If we have n functions that are n-1 times differentiable on an interval [a,b], the Wronskian is defined as:
So, basically, you create a matrix where each column vector is the successive derivatives of fi and you take the determinant.
The significance is the following theorem:
If there exists x in [a,b] such that W[f1, ..., fn](x) ≠ 0, then f1, ..., fn are linearly independent.
Why does that matter? Well, it means they form a basis for the subspace they span, which is sometimes a useful thing to know. (At least, it's useful if you're staring at a Q question that asks you to find the basis of a subspace. Maybe not that useful any other time.)
Apparently, it's also occasionally useful for solving higher order systems of differential equations, though the methods I dug up on line didn't seem a whole lot easier than just solving the equations the normal way. At any rate, not too many people do that anymore. Numerical packages are so powerful now, it's just not worth the effort except in trivial cases. Wronski developed this idea in 1812, a decade before Babbage secured funding for building the world's first computer.
So, probably won't be using this result very much unless it comes up on the Q, which it might.
I should probably also remember that the converse is not necessarily true. You can have independent functions with a Wronskian that is zero everywhere.
No comments:
Post a Comment