Bézout's Theorem and Cayley-Bacharach

This lecture covers the proof of Bézout's theorem, which states that for two projective plane curves of respective degrees with no common components, the intersection multiplicity is the product of their degrees. It also delves into the Cayley-Bacharach theorem, discussing how two projective cubics intersecting at distinct points lead to linear combinations. The lecture explores applications to incidence geometry, emphasizing the non-collinearity of four points and the implications for components. Various mathematical concepts such as basis, isomorphism, and dimensionality are elucidated through detailed examples and derivations.