February 17 2009
Wednesday, February 18th, 2009On this day, I learned how to use the Wronksian to determine linear independence of a set of functions. It’s something I remember hearing about in Linear Algebra, but we just kinda glazed over it. For a Wronskian, you take your functions of x, then however many of them there are minus one, you take that many derivatives of them. If you have three functions, take two derivatives. Then you set them up in a 3×3 matrix like so:
|f1(x) f2(x) f3(x)| |f1'(x) f2'(x) f3'(x) | |f1''(x) f2''(x) f3''(x) |
Take the determinant, then if this ends up as a function of x, find any zeros in the [hopefully] given interval. If there are no zeros or it is a nonzero constant, congratulations, your functions are linearly independent. If there are zeros or your function ends up as 0, sadly, it is linearly dependent.
