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### Vandermonde Matrix

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. Factorise $p$. 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?

 2 None slelievre 17654 ●22 ●160 ●348 http://carva.org/samue...

### Vandermonde Matrix

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}$$\end{pmatrix}$$

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

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

Hint: work out what the method number_of_operandsnumber_of_operands does).
does).
2. Factorise $p$.
3. 
2.  Factor $p$. Based on the above calculations calculations, make a guess for a general fact 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?


 3 None updated 2022-03-19 14:43:11 +0200 slelievre 17654 ●22 ●160 ●348 http://carva.org/samue... Vandermonde Matrix 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. How many terms does the polynomial $p$ have? Hint: work out what the method number_of_operands does). Factor $p$. 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? 4 None updated 2022-03-19 14:45:23 +0200 slelievre 17654 ●22 ●160 ●348 http://carva.org/samue... Vandermonde Matrix 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. How many terms does the polynomial $p$ have? Hint: work out what the method number_of_operands does). number_of_operands does). Factor $p$. 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? 5 None updated 2022-03-19 14:47:21 +0200 slelievre 17654 ●22 ●160 ●348 http://carva.org/samue... Vandermonde Matrix The $n \times n$ Vandermonde matrix is the matrix $$V_n = \begin{pmatrix} 1 & x_0 & x_0^2 & \dots & x_0^{n-1} \\ 1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\ \vdots & \vdots & \vdots & & \vdots \\ 1 & x_{n-1} & x_{n-1}^2 & \dots & x_{n-1}^{n-1} \\ \end{pmatrix}$$ Calculate $p =\det(V_7)$. (Hint: =\det(V_7)$. Hint: work out what the code x = var('x', n=7) does. How many terms does the polynomial$p$have? Hint: work out what the method number_of_operands does). Factor$p$. 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 Or how would I at the least get started on it?       Copyright Sage, 2010. Some rights reserved under creative commons license. 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