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I got to the end of Geometric Duality Theorem which is

$$\vec c = \sum y_i^*\vec \alpha+\sum w_j^*(-\vec e_j)$$

$\vec c$ = Original objective function being maximized.
$y_i^*$ = Optimal solution to dual problem
$w_j^*$ = Optimal slack variable solutions to primal problem
$\vec e_j$ = A unit vector where the jth index is 1

I don't really understand what the negative on $e_j$ means. How does that help get the original objective function when multiplied by $w_j^*$? Also when you sum them do they just become a vector? So you just do a vector sum(for example, [1,2,3][1,2,1]+[1,2,3][1,2,1]=[1,4,3]+[1,4,3]=[2,8,6]?)

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