Scalar input or scalar output systems; two or three input, two output systems - some examples; the transfer and Hankel matrices; polynomial matrices; projective space; projective algebraic geometry I - basic concepts; projective algebraic geometry II - regular functions, local rings, morphisms; exterior algebra and grassmannians; the Laurent isomorphism theorem I; projective algebraic geometric III - products, projections, degree; the Laurent isomorphism theorem II; projective algebraic geometry IV - families, projections, degree; the state space - realizations, controllability, observability, equivalence; projective algebraic geometry V - fibres of morphisms; projective algebraic geometry VI - tangents, differentials, simple subvarieties; the geometry quotient theorem; projective algebraic geometry VII -divisors; projective algebraic geometry VIII - intersections; state feedback; output feedback; formal power series, completions, regular local rings, and Hilbert polynomials; specialization, generic points and spectra; differentials; the space nm; review of affine algebraic geometry.
