I have the following problem:
In Section $4.2$ it was shown that if $\mathbf x_1$ and $\mathbf x_2$ are solutions of
$$ \mathbf x' = \begin{pmatrix} p_{11}(t) & p_{12}(t) \\ p_{21}(t) & p_{22}(t) \\ \end{pmatrix}\mathbf x $$
on an interval I, then the Wronskian W of $\mathbf{x_1}$ and $\mathbf{x_2}$ satisfies the differential equation $W' = (p_{11}+p_{22})W$. A generalization of that proof shows that if $\mathbf{x_1,\cdots,x_n}$ are solutions of $$\mathbf{x'=Ax,\ \ \ \ \ x}(0) = \mathbf{x}_0$$ on I, then the Wronskian of $\mathbf{x_1,\cdots,x_n}$, denoted by W, satisfies the differential equation
$$W' = (p_{11} +p_{22}+ \cdots + p_{nn})W = tr(\mathbf{P}(t))W~.\tag{i}$$
(a) Explain why Eq. (i) also implies that W is either identically zero or else never vanishes on I in accordance with Theorem 6.2.5 (shown below).
(b) If $y_1,\cdots,y_n$ are solutions of
$$\frac{d^ny}{dt^n}+p_1(t)\frac{d^{n-1}y}{dt^{n-1}}+ \cdots +p_{n-1}(t)\frac{dy}{dt}+p_n(t)y =0,$$
find a counterpart to Eq. (i) satisfied by $W = W[y_1,\cdots,y_n](t)$.
Theorem $6.2.5$ :
Let $x_1,\cdots,x_n$ be solutions of $\mathbf{x' = p(t)x}$ on an interval $I = (\alpha , \beta)$ in which P(t) is continuous.
(i) If $\mathbf{x}_1,\cdots,\mathbf{x}_n$ are linearly independent on I, then $W[\mathbf{x}_1,\cdots,\mathbf{x}_n](t) \ne0$ at every point in I,
(ii) If $\mathbf{x}_1,...,\mathbf{x}_n$ are linearly dependent on I, then $W[\mathbf{x}_1,...,\mathbf{x}_n](t) =0$ at every point in I.
$$$$ $$$$
For (a), I'm having trouble trying to figure out where to start. I know what a Wronskian is, but I don't know how I would prove that W is either zero or never vanishes using (i)
For (b), I'm not sure what the problem is asking.
(a) Hint: What is the solution to $y' = ky$? This solution has an arbitrary constant: for which values of this constant does the solution vanish?
(b) Recall that an $n^\text{th}$ order linear differential equation can be turned into a system of $n$ first order differential equations by letting $x_1 = y, x_2 = y', \ldots, x_n = y^{(n-1)}$. What is the resulting system of differential equations? If you write this system in matrix form as $\mathbf{x'} = \mathbf{Ax}$, what is the trace of $\mathbf{A}$?