than the order of the differential equation. By a linear change of variables, any equation of the form A ∂ 2 u ∂ x 2 + 2 B ∂ 2 u ∂ x ∂ y +
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
are further classified by degree. Linear equation – algebraic equation of degree one. Polynomial equation – equation in which a polynomial is set equal
Outline_of_algebra
Polynomial whose roots are the eigenvalues of a matrix
oscillations. Secular equation may have several meanings. In linear algebra it is sometimes used in place of characteristic equation. In astronomy it is
Characteristic_polynomial
Field-equations in general relativity
leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves. The Einstein field equations (EFE) may be written
Einstein_field_equations
linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear equations and
Glossary_of_linear_algebra
Class of partial differential equations
Elliptic differential equations appear in many different contexts and levels of generality. First consider a second-order linear PDE for an unknown function
Elliptic partial differential equation
Elliptic_partial_differential_equation
Algebraic equation on which the solution of a differential equation depends
differential equation or difference equation. The characteristic equation can only be formed when the differential equation is linear and homogeneous, and has constant
Characteristic equation (calculus)
Characteristic_equation_(calculus)
Ordinary differential equation
Euler–Cauchy equation, also known as a Cauchy–Euler equation, equidimensional equation, or Euler's equation, is a linear ordinary differential equation for which
Cauchy–Euler_equation
Mathematical concept
mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast
Underdetermined_system
Partial differential equation with nonlinear terms
for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear
Nonlinear partial differential equation
Nonlinear_partial_differential_equation
differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations. A general PDAE
Partial differential algebraic equation
Partial_differential_algebraic_equation
Extension of evidence theory to continuous variables of interest
vacuous on which no knowledge bears. Logical knowledge is represented by linear equations, or geometrically, a certainty hyperplane. Probabilistic knowledge
Linear_belief_function
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