Chapter 3.2, Problem 34P

Assume as known that the Vandermonde determinant $V=\left|\begin{array}{l}11\mathrm{...}1\\ r_1{r}_2\mathrm{...}{r}_3\\ r_1^2{r}_2^2\mathrm{...}{r}_n^2\\ ⋮⋮⋮\\ r_1^{n-1}{r}_2^{n-1}\mathrm{...}{r}_n^{n-1}\end{array}\right|$ is nonzero if the numbers $r,r_2,\mathrm{...},r_n$, are distinct. Prove by the method of Problem 33 that the functions $f_i\left(x\right)=\exp \left(r_{i}x\right),1\underset{\_}{\underset{\_}{<}}i\underset{\_}{\underset{\_}{<}}n$ are linearly independent.