Explore the fundamental concepts of duality in linear programming through this comprehensive lecture from MIT's Principles of Discrete Applied Mathematics course. Learn how to construct dual problems for arbitrary linear programs beyond the standard canonical form, and discover the powerful concept of complementary slackness that connects primal and dual solutions. Examine a physics-inspired proof of the strong duality theorem, which establishes the fundamental relationship between optimal values of primal and dual linear programs. Apply these theoretical insights to prove Koenig's theorem, demonstrating the practical applications of duality theory in discrete mathematics. Master the mathematical techniques for transforming optimization problems and understanding the deep connections between different formulations of linear programming problems.