Entities which process inputs $\mathbf{u} \in \R^{m}$ and produce outputs $\mathbf{y} \in \R^{k}$ are known as systems. If a system is known to be linear it can be modeled with the state space equations and if it is time-invariant, then the operators involved do not change with time.
$$ \mathbf{\dot{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t) \\ \mathbf{y}(t) = C\mathbf{x} (t) \\ \text{Continuous Time} $$
$$ \mathbf{x}[i+1] = A\mathbf{x}[i] + B\mathbf{u}[i] \\ \mathbf{y}[i] = C\mathbf{x}[i] \\ \text{Discrete Time} $$
A system is said to be static if its output depends only on its current input (i.e. transformations and mappings) and dynamic if its output depends on all past inputs and outputs.
Features useful about a system can be encoded in a state vector $\mathbf{x} \in \R^{n}$, which should form a minimal set, of which the output $\mathbf{u}$ is always a subset of. The space from which a state vector can be identified from is known as the state space.
State at which the system ceases to evolve. In the context of first-order differential equations, this occurs when $\dot{x} = 0$. In the context of first-order difference equations, this occurs when $x[i+1] = x[i]$.
The input $\mathbf{u}$ is also known as the forced response, as it forces the system to respond to a stimulus. This can be shown by first state space equation, where the difference between the change in state and the system must always match the input.
States that the general solution to any linear differential or difference equation can be found by finding the homogeneous solution and particular solutions separately.
$$ x=x_h +x_p $$
This is a direct corollary of linear systems. Since we can model any linear differential or difference equation in terms of a linear operator $A$, we have $A(\mathbf{x_h} + \mathbf{x_p}) = A\mathbf{x_h} + A\mathbf{x_p} = 0 + \mathbf{u} = \mathbf{u}$ in the context of:
$$ \sum {k=1} ^{\ell} c_k x^{(k)} = u \hspace{15pt} \leftrightarrow \hspace{15pt} \begin{bmatrix} c_1 & \dots & c{\ell} \end{bmatrix} \begin{bmatrix} x \\ \vdots \\ x^{(\ell)} \end{bmatrix} =\begin{bmatrix} u \end{bmatrix} $$
The analog to difference equations follows similarly.