Course program
1. Vectors in the plane and in space.
2. Operations between vectors.
3. Linear combinations.
4. Dot product.
5. Cross product.
6. Mixed product.
7. Cartesian coordinates.
8. Equations of lines and planes.
9. Angles.
10. Distances.
11. Projections.
12. Parametric equations of lines.
13. Parametric equations of curves in space.
14. Tangents.
15. Systems of linear equations.
16. Row echelon form.
17. Gauss algorithm.
18. Matrices.
19. Operations between matrices.
20. Matrix multiplication.
21. Invertible matrices.
22. Inverse of a matrix.
23. Gaussian algorithm.
24. Determinants.
25. Properties of determinants.
26. Determinant of a product.
27. Laplace expansion.
28. Cramer's Rule.
29. Vector spaces over ℝ.
30. Linear combinations.
31. Subspaces.
32. Sum of subspaces.
33. Span of a set of vectors.
34. Linear independence.
35. Bases.
36. Dimension.
37. Coordinates.
38. Row space of a matrix.
39. Column space of a matrix.
40. Rank of a matrix.
41. Applications to systems of linear equations.
42. Linear maps.
43. Kernel and image.
44. Dimension theorem.
45. Composition of linear maps.
46. Matrix of a linear map.
47. Change of basis.
48. Similar matrices.
49. Eigenvalues.
50. Eigenvectors.
51. Characteristic polynomial.
52. Eigenspaces.
53. Diagonalization of matrices.
54. Diagonalization of linear maps.
55. Bilinear forms.
56. Quadratic forms.
57. Matrix associated to a bilinear form.
58. Symmetric matrices.
59. Diagonalization of quadratic forms.
60. Conics in the plane and their classification.
Prerequisites
The usual high-school matematics that includes elementary geometry, algebra (polynomials, equations, inequalities) and trigonometry
Books
S. Capparelli – A. Del Fra: Geometria, Nuova edizione, (Esculapio, 2015)
S. Capparelli: Esercitazioni di Geometria, Esculapio, 2019
Teaching mode
The course is presented via traditional classroom lectures containing a suitable amount of motivation and new material, as well as the necessary examples and exercises. Both new material and exercises are presented together.
Frequency
Attendance is optional
Exam mode
The written test will last 2-3 hours and contain a suitable number of exercises concerning THE BASIC NOTIONS OF LINEAR ALGEBRA (MATRICES, DETERMINANTS, SYSTEMS OF LINEAR EQUATIONS, VECTOR SPACES, LINEAR APPLICATIONS) AND OF ANALYTIC GEOMETRY IN DIMENSION TWO AND THREE (LINES AND PLANES, BRIEF INTRO TO CURVES AND SURFACES.) There might also be problems requiring the student TO TRANSLATE SIMPLE GEOMETRIC PROBLEMS IN ANALYTIC FORM AND TO INTERPRET THE ALGEBRAIC RESULTS as well as simple proofs. The test is aimed at checking the understanding of the material and the ability to apply it in simple cases. The written test is then followed by an optional oral examination only for those who completely passed the written test and aspire to a top grade.
The final grade will not be an average between the written test and the oral exam, rather a numerical expression of a final evaluation.
Bibliography
S. Capparelli – A. Del Fra: Geometria, Nuova edizione, (Esculapio, 2015)
S. Capparelli: Esercitazioni di Geometria, Esculapio, 2019
Lesson mode
The course is presented via traditional classroom lectures containing a suitable amount of motivation and new material, as well as the necessary examples and exercises. Both new material and exercises are presented together.
