Linear dependence of polynomical functions

[Fernando Revilla]

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.

 

[Fernando Revilla]

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.