MHB Linear dependence of polynomical functions

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The discussion centers on the linear dependence of polynomial functions using the Wronskian determinant. It states that a set of solutions is linearly independent if the Wronskian is non-zero across an interval. However, a zero Wronskian does not definitively indicate linear dependence, as noted in a YouTube comment. The example provided with functions f1(x) = x, f2(x) = x^2, and f3(x) = 4x - 3x^2 shows a Wronskian of zero, leading to the conclusion that these functions are indeed linearly dependent. The rank of the corresponding matrix confirms this dependency, illustrating the relationship between polynomial functions and their vector space representation.
Fernando Revilla
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I quote a question from Yahoo! Answers

Trying to understand the material here. It says that "...the set of solutions is linearly independent on I if and only if W(y1, y2...yn) doesn't = 0 for every x in the interval. (W(y1, y2...yn) being the Wronskian.)

But then I read a comment on youtube: "your first example is wrong, the wronsky is only used to show linear independence. if your determinant is 0 , it doesn't always mean ur your vectors are linear dependent." I guess the wronskian was used for vectors here but I imagine the concept is same for DE's?

So I have this set of functions f1(x) = x, f2(x) = x^2, f3(x) = 4x - 3x^2

and I get the wronskian to = 0. So by the youtuber's comment does this mean these set of functions could either be linearly independent or dependent? How do you determine whether they're independent or dependent?

I have given a link to the topic there so the OP can see my response.
 
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How do you determine whether they're independent or dependent?

Consider the vector space $\mathbb{R}_2[x]$ (polynomical functions with degree $\le 2$) and the canonical basis $B=\{1,x,x^2\}$. The respective coordinates are: $$[x]_B=(0,1,0)\;,\;[x^2]_B=(0,0,1)\;,\;[ 4x - 3x^2]_B=(0,4,-3)$$ But $\mbox{rank } \begin{bmatrix} 0 & 1 &\;\; 0\\ 0 & 0 & \;\;1 \\ 0 & 4 &-3\end{bmatrix}=2.$ We have no maximum rank, so the rows are linearly dependent. Using the standard isomorphism between vectors and coordinates, we conclude that $f_1(x)=x$, $f_2(x)=x^2$ and $f_3(x)=4x - 3x^2$ are linearly dependent.
 
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