In Autumn 2012 Sasha Ananin asked me to give an intensive IUM-style mini-course on basic projective geometry and attendant algebra at the São Carlos branch of the Universidade de São Paulo (USP, Brasil). In a preliminary program that appeared after discussions with Sasha I have made an attempt to combine a review of classical projective varieties: quadrics, Segre, Veronese, and Grassmannian with a self-contained introduction to linear, multilinear, and polynomial algebra and supplement this mixture with real affine convex geometry. The important part of the course should consist of exercises and home task problems like it is usual at IUM. The principal organizer of my visit to São Carlos was Carlos Henrique Grossi Ferreira. Without his support these lectures would be impossible.
I've combined the content of all the lectures in a small textbook (1.2 Mb PDF) (equipped with comments to some exercises).
Multilinear algebra and geometry.
Affine convex geometry.