state and prove the principle of linear superposition of eingn staes
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In physics and systems theory, thesuperposition principle,[1] also known assuperposition property, states that, for alllinear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input Aproduces response X and input B produces response Y then input (A + B) produces response (X + Y).
The homogeneity and additivity properties together are called the superposition principle. A linear function is one that satisfies the properties of superposition. It is defined as
{\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\,} Additivity{\displaystyle F(ax)=aF(x)\,} Homogeneityfor scalar a.
This principle has many applications inphysics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. 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, Laplace transforms, andlinear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.
The superposition principle applies to anylinear system, including algebraic equations,linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as avector sum.
The homogeneity and additivity properties together are called the superposition principle. A linear function is one that satisfies the properties of superposition. It is defined as
{\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\,} Additivity{\displaystyle F(ax)=aF(x)\,} Homogeneityfor scalar a.
This principle has many applications inphysics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. 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, Laplace transforms, andlinear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.
The superposition principle applies to anylinear system, including algebraic equations,linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as avector sum.
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