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  "body": "\n## Definition\n\nLet $I$ be an interval and let $f_1, \\dots, f_n : I \\to \\mathbb{R}$ be $n - 1$ times differentiable functions.\nThe Wronskian of $f_1, \\dots, f_n$ is the function $W(f_1, \\dots, f_n) : I \\to \\mathbb{R}$ given by\n$$W(f_1, \\dots, f_n)(x) = \\det \\begin{pmatrix} f_1(x) \u0026 f_2(x) \u0026 \\cdots \u0026 f_n(x) \\\\ f_1'(x) \u0026 f_2'(x) \u0026 \\cdots \u0026 f_n'(x) \\\\ \\vdots \u0026 \\vdots \u0026 \\ddots \u0026 \\vdots \\\\ f_1^{(n-1)}(x) \u0026 f_2^{(n-1)}(x) \u0026 \\cdots \u0026 f_n^{(n-1)}(x) \\end{pmatrix}.$$\n\n## Remarks\n\nFor two functions, $W(f, g) = f g' - f' g$.\nExample: $W(\\cos, \\sin) = \\cos^2 + \\sin^2 = 1$.\nThe [[Wronskian test|0MHE4B1QCQ]] uses $W$ to decide whether solutions of a linear equation are linearly independent.\n\n## Explanations\n\n### L1 human\n\nThe rows of the matrix are the functions, then their first derivatives, and so on.\nThe determinant measures whether the vectors of starting data of the functions are independent at the point $x$.\n",
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