Show (1,a,a^2) (1,b,b^2) (1,c,c^2) are linearly independant

Hi there,

I'm attempting to solve the following question: Show {1,a,a^2),(1,b,b^2),(1,c,c^2)} are linearly independant if a, b and c are distinct.

I'm sure I'm so close!

What I've come up with so far is putting the three vectors in rows in a matrix, and finding the determinant with row expansion, which gives a(b^2-c^2)+b(c^2-a^2)+c(a^2-b^2)

I know that they are linearly independant when this determinant doesn't equal zero, and it's simple to show it DOES equal zero if a=b, b=c or c=a. For example, when a=b, b^3-bc^2+bc^2-b^3+c(b^2-b^2)=0

However, I don't think showing this is sufficient proof that when they are distinct, the determinant will never equal zero.

Could you point me in the right direction?

Thanks!