1. Knowledge and understanding:
definitions and theorems about vector spaces, linear applications and
endomorphisms, fundamental constructions and theorems about lines and
planes in the 3-dimensional space and conics in the plane, definitions
and theorems about quadrics.
2. Applying knowledge and understanding: being able
to compute the rank of a matrix, to study a vector space, to study a
linear application, to determine eigenvalues and eigenvectors of
endomorphisms, to diagonalize a matrix, being able to solve linear
geometry problems about points, lines and planes in the 3-dimensional
space, to classify conics and quadrics and to study pencils of conics in
the plane.
4. Communication skills:
the frequence of the lessons and the suggested books will help the
student to familiarize with the rigour of mathematical language and to
learn the specific language of linear algebra and geometry. Through the
continuous interaction with the teacher, the student will learn to
communicate with rigour and clarity his/her acquired knowledge, both in
an oral and in a written way. At the end of the course, the student will
have learnt that the mathematical language is useful in order to be
able to communicate in a clear way in a scientific setting.
During the lessons topics and concepts will be proposed in a formal way, together with meaningful examples, applications and exercises. A tutor will carry classroom exercises. The student will be sollicited to carry out exercises autonomously, even during the lessons.
Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.
Learning assessment may also be carried out on line, should the
conditions require it.
Linear Algebra:
Geometry
The proofs of the theorem signed with * can be ometted.
More didactic material is available at https://studium.unict.it/ and https://www.dmi.unict.it/bonacini/didattica/
Subjects | Text References | |
---|---|---|
1 | Introduction to set theory. introduction to fields and vector spaces. Determinant of a matrix. Rank and reduction of a matrix. Resolution of alinear system. Required time: 9 hours. | Theory book: chapters 1,3. Exercise book: chapter 1. |
2 | Operations with matrices. Required time: 2 hours. | Theory book: chapter 3. Exercise book: chapter 1 |
3 | Vector spaces. Generators and linear independence. Subspaces. Base and components with respect to a base. Dimension of a vector space. Required time: 9 hours. | Theory book: chapter 2. Execrcise book: chapter 2. |
4 | Sum and intersection of vector spaces. Extracting a base from a set of generators and expanding a linearly independent set to a base. Required time: 2 hours. | Theory book: chapter 2. Exercise book: chapter 2. |
5 | Linear applications and their assignment. Studying a linear application. Computation of images and inverse images. Required time: 10 hours. | Theory book: chapter 4. Exercise book: chapters 3,4. |
6 | Base change matrices and similar matrices. Operations with linear applications. required time: 2 hours. | Theory book: chapter 4. Exercise book: chapter 5. |
7 | Eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Endomorphisms and diagonalization. Required time: 9 hours. | Theory book: chapter 5. Exercise book: chapter 6. |
8 | Applications under conditions. Restrictions and extensions of linear applications. Required time: 2 hours. | Theory book: chapter 5. Exercise book: chapters 7,8. |
9 | Generalities on vector calculus. Cartesian coordinates and homogeneous coordinates. Assignment of lines and planes and their equations. Points at infinity. Intersections. Parallelism and orthogonality. Pencils of lines and planes. Distances. Required time: 10 hours. | Theory book: chapters 1,2,3. Exercise book: chapter 1. |
10 | Angles. Orthogonal projections. Bisecting lines and planes. Symmetries. Locus of lines. | Theory book: chapters 1,2,3. Exercise book: chapter 1. |
11 | Conics and associated matrices. Change of coordinates in the plane, orthogonal invariants and reduced equations of a conic. Classification of conics. Circles. Tangent lines. Pencils of conics. Required time: 8 hours. | Theory book: chapter 4. Exercise book: chapter 2. |
12 | Complete study of conics. Conics under conditions. Required time: 4 hours. | Theory book: chapter 4. Exercise book: chapter 2. |
13 | Quadrics and associated matrices. Irreducible quadrics. Vertices of a quadrics and degenerate quadrics. Conic at infinity. Cones and cylinders. Reduced equations of a quadric. Classification of non degenerate quadrics. Required time: 7 hours. | Theory book: chapter 5. Exercise book: chapter 3. |
14 | Tangency. Conic sections of a quadric. Spheres. required time: 2 hours. | Theory book: chapter 5. Exercise book: chapter 3. |
The examination is written and oral. The written examination, which usually lasts 3 hours, is compulsory to take the oral examination.
Exminations may also be carried out on line, should the conditions require it. In such a case, the length of the written examination might change.
Linear Algebra exercises:
1. study of a linear application with parameter, determining kernel and image.
2. study of the diagonalization of an endomorphism with parameters, determining, if possible, a bases of eigenvectors.
3. inverse image of a vector, resolution of a linear system with parameter, inverse image of a vector space, image of a vector space.
4. exercises on vector spaces and their dimension, direct sum, operations on linear applications, induced linear applications, restrictions and extensions.
Geometry exercise:
1. linear geometry exercises in the 3-dimensional space: parallelism and orthogonality, distances, orthogonal projections, angles.
2. pencil of conics, complete study of a conic, conics under conditions.
3. classification of quadrics with parameter, quadrics under conditions, intersection of quadrics with planes.