Proposition 4

For the $n$ integrals $\mathbf{u}_j$ ($1 \leq j \leq n$) of equation (4) to form a fundamental system it is necessary and sufficient that their values $\mathbf{u}_j(t_0)$ at a point $t_0 \in J$ be linearly independent vectors in $E$.

Indeed, the map which to every $\mathbf{x}_0 \in E$ associates the integral $t \mapsto C(t, t_0).\mathbf{x}_0$ is an isomorphism of $E$ onto $\mathscr{I}$ (IV, p. 180, cor. 1 and IV, p. 181, prop. 2).

If $(\mathbf{e}_j)_{1 \leq j \leq n}$ is any basis of $E$ over $\mathbf{C}$, the $n$ integrals

thus form a fundamental system; if one identifies $C(t, t_0)$ with its matrix with respect to the basis $(\mathbf{e}_j)$ the integrals $\mathbf{u}_j$ are precisely the columns of the matrix $C(t, t_0)$. The integral of (4) that takes the value $\mathbf{x}_0 = \sum_{j=1}^n \lambda_j \mathbf{e}_j$ at the point $t_0$ is then $C(t, t_0).\mathbf{x}_0 = \sum_{k=1}^n \lambda_k \mathbf{u}_k(t)$.

Given any $n$ integrals $\mathbf{u}_j$ ($1 \leq j \leq n$) of (4) one terms the determinant of these $n$ integrals at a point $t \in J$ with respect to a basis $(\mathbf{e}_j)_{1 \leq j \leq n}$ of $E$, the determinant

of the $n$ vectors $\mathbf{u}_j(t)$ with respect to the basis $(\mathbf{e}_j)$ (Alg., III, p. 522). One has (Alg., III, p. 523, prop. 2)

By prop. 4 of IV, p. 184, for $(\mathbf{u}_j)_{1 \leq j \leq n}$ to be a fundamental system of integrals of (4) it is necessary and sufficient that the determinant $\Delta(t)$ of the $\mathbf{u}_j$ be $\neq 0$ at some one point $t_0$ of $J$; the formula (14) then shows that $\Delta(t) \neq 0$ at every point of $J$, in other words, that the vectors $\mathbf{u}_j(t)$ ($1 \leq j \leq n$) are always linearly independent.