Example 1

Tag 08CO, permanent. Cite it as [Bourbaki, Tag 08CO].

Functions of a Real Variable, chapter IV, § 2. LINEAR DIFFERENTIAL EQUATIONS

When $E$ is of finite dimension $n$ over $\mathbf{R}$ one can identify the endomorphism $A(t)$ with its matrix $\left(a_{ij}(t)\right)$ with respect to any basis of $E$ (Alg., II, p. 343); when one identifies a vector $x \in E$ with the column matrix $(x_j)$ of its components with respect to the basis of $E$ under consideration the expression $A(t).x$ conforms to the general conventions of Algebra (Alg., II, p. 343, prop 2). In this case, equation (2) is equivalent to the system of scalar differential equations

\frac{dx_i}{dt} = \sum_{j=1}^n a_{ij}(t)x_j + b_i(t) \qquad (1 \leq i \leq n).

(3)

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

*Alg.*, II, p. 343