I got stuck on the following problem:
Let $\boldsymbol{x}^{(1)},...,\boldsymbol{x}^{(m)}$ be solutions of $\boldsymbol{x}'=\boldsymbol{P}(t)\boldsymbol{x}$ on the interval $\alpha<t<\beta$. Assume that $\boldsymbol{P}$ is continuous, and let $t_0$ be an arbitrary point in the given interval. Show that $\boldsymbol{x}^{(1)},...,\boldsymbol{x}^{(m)}$ are linearly dependent for $\alpha<t<\beta$ if (and only if) $\boldsymbol{x}^{(1)}(t_0),...,\boldsymbol{x}^{(m)}(t_0)$ are linearly dependent. In other words $\boldsymbol{x}^{(1)},...,\boldsymbol{x}^{(m)}$ are linearly dependent on the interval $(\alpha,\beta)$ if they are linearly dependent at any point in it.
Hint: There are constants $c_1,...,c_m$ that satisfy $c_1\boldsymbol{x}^{(1)}(t_0),...,c_m\boldsymbol{x}^{(m)}(t_0)=\boldsymbol{0}$. Let $\boldsymbol{z}(t)=c_1\boldsymbol{x}^{(1)}(t),...,c_m\boldsymbol{x}^{(m)}(t)$, and use the uniqueness theorem to show that $\boldsymbol{z}(t)=\boldsymbol{0}$ for each $t$ in $\alpha<t<\beta$.
Now, I have trouble understanding the uniqueness theorem, because I do not understand how it relates to these functions being zero on the rest of the interval except at $t_0$. Maybe if I presume $\boldsymbol{z}(t)$ to be a unique solution on the interval, then we somehow deduce it is the zerovector(?)
I do not have much to go on really, appreciate any help.
let everything be as you have defined it and consider that $\{x^{(j)}(t_0)\}_{j=1}^m$ are linearly dependent. Then $$\exists \textrm{ scalars } \{c_j\}_{j=1}^m \textrm{ and some specific }c_k \neq 0 \hspace{.5in}(1 \leq k \leq m)$$ so that $$\sum_j c_jx^{(j)}(t_0) = 0.$$
Hence $$x^{(k)}(t_0) = -\left(\sum_{s=1}^{k-1} \frac{c_s}{c_k}x^{(s)}(t_0) + \sum_{r = k+1}^m\frac{c_r}{c_k}x^{(r)}(t_0)\right).$$
Now the existence and uniqueness theorem for linear systems tells us that, as $P(t)$ is continuous, that there exists one, and only one solution on the open interval $(\alpha, \beta)$. So, because this equality holds at the point $t_0 \in (\alpha, \beta)$, it must hold at all points in $(\alpha, \beta)$. Therefore the functions are linearly dependent on the entire interval.