Wronskian

Definition

Let $I$ be an interval and let $f_1, \dots, f_n : I \to \mathbb{R}$ be $n - 1$ times differentiable functions. The Wronskian of $f_1, \dots, f_n$ is the function $W(f_1, \dots, f_n) : I \to \mathbb{R}$ given by $$W(f_1, \dots, f_n)(x) = \det \begin{pmatrix} f_1(x) & f_2(x) & \cdots & f_n(x) \\ f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{pmatrix}.$$

Remarks

For two functions, $W(f, g) = f g' - f' g$. Example: $W(\cos, \sin) = \cos^2 + \sin^2 = 1$. The Wronskian test uses $W$ to decide whether solutions of a linear equation are linearly independent.

Explanations

L1 human

The rows of the matrix are the functions, then their first derivatives, and so on. The determinant measures whether the vectors of starting data of the functions are independent at the point $x$.