Prove the following theorem:
If are distinct points in an interval , and if distinct points are outside that interval, then the matrix having elements is invertible.
for n = 1, there's nothing to prove. so, from now on i'll assume that n > 1. we want to prove that your matrix has a non-zero
determinant. so suppose is the determinant of your matrix. first see that:
where
so we just need to prove that the proof is by induction on if then:
now suppose that whenever , and now define:
clearly is a polynomial of degree and
it's also evident that thus where is a constant.
so we only need to prove that since we only need to prove that this is easy to prove
because from the definition of it's easy to see that: but by
induction hypothesis thus