Linear Independence of Functions with Wronskian | Linear Algebra

We introduce the Wronskian of a set of n-1 continuously differentiable functions. We'll see that the Wronskian, which is the determinant of a matrix made up of the functions and their derivatives, can tell us about the linear independence of the set of functions. We'll also do an example of using the Wronskian to show a set of functions is linearly independent. #linearalgebra
Linear Algebra course: • Linear Algebra
Linear Algebra exercises: • Linear Algebra Exercises
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0:00 Intro
0:15 Are the Functions Linearly Independent?
1:08 The Wronskian
2:22 Linear Dependence and the Wronskian
6:02 An Important Wronskian Theorem
6:53 Show 1, e^x, and e^(2x) are Linearly Independent with Wronskian
8:28 Conclusion
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[@kingbeauregard]

For fun, I just did a Wronskian of the functions "[sin(x)]^2" and "1 - [cos(x)]^2". The determinant worked out to be uniformly 0, so, they're linearly dependent. I just thought it'd be entertaining to do an example where the Wronskian actually detects linear dependence.

[@talastra]

yes, good idea

[@WrathofMath]

Yup, it's a fascinating result, kind of fun to use! Edit: Per @scollyer.tuition 's comment, he is correct that W=0 does not necessarily imply linear dependence. We saw in the video, and you see in your example, how linear dependence forces W=0, but the converse is not true. We of course used the contrapositive in the video, which is true since it is equivalent to linear dependence -> W=0 .

[@scollyer.tuition]

IIRC, you need additional conditions on your functions to guarantee linear dependence - W = 0 does not imply linear dependence, in general.

[@WrathofMath]

@@scollyer.tuition That's correct, added edit to previous comment to clarify

[@user-wr4yl7tx3w]

great content, thanks

[@WrathofMath]

Thanks for watching!

[@kingbeauregard]

Homer Simpson once said, "Donuts ... is there anything they can't do?" That's how I feel about determinants.

[@WrathofMath]

Same