How can i show that two polynomials are linearly independent How can I show that two polynomials, specifically : $x^3 - x$ , $x^2 - x$ are linearly independent?
Intuitively I know they are since one is cubed and the other is squared. But how do I prove it?

Suppose $a(X^3 - X) + b(X^2 -X) = 0$. We want to show that $a = b = 0$.I 
Suppose we are working in $F[X]$ where $F$ is some field. Then equality is defined as equality of coefficients ($\sum_i a_i x^i = \sum_i b_i x^i$ iff for all $i$ we have $a_i = b_i$. Then we have $aX^3 + bX^2 + (-a + -b)X = 0 = 0X + 0X^2 + 0X^3$, so $a = 0, b = 0$ by comparing the coefficients of $X^3$ and $X^2$ directly.
If we are working in the space of polynomial functions on the reals, say, the equality $aX^3 + bX^2 + (-a -b)X = 0$ should hold as functions, so the identity should hold for all values of $X$ (in the reals say). So pick e.g. $X= 2$ and $X=3$ to get two equations in $a$ and $b$ and solve to show that $a=b=0$.   
