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In linear algebra, a linear function is a map f between two vector spaces such that. Here a denotes a constant belonging to some field K of scalars (for example, the real numbers) and x and y are elements of a vector space, which might be K itself. In other terms the linear function preserves vector addition and scalar multiplication.
The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that has been defined in the preceding section. If b = 0, the line is a vertical line (that is a line parallel to ...
Linear function (calculus) In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates) is a non-vertical line in the plane. [1] The characteristic property of linear functions is that when the input variable is changed, the change in the output is ...
Nonlinear dynamics. Game theory. v. t. e. In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1][2] Nonlinear problems are of interest to engineers, biologists, [3][4][5] physicists, [6][7] mathematicians, and many other scientists ...
In mathematics, the term linear is used in two distinct senses for two different properties: . linearity of a function (or mapping);; linearity of a polynomial.; An example of a linear function is the function defined by () = (,) that maps the real line to a line in the Euclidean plane R 2 that passes through the origin.
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral = is a linear functional from the vector space [,] of continuous functions on the interval [,] to the real numbers.
Sometimes the term linear operator refers to this case, [1] but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that and are real vector spaces (not necessarily with =), [citation needed] or it can be used to emphasize that is a function space, which is a common ...
Linear dynamical systems are dynamical systems whose evolution functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general ...
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