*Note: This question may seem extremely elementary, but I am **not** a beginner to signal processing, linear system theory, control theory, etc.*
*I believe my confusion is over a subtle point, so, before responding, please read the entire question so you understand the source of my confusion first.*

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Consider this simple autonomous system ([autonomous means it has no direct dependence on $t$](https://en.wikipedia.org/wiki/Autonomous_system_(mathematics)#Definition)):

$$\dot{x}(t) = A x(t)$$

Assume the output is the state ($y(t) = x(t)$) and $A \neq 0$. My questions are:

1. Is this system necessarily linear?
2. Is this system necessarily time-invariant?
3. What is the *zero-input response* of this system?  
4. Would the answers above change if we assumed the system was causal?

Easy, right?

1. [Obviously yes, since it is in the form $\dot{x}(t) = Ax(t)$](https://see.stanford.edu/materials/lsoeldsee263/09-auto-sys.pdf#page=2)
2. [Obviously yes, since $A$ does not depend on $t$](https://see.stanford.edu/materials/lsoeldsee263/09-auto-sys.pdf#page=2)
3. The zero-input response is: $x(t) = x(0)\,e^{At}$
4. No, these are unrelated properties.

**But wait**!

1. Linearity means a linear combination of the inputs must give the same linear combination of the outputs. If I scale the input, $u$, then $x$ is not affected at all. So how can it be linear?!
2. Time-invariance means that if I delay the input, the output is delayed by the same amount of time. But if I delay the input, $u$, then $x$ is not affected at all. So how can it be time-invariant?!  
3. If the initial state is an "input": Doesn't linearity imply $x(0) = 0 \implies x(t) = 0$?  
   If the initial state isn't  an "input": Isn't the system nonlinear unless $x(0) = 0$? (see #1)
4. Wouldn't *zero-input* give *zero-output* for a causal LTI system (the "ZIZO" property)?


Hopefully it's clear why I'm confused.  
Where am I going wrong?