LINEARITY
Linearity, in my definition, means that superposition always works. In other words, summation of inputs yield summation of outputs.
Example of Linearity and its proof
$ \,y(t)=x(2t)\, $
Proof:
$ x1(t) \to System \to y1(t)=x1(2t) \to Scalar multiplication(*a) \to ax1(2t) $
$ x2(t) \to System \to y2(t)=x2(2t) \to Scalar multiplication(*b) \to bx2(2t) $
$ ax1(2t) and bx2(2t) \to SUM \to '''ax1(2t)+bx2(2t)''' $
$ x1(t) \to Scalar multiplication(*a) \to ax1(t) $
$ x2(t) \to Scalar multiplication(*b) \to bx2(t) $
$ ax1(t) and bx2(t) \to SUM \to \to System \to'''ax1(2t)+bx2(2t)''' $
Those two yielded the same outputs thus it is linear.
Example of non-linearity and its proof
$ \,y(t)=e^{x(t)}\, $
Proof:
$ x(t) \to System \to y(t)=e^{x(t)} \to Time Shift(t0) \to z(t)=y(t-t0) $
$ \, =e^{x(t-t0)}\, $
$ x(t) \to Time Shift(t0) \to y(t)=x(t-t0) \to System \to z(t)=e^{y(t)} $
$ \, =e^{x(t-t0)}\, $
Both cascades yielded the same outputs, thus $ \,y(t)=e^{x(t)}\, $ is time invariant.
