The set of all solution of $n$-th ordinary differential equation is basis of dimension $n$

Question

Prove that the set of solution of $n^\text{th}$ order homogeneous differential equation is subspace with dimension $n$.

What I'm doing

let $S=\{y:L(y)=0\}$ where $L(y)=y^{(n)}+a_1y^{(n-1)}+.....a_ny$

then $S$ is subspace of set of all $n-$ times differential function?

[I proved this $S$ subspace as $ay_1+by_2 \in S$ as if $y_1$ and $y_2$ are two solutions ]

and how to prove this subspace has dimension $n$?