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== Example of a linear system == | == Example of a linear system == | ||
System is: <math> f(x) = 23x \,</math> | System is: <math> f(x) = 23x \,</math> | ||
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<math>X_1(t) = t^2 \,</math> | <math>X_1(t) = t^2 \,</math> | ||
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<math>X_2(t) = 2t^2 \,</math> | <math>X_2(t) = 2t^2 \,</math> | ||
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<math>f(aX_1 + bX_2) = af(X_1) + bf(X_2) \,</math> | <math>f(aX_1 + bX_2) = af(X_1) + bf(X_2) \,</math> | ||
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<math>f(at^2 + 2bt^2) = af(t^2) + bf(t^2) \,</math> | <math>f(at^2 + 2bt^2) = af(t^2) + bf(t^2) \,</math> | ||
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<math>f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \,</math> | <math>f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \,</math> | ||
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<math>f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \,</math> | <math>f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \,</math> | ||
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<math> f(x) = 23x \,</math> | <math> f(x) = 23x \,</math> | ||
== Example of a non-linear system == | == Example of a non-linear system == | ||
Revision as of 16:14, 10 September 2008
Linearity
A system is called linear if and only if:
$ f(ax_1 + bx_2) = af(x_1) + bf(x_2) $
Example of a linear system
System is: $ f(x) = 23x \, $
$ X_1(t) = t^2 \, $
$ X_2(t) = 2t^2 \, $
$ f(aX_1 + bX_2) = af(X_1) + bf(X_2) \, $
$ f(at^2 + 2bt^2) = af(t^2) + bf(t^2) \, $
$ f(at^2 + 2bt^2) = a*23t^2 + b*46t^2 \, $
$ f(at^2 + 2bt^2) = 23(at^2 + 2bt^2) \, $
$ f(x) = 23x \, $
