Trying to better understand the structure of solutions to an arbitrary system of homogeneous first order linear differential equations.

Question

Hello Math Stack Exchange, this is my first post so I apologize if I am not using this platform as intended. If so, please let me know so that I can better participate in this community.

Currently, I am trying to work through Linear Ordinary Differential Equations by Earl Coddington. In particular, I am currently on the section about first order homogeneous systems. To get into the thick of it: Let $\mathcal{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and consider the homogeneous system $X'=A(t)X$, $t\in I$, where $I$ is an interval on the real line, $X:I\rightarrow \mathcal{F}^n$ is differentiable, and $A:I\rightarrow M_{n\times n}(\mathcal{F})$ is continuous.

1. Coddington notes that by the existence uniqueness theorem for the initial value problem that given any $\tau \in I$, if $X(t)$ is any solution such that $X(\tau)=(0,\cdot\cdot\cdot,0)^T$, then $X(t)=(0,\cdot\cdot\cdot,0)^T$ for all $t\in I$. In other words this $X$ is the trivial solution. I do not quite see why the existence uniqueness theorem implies that if the solution equals the zero vector at one point $\tau\in I$ then it must be equal to zero for $\textbf{all}$ $t\in I$. He also uses this observation to motive the theorem that the solutions to this system form an $n$-dimensional vector space over $\mathcal{F}$, and I am curious as to why this particular observation leads us to that.

2. After proving that a set of $k$ solutions $\{X_1,\cdot\cdot\cdot, X_k\}$ to the above system is linearly independent iff the vectors $\{X_1(\tau)=F_1, \cdot\cdot\cdot, X_k(\tau)=F_k\}$ for some $\tau\in I$ are linear independent in $\mathcal{F}^n$, Coddington notes that this result plus the theorem that the set of solutions to the same system form a vector space can be rephrased as follows. Denote the set of solutions to the above system by $S$ and let $\tau \in I$ be fixed and for any $\xi\in \mathcal{F}^n$ denote by $X_{\xi}(t)$ that solution of the above system such that $X_{\xi}(\tau)=\xi$. Then the map $\mathcal{S}:\xi\in \mathcal{F}^n\rightarrow X_{\xi}\in S$ is an isomorphism. I want to prove this myself, but I don't quite understand what the map in question ($\mathcal{S}$) is...?

Any help is appreciated! If anybody happens to have used this book, all of the material I am using is taken basically verbatim from pages 26-29.

Answer 1

1. The $X\equiv 0$ is a solution to the initial value problem $X’(t)=A(t)X(t), X(\tau)=0$. By the existence uniqueness theorem (Picard–Lindelöf theorem) this must be the only solution. We can find to every initial value a solution. Since there are $n$ linearly independent initial values at maximum we may suggest that there are only $n$ linearly independent solutions.

2. The map $\mathcal S$ means that to every initial value $\xi\in\mathcal F^n$ you can find a unique solution $X_\xi\in S=\operatorname{span}\{X_1,\dots,X_n\}$ and vice versa. Additionally, the vector space structure is respected.