Value of Vandermonde Determinant/Formulation 1/Proof 4

Proof
Let:


 * $V_n = \begin{vmatrix}

1     & x_1     & x_1^2     & \cdots & x_1^{n-2}     & x_1^{n-1} \\ 1     & x_2     & x_2^2     & \cdots & x_2^{n-2}     & x_2^{n-1} \\ \vdots & \vdots & \vdots    & \ddots & \vdots        & \vdots \\ 1     & x_n     & x_n^2     & \cdots & x_n^{n-2}     & x_n^{n-1} \end{vmatrix}$. Let $\map f x$ be any monic polynomial of degree $n-1$:

Apply Elementary column operations to $V_n$ repeatedly to show:

where

Select a specific degree $n-1$ monic polynomial:

The selected polynomial is zero at all values $x_1,\ldots,x_{n-1}$.

Then the last column of $W$ is all zeros except the entry $\map f {x_n}$.

Expand $\det \paren {W}$ by cofactors along the last column to prove

Let statement $\map P n$ be:


 * $\displaystyle V_n = \prod_{1 \mathop \le i \mathop \lt j \mathop \leq n} \paren { x_j - x_i }\quad$ for $n \ge 2$

Definition:Mathematical Induction will be applied.

Basis for the Induction

By definition, determinant $V_1 = 1$.

To prove $\map P 2$ is true, use equation (1) with $n=2$:


 * $\displaystyle V_2 = \paren {x_2 - x_1} V_1$

Induction Step

Assume $\map P n$ is true. To prove $\map P {n+1}$ is true:

The induction is complete.