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In advanced mathematics texts, the term linear function often denotes specifically homogeneous linear functions, while the term affine function is used for the general case, which includes . The natural domain of a linear function f ( x ) {\displaystyle f(x)} , the set of allowed input values for x , is the entire set of real numbers , x ∈ R ...
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 variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
which are functions of the principal invariants above. These are the coefficients of the characteristic polynomial of the deviator (() /), such that it is traceless. The separation of a tensor into a component that is a multiple of the identity and a traceless component is standard in hydrodynamics, where the former is called isotropic ...
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.
The functions whose graph is a line are generally called linear functions in the context of calculus. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the
In linear algebra, the identity matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1.
Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function = at any = based on the value and slope of the function at =, given that () is differentiable on [,] (or [,]) and that is close to .
The defining properties of any LTI system are linearity and time invariance.. Linearity means that the relationship between the input () and the output (), both being regarded as functions, is a linear mapping: If is a constant then the system output to () is (); if ′ is a further input with system output ′ then the output of the system to () + ′ is () + ′ (), this applying for all ...