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Linear equation. In mathematics, a linear equation is an equation that may be put in the form where are the variables (or unknowns), and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation and may be arbitrary expressions, provided they do not contain any of the variables.
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously ...
Graph of points and linear least squares lines in the simple linear regression numerical example The 0.975 quantile of Student's t -distribution with 13 degrees of freedom is t * 13 = 2.1604 , and thus the 95% confidence intervals for α and β are
In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space , the vector space V* consisting of linear maps f : V → F where F is the field of scalars.
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations. It is named after the German mathematicians Carl Friedrich Gauss and Philipp Ludwig von Seidel. Though it can be applied to any matrix with non ...
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form + ′ + ″ + () = where a 0 (x), ..., a n (x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y (n) are the successive derivatives of an unknown function y of ...
Linearity of a homogenous differential equation means that if two functions f and g are solutions of the equation, then any linear combination af + bg is, too. In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work.
is a horizontal line with y-intercept a0. The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x, where a1 ≠ 0, is an oblique line with y-intercept a0 and slope a1. The graph of a degree 2 polynomial. f(x) = a0 + a1x + a2x2, where a2 ≠ 0. is a parabola. The graph of a degree 3 polynomial.