Existence and Uniqueness of the State-Transition Matrix of a Linear System

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b\in\mathbb{R}$ with $a<b$, let $n\in\mathbb{N}$, and let $A:[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ be continuous on $[a,b]$, for $d_{\mathbb{R}}$ on $[a,b]$ and $d_{n}$ on $\mathcal{M}_{n}(\mathbb{R})$. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing the two distinct points $a$ and $b$; differentiability and derivatives of maps from $[a,b]$ into $\mathbb{R}^{n}$ or into $\mathcal{M}_{n\times q}(\mathbb{R})$ ($q\in\mathbb{N}$) are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii), taken relative to $[a,b]$. For a map $\Phi:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ and $s\in[a,b]$, $\Phi(\cdot,s):[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ is the map $t\mapsto\Phi(t,s)$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii)).

(i) Let $s\in[a,b]$ and $q\in\mathbb{N}$. If $X:[a,b]\to\mathcal{M}_{n\times q}(\mathbb{R})$ is differentiable, with $X'(t)=A(t)X(t)$ for every $t\in[a,b]$, and $X(s)=0_{n\times q}$, then $X(t)=0_{n\times q}$ for every $t\in[a,b]$. Likewise, if $x:[a,b]\to\mathbb{R}^{n}$ is differentiable, with $x'(t)=A(t)x(t)$ for every $t\in[a,b]$, and $x(s)=0$, then $x(t)=0$ for every $t\in[a,b]$.

(ii) There is exactly one map $\Phi:[a,b]\times[a,b]\to\mathcal{M}_{n}(\mathbb{R})$ such that, for every $s\in[a,b]$, the map $\Phi(\cdot,s)$ is differentiable, with derivative $A(t)\Phi(t,s)$ at every $t\in[a,b]$, and $\Phi(s,s)=I_{n}$.