User:Christopherlumb/Sandbox

$$\int_0^{30} \mathcal{L}(t) dt$$

An equivalent proof for Rn starts with the summation below.

Expanding the brackets we have:


 * $$ \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2

= \sum_{i=1}^n x_i^2 \sum_{j=1}^n y_j^2 + \sum_{j=1}^n x_j^2 \sum_{i=1}^n y_i^2 - 2 \sum_{i=1}^n x_i y_i \sum_{j=1}^n x_j y_j $$,

collecting together identical terms (albeit with different summation indices) we find:


 * $$ \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2

= \sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2. $$

Because the left-hand side of the equation is a sum of the squares of real numbers it is greater than or equal to zero, thus:



\sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2 \geq 0 $$.