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The Wronskian of two functions is $W(t) = t^2 - 4$. Are these functions linearly dependent?

I don't think they are, since the Wronskian is only equal to zero when $t = 2$ or $t = -2$. I'm not sure though, since the Wronskian has thus far only been used in my class for functions of which we know they're solutions to a differential equation.

Question: Can you conclude that these functions are linearly independent because their Wronskian is only equal to zero at a couple of points?

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  • $\begingroup$ Isn't $W(\cdot)$ the Lambert W Function? At least, that's what I have been taught. $\endgroup$ – Mr Pie Feb 27 '18 at 14:10
  • $\begingroup$ The notation can also be used to denote the Wronskian. $\endgroup$ – Adrian Keister Feb 27 '18 at 14:12
  • $\begingroup$ @user477343 Really? And what is $\pi$? Is it the quotient between the perimeter and the diameter of any circle? Or is it the prime counting function? $\endgroup$ – José Carlos Santos Feb 27 '18 at 14:12
  • $\begingroup$ @JoséCarlosSantos Hahah well in that case, I have seen the two different uses of $\pi$. I have never heard of the Wronskian function before, so it is new to me and I was just clarifying. But perhaps I asked a dumb question anyway since there are only so many letters in the alphabet :) $\endgroup$ – Mr Pie Feb 27 '18 at 14:14
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If the wronskian of a set of function is different from zero at at least a point of the domain than the functions are linearly independent

This is a consequence of the fact that the zero element of the vector space is the null function, that is the function that has always value zero.

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Of course we can conclude that. If they were linearly dependent, the Wronskian would be $0$ everywhere.

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  • $\begingroup$ Thanks for your reply! Suppose that both $f_1$ and $f_2$ are one-dimensional functions, in the sense that $f_1(x) = y$ and $f_2(x) = z$, where $y,z$ are scalars. What would the Wronskian be? $\endgroup$ – Mr. President Mar 6 '18 at 14:34
  • $\begingroup$ @Mr.President It would be $0$, of course. $\endgroup$ – José Carlos Santos Mar 6 '18 at 15:21

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