| Version 4 |
Version 3 |
| To begin, note that the determinant of the $n \times n$ Vandermonde |
To begin, note that the determinant of the $n \times n$ Vandermonde |
| matrix (which we shall denote as `$\Delta$') is a homogenous |
matrix (which we shall denote as `$\Delta$') is a homogenous |
| polymonial of order $n(n-1)/2$ because every term in the determinant |
polymonial of order $n(n-1)/2$ because every term in the determinant |
| is, up to sign, the product of a zeroth power of some variable times the first |
is, up to sign, the product of a zeroth power of some variable times the first |
| power of some other variable , $\ldots$, the $n-1$-st power of some |
power of some other variable , $\ldots$, the $n-1$-st power of some |
| variable and $0 + 1 + \cdots + (n-1) = n(n-1)/2$. |
variable and $0 + 1 + \cdots + (n-1) = n(n-1)/2$. |
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| Next, note that if $a_i = a_j$ with $i \neq j$, then $\Delta = 0$ |
Next, note that if $a_i = a_j$ with $i \neq j$, then $\Delta = 0$ |
| because two columns of the matrix would be equal. Since $\Delta$ is a |
because two columns of the matrix would be equal. Since $\Delta$ is a |
| polynomial, this implies that $a_i - a_j$ is a factor of $\Delta$. |
polynomial, this implies that $a_i - a_j$ is a factor of $\Delta$. |
| Hence, |
Hence, |
| \[ \Delta = C \prod_{1 \leq i < j \leq n}(a_j - a_i) \] |
\[ \Delta = C \prod_{1 \leq i < j \leq n}(a_j - a_i) \] |
| where C is some polynomial. However, since both $\Delta$ and the |
where C is some polynomial. However, since both $\Delta$ and the |
| product on the right hand side have the same degree, $C$ must have |
product on the right hand side have the same degree, $C$ must have |
| degree zero, i.e. $C$ must be a constant. So all that remains is the |
degree zero, i.e. $C$ must be a constant. So all that remains is the |
| determine the value of this constant. |
determine the value of this constant. |
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| One way to determine this constant is to look at the coefficient of |
One way to determine this constant is to look at the coefficient of |
| the leading diagonal, $\prod_n (a_n)^{n-1}$. Since it equals 1 in both |
the leading diagonal, $\prod_n (a_n)^{n-1}$. Since it equals 1 in both |
| the determinant and the product, we conclude that $C = 1$, hence |
the determinant and the product, we conclude that $C = 1$, hence |
| \[ \Delta = \prod_{1 \leq i < j \leq n}(a_j - a_i). \] |
\[ \Delta = \prod_{1 \leq i < j \leq n}(a_j - a_i). \] |
