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The $n \times n$ Vandermonde matrix is the matrix $$ V_n = \begin{pmatrix} 1 & x_0 & x_0^2 \dots & x_0^{n-1} \\ 1 & x_1 & x_1^2 \dots & x_1^{n-1} \\ \vdots & & & \vdots \ \\ 1 & x_{n-1} & x_{n-1}^2 \dots & x_{n-1}^{n-1} \end{pmatrix} $$

• Calculate $p =\det(V_7)$. (Hint: work out what the code x = var('x', n=7) does.

1. How many terms does the polynomial $p$ have?

   Hint: work out what the method `number_of_operands` does).
   

2. Factor $p$.

3. Based on the above calculations, make a guess for a general fact about the determinant of any Vandermonde matrix.

How would I solve this question? or How would I at the least get started on it?

The $n \times n$ Vandermonde matrix is the matrix $$ V_n = \begin{pmatrix} \begin{pmatrix} 1 & x_0 & x_0^2 & \dots & x_0^{n-1} \\ \\ 1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\ \\ \vdots & \vdots & & & \vdots \\ \\ 1 & x_{n-1} & x_{n-1}^2 & \dots & x_{n-1}^{n-1} \\ \end{pmatrix} $$

• Calculate $p =\det(V_7)$. (Hint: work out what the code x = var('x', n=7) does.

1. How many terms does the polynomial $p$ have?

   Hint: work out what the method `number_of_operands` does).
   number_of_operands

   does).

2. Factor $p$.

3. Based on the above calculations, make a guess for a general fact about the determinant of any Vandermonde matrix.