Solution space of a homogeneous linear equation has dimension n

Statement

Let $n \ge 1$, let $I$ be an open interval, and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous. Let $S$ be the set of solutions $y : I \to \mathbb{R}$ of $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$. Then $S$ is a real vector space of dimension $n$. More precisely, for each $x_0 \in I$, the map $S \to \mathbb{R}^n$, $y \mapsto \left(y(x_0), y'(x_0), \dots, y^{(n-1)}(x_0)\right)$ is a linear bijection.

Remarks

The superposition principle gives that $S$ is a vector space. The map is injective by the uniqueness of solutions of the initial value problem, and surjective by their existence on all of $I$. A basis of $S$ is a fundamental system of solutions. Example: the solutions of $y'' + y = 0$ on $\mathbb{R}$ are $C_1 \cos x + C_2 \sin x$, a space of dimension $2$.

Explanations

L1 human

A solution is fixed by its starting data: the value and the first $n - 1$ derivatives at one point. These are $n$ numbers that you can choose freely. So there are exactly $n$ independent solutions, and each solution is a combination of them.