# Methodology for the Wronskian

MathewsMD
I was just curious and had a question: why does the Wronskian indicate linear independence if ## W ≠ 0 ## but is linearly dependent if ## W = 0 ##? Is there a proof to help understand the exact operations of the Wronskian and why it conveys these properties based on these results alone? Thank you!

First, think about systems of simultaneous linear equations and how they analyzed with matrices. Traditionally, we write the system in the form $M\ x = b$ with the left hand side being the product of a matrix multiplied on the right by a column vector (as opposed to a matrix multiplied on the left by a row vector). The product $M\ x$ of a matrix times a column vector can be viewed as a linear combination of the columns of $M$ with coefficients taken from the entries of $x$.

Writing the jth column of $M_{*.j}$

$M\ x = x_1 M_{*,1} + x_2 M_{*.2} + ... x_n M_{*,n}$

If we have more or fewer unknowns than equations, the matrix $M$ isn't square, so we can't do an analysis by taking its determinant. When $M$ is a square matrix and its derminant is non-zero, we can find a unique solution for variables $x$. In particular , we can find a unique solution for the system $M\ x = b$ when $b$ is the column vector of zeroes. The unique solution to $M\ x = 0$ would be $x_j = 0$ for all $j$..

If the column vectors of $M$ were dependent then solution for $M\ x = 0$ would not be unique. For example if $M_{*,1} = \sum_{j=2}^n a_j M_{*,j}$ with at least one of the $a_j$ nonzero then the values $x_1 = 1$ and $x_j = -a_j$ for $j > 1$ would be a nonzero solution to $M\ x = 0$.

By analogy, the Wronskian $W$ is the derminant of a square matrix of functions $M$ . A column vector of $M$ gives a function and it's successive derivatives. If $W$ is nonzero then $M\ x = 0$ has the unique solution $x= 0$ , so the column vectors of $M$ are independent.

That's just "by analogy". There would be lots of technicalities to consider if we want to prove anything about solutions to a differential equation. At least the analogy reminds us that a given column has entries all relate to the same function ( and a given row has entries that all relate to the same order of differentiation).

MathewsMD