# Math Help - Show (1,a,a^2) (1,b,b^2) (1,c,c^2) are linearly independant

1. ## 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!

2. You can factorise the determinant as:
det=(a-b)(b-c)(c-a)

The result follows immediately.

3. Wow, I was trying so hard to factorise it but missed that completely. :|
Thanks!

4. Originally Posted by genericguy
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!
Just as a point of interest, in general (1,x_1,s,x_1^n),...,(1,x_n,...,x_n^n) are linearly independent if x_j != x_k k,j in [n] This is the Vandermonde matrix.