Linearity
Because engineers talk in symbols instead of words...the following describes linearity.
- A/*B = multiply by any complex number
x(t) --> system --> *A
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+ --> w(t)
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y(t) --> system --> *B
x(t) --> *A
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+ --> system --> z(t)
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y(t) --> *B
If z(t) == w(t) then the system is linear.
In linearity, the values are scalable and follow the rules of superposition.
An example of a non-linear system
$ y(t)=t^2*sqrt(x(t)) $
x1(t)-->system--> *A =x1'(t) x2(t)-->system--> *B =x2'(t)
w(t)=x1'(t)+x2'(t)= $ A*t^2*sqrt(x1(t))+B*t^2*sqrt(x2(t)) $
x3(t)--> *A -->system--> = x3'(t)
x4(t)--> *B -->system--> = x4'(t)
z(t)=x3'(t) + x4'(t)= $ t^2*sqrt(A*x1(t))+t^2*sqrt(B*x2(t)) $
Since z(t) is not equal to w(t) (because one equation has root A and root B and the other just has A and B) the system is non-linear.
An example of a linear system
$ y(t)=t^2*x(t) $
x1(t)-->system--> *A =x1'(t) x2(t)-->system--> *B =x2'(t)
w(t)=x1'(t)+x2'(t)= $ A*t^2*x1(t)+B*t^2*x2(t) $
x3(t)--> *A -->system--> = x3'(t)
x4(t)--> *B -->system--> = x4'(t)
z(t)=x3'(t) + x4'(t)= $ t^2*A*x1(t)+t^2*B*x2(t) $
Since z(t) is equal to w(t) the system is linear.
