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A linear system is a mathematical model of a system based on the use of a linear operator. A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions.

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Complex representation of electric field

One simple way to deal with "i" thing is to use Euler's identity $$ i = e^{i\pi/2} $$ so you can write $$ E_{out2}=\frac{A}{\sqrt2}e^{i(\omega t+ \pi/2)} $$ I hope it helps
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Why do people say the dynamics of quantum mechanics is always linear?

The linearity of quantum comes from The Principle of Superposition and is one of the most fundamental ideas in Quantum Theory. In the book "The Principles of Quantum Theory" chapter I, Paul Dirac argu …
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