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A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.) [1] [2] [3] [4]
If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system. These are exactly the properties required for the solution set to be a linear subspace of R n. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation. [ 3 ] 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.
The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency-domain linear transform methods such as Fourier and Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the ...
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 ...
As the computation of the kernel of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the Lapack library. [citation needed]
Homogeneous coordinates, used in projective spaces; Homogeneous differential equation; Homogeneous distribution; Homogeneous element and homogeneous ideal in a graded ring; Homogeneous equation (linear algebra): systems of linear equations with zero constant term; Homogeneous function; Homogeneous graph; Homogeneous (large cardinal property)