What Is a System of First-Order Linear ODEs?

A system of first-order linear ODEs couples several unknown functions $x_1(t), x_2(t), \ldots, x_n(t)$ so that each derivative $x_i'$ depends linearly on all the unknowns. In the simplest (constant-coefficient, homogeneous) case the system looks like:

$$x_1' = a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n$$$$x_2' = a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n$$$$\vdots$$$$x_n' = a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n$$

We immediately dress this in its matrix costume. Define the unknown vector and the coefficient matrix:

$$\mathbf{x}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \\ \vdots \\ x_n(t) \end{pmatrix}, \qquad A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

Then the entire system collapses to the compact form:

$$\boxed{\mathbf{x}' = A\mathbf{x}}$$

This is the gateway: every tool from linear algebra — eigenvalues, eigenvectors, determinants, diagonalization — now applies directly to solving differential equations.