Geometry 2014: Notes and Exercises Sophie M. Fosson
[email protected] June 5, 2014 ii These notes follow my exercise classes in chronological order; • are a support to student practice; • do not cover the whole course; • include some theory reminders; • include the exercises proposed during my exercise classes; • include further material and exercises; • are sure to contain several mistakes; • shall be updated every two-three weeks; • are expected to be enhanced by student feedback (that is, if you find any • mistakes please tell me! by email). Thank you, s.m.f. Contents 1 Week 1 1 1.1 Introduction to matrices . .1 1.2 Homogeneous linear systems . .2 1.3 Determinant . .4 1.4 Cramer's rule . .4 2 Week 2 7 2.1 Matrix reduction and rank . .7 2.2 Linear systems . .8 2.3 Matrix inversion . .9 3 Week 3 11 3.1 Rn as vector space . 11 3.2 Bases and dimension . 12 3.3 Products between vectors . 13 4 Week 4 15 4.1 Vector spaces . 15 4.2 Back to RN : straight lines and planes . 17 4.3 Lines and planes in R3: Cartesian Equations . 17 4.4 When a plane is a subspace? . 18 4.5 Geometric description of linear systems' solutions . 18 4.6 Geometric description of linear systems' solutions . 19 5 Week 5 21 5.1 Linear mappings . 21 5.2 Matrix associated with a linear mapping . 23 6 Week 6 25 6.1 Endomorphisms . 25 6.2 Eigenvectors, eigenvalues . 25 6.3 Multiple choice quizzes . 26 7 Week 7 29 7.1 Extra material .