| Line 3: | Line 3: | ||
A system is called linear if and only if: | A system is called linear if and only if: | ||
| − | <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> |
== Example of a linear system == | == Example of a linear system == | ||
| Line 15: | Line 15: | ||
<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> | ||
| − | <math>f(at^2 + 2bt^2) = af(t^2) + bf( | + | <math>f(at^2 + 2bt^2) = af(t^2) + bf(2t^2) \,</math> |
<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> | ||
| Line 28: | Line 28: | ||
== Example of a non-linear system == | == Example of a non-linear system == | ||
| − | System is: <math> f(x) = 23x + 1<math> | + | System is: <math> f(x) = 23x + 1\,<math> |
<math>X_1(t) = t^2 \,</math> | <math>X_1(t) = t^2 \,</math> | ||
| Line 34: | Line 34: | ||
| − | <math>f(aX_1 + bX_2) | + | <math>f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \,</math> |
| − | <math>f(at^2 + 2bt^2) | + | <math>f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \,</math> |
| − | <math>f(at^2 + 2bt^2) | + | <math>f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \,</math> |
| − | <math>f(at^2 + 2bt^2) | + | <math>f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \,</math> |
| − | <math> f( | + | <math> f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \,</math> |
| + | |||
| + | <math> f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \,</math> | ||
Revision as of 17:27, 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(2t^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 \, $
Example of a non-linear system
System is: $ f(x) = 23x + 1\,<math> <math>X_1(t) = t^2 \, $
$ X_2(t) = 2t^2 \, $
$ f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \, $
$ f(at^2 + 2bt^2) \neq af(t^2) + bf(2t^2) \, $
$ f(at^2 + 2bt^2) \neq a(23t^2+1) + b(23*(2t^2)+1) \, $
$ f(at^2 + 2bt^2) \neq 23 at^2 + 1 + 46 bt^2 + b \, $
$ f(at^2 + 2bt^2) \neq 23 (at^2 + 2bt^2) + a + b \, $
$ f(aX_1 + bX_2) \neq af(X_1) + bf(X_2) \, $
