Wronskian

With the help of the Wronskian, which was named after the Polish mathematician Josef Hoene - Wronski, you can test scalar functions on linear independence, if they are sufficiently often differentiable. This can be a useful tool in particular when solving an ordinary differential equation.

Definition

For real - or complex-valued functions on an interval, the Wronskian is defined by

Where in the first line of the functions and the superscript numbers in parentheses indicate the first through -th derivative in the other rows.

The calculation of the Wronskian of linear ordinary differential equations of second order can be simplified by the use of Abelian identity.

Criterion for linear independence

Criterion

Applies for one, so the functions on the interval are linearly independent.

Counterexample for the converse

Caution: For not follow the linear dependence of functions, that is, the converse is false. It is, however, that the functions of a sub-region are linearly dependent. An example of this are the functions defined on the real numbers

For all

But leads to too and for what implies the linear independence of, respectively, for the two functions. For true and what is in linear dependence.