 | 1 | initial version | |
The n×n Vandermonde matrix is the matrix Vn=(1x0x20…xn−10 1x1x21…xn−11 ⋮⋮ 1xn−1x2n−1…xn−1n−1)
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 |
The n×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}$$\end{pmatrix}
$$
x = var('x', n=7) does.x = var('x',n=7)
does.
1. How many terms does the polynomial p have? have?
Hint: work out what the method number_of_operands`number_of_operands` does).
does).
2. Factorise $p$.
3. 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
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
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
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?
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