(New page: A system is linear if for specific inputs <math>x_1(t)</math> and <math>x_2(t)</math> yielding the outputs <math>y_1(t)</math> and <math>y_2(t)</math>, respectively, the combination <math>...) |
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A system is linear if for specific inputs <math>x_1(t)</math> and <math>x_2(t)</math> yielding the outputs <math>y_1(t)</math> and <math>y_2(t)</math>, respectively, the combination <math>ax_1(t) + bx_2(t)</math>, where <math>a</math> and <math>b</math> are any complex numbers, yields the output <math>ay_1(t) + by_2(t)</math>. | A system is linear if for specific inputs <math>x_1(t)</math> and <math>x_2(t)</math> yielding the outputs <math>y_1(t)</math> and <math>y_2(t)</math>, respectively, the combination <math>ax_1(t) + bx_2(t)</math>, where <math>a</math> and <math>b</math> are any complex numbers, yields the output <math>ay_1(t) + by_2(t)</math>. | ||
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| + | Example of a non-linear system: | ||
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| + | <math>System = \sqrt(t)</math> | ||
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| + | <math>x = 1</math>, <math>y = 2</math>, <math>a = 2</math>, <math>b = 1</math> | ||
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| + | <math>x \Longrightarrow System \Longrightarrow 1</math> | ||
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| + | <math>y \Longrightarrow System \Longrightarrow 1.41421</math> | ||
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| + | <math>a(1) + b(1.41421) = 3.41421</math> | ||
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| + | <math>ax + by \Longrightarrow System \Longrightarrow 2 \neq 3.41421 \therefore </math> the system is not linear. | ||
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| + | Example of a linear system: | ||
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| + | <math>System = t</math> | ||
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| + | <math>x = 1</math>, <math>y = 2</math>, <math>a = 2</math>, <math>b = 1</math> | ||
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| + | <math>x \Longrightarrow System \Longrightarrow 1</math> | ||
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| + | <math>y \Longrightarrow System \Longrightarrow 2</math> | ||
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| + | <math>a(1) + b(2) = 4</math> | ||
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| + | <math>ax + by \Longrightarrow System \Longrightarrow 4 \therefore</math> the system is linear. | ||
Latest revision as of 04:31, 12 September 2008
A system is linear if for specific inputs $ x_1(t) $ and $ x_2(t) $ yielding the outputs $ y_1(t) $ and $ y_2(t) $, respectively, the combination $ ax_1(t) + bx_2(t) $, where $ a $ and $ b $ are any complex numbers, yields the output $ ay_1(t) + by_2(t) $.
Example of a non-linear system:
$ System = \sqrt(t) $
$ x = 1 $, $ y = 2 $, $ a = 2 $, $ b = 1 $
$ x \Longrightarrow System \Longrightarrow 1 $
$ y \Longrightarrow System \Longrightarrow 1.41421 $
$ a(1) + b(1.41421) = 3.41421 $
$ ax + by \Longrightarrow System \Longrightarrow 2 \neq 3.41421 \therefore $ the system is not linear.
Example of a linear system:
$ System = t $
$ x = 1 $, $ y = 2 $, $ a = 2 $, $ b = 1 $
$ x \Longrightarrow System \Longrightarrow 1 $
$ y \Longrightarrow System \Longrightarrow 2 $
$ a(1) + b(2) = 4 $
$ ax + by \Longrightarrow System \Longrightarrow 4 \therefore $ the system is linear.
