In general a system of coupled first order linear equations is of the form
$$ x_1'=p_{11}(t)x_1+\dots+p_{1n}x_n+g_1(t)\\ x_2'=p_{21}(t)x_1+\dots+p_{2n}x_n+g_2(t)\\ \vdots\\ x_n'=p_{n1}x_1+\dots+p_{nn}x_n+g_n(t) $$
To improve notation we define:
$$
\bold{x}=\begin{bmatrix}
x_1\\
x_2\\
\vdots\\
x_n
\end{bmatrix}\bold{x}'=\begin{bmatrix}
x_1'\\
x_2'\\
\vdots\\
x_n'
\end{bmatrix} \bold{g}=\begin{bmatrix}
g_1\\
g_2\\
\vdots\\
g_n
\end{bmatrix}~~P=\begin{bmatrix}
p_{11}&\dots&p_{1n}\\
p_{21}&\dots&p_{2n}\\
\vdots&&\vdots\\
p_{n1}&\dots&p_{nn}
\end{bmatrix}
$$
$$ \bold{x}'=P\bold{x}+\bold{g} $$
Next we defined the solutions to this system as $\bold{x}^{(k)}$
Where the second number of the index represents the $k^{th}$ solution
$$ X=\begin{bmatrix}
|&&|\\ \bold x^{(1)}&\dots&\bold x^{(n)}\\ |&&|\\
\end{bmatrix}= \begin{bmatrix} x_{11} & x_{12} & \dots&x_{1n}\\ x_{11} & x_{22} & \dots&x_{2n}\\ \vdots&&&\vdots\\ x_{n1} & x_{n2} & \dots&x_{nn}\\
\end{bmatrix} $$
Theorem 7.4.1: Principle of Superposition
<aside> ▫️ If $\bold x^{(1)}$ and $\bold x^{(2)}$ are solutions, then so is $c_1\bold x^{(1)}+c_2\bold x^{(2)}$
</aside>
<aside> ▫️ In general, if $\bold x^{(1)},\dots,\bold x^{(k)}$ are solutions, then so any linear combination of them
</aside>
This essentially demonstrates that $U=\{ y~\epsilon~V_n(I)|~~~\bold y'=A\bold y\}$ is a subspace of $V_n$, the collection of all continuously differentiable functions on $\R^n$
Theorem 7.4.2: Uniqueness of Fundamental Solutions
<aside> ▫️ If $\{\bold x^{(1)},\dots,\bold x^{(n)}\}$ are linearly independent solutions of $\bold x'=P\bold x$ on an interval $I$, then for any solution $\bold x = \bold{\Phi(t)}$ there exists a unique set of constants $\bold c = c_1,\dots,c_n$ such that $\bold\Phi = c_1\bold x^{(1)}+\dots+c_n\bold x^{(n)}$
</aside>
This theorem demonstrates that $U$ is n-dimensional. If we can find n linearly independent vectors, then these vectors span the whole subspace of solutions
If $\{\bold x^{(1)},\dots,\bold x^{(n)}\}$ is linearly independent on $I$, then $X(t)$ (notated above) is a fundamental matrix whose span is the fundamental set of solutions