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Algebra Lineare e Geometria
Linear Algebra and Geometry
Maria Chiara Brambilla

Seat Ingegneria
A.A. 2016/2017
Credits 6
Hours 48
Period I
Language ENG

Prerequisites
none

Learning outcomes
KNOWLEDGE AND UNDERSTANDING:
The course gives the students the basic knowledges in geometry and in the field of complex numbers, using in particular tools in linear algebra and analytic geometry such as vector spaces, linear applications and their representations in terms of vectors and matrices. These knowledges cooperate with the ones learned in the courses of Mathematical Analysis I and II so that the student acquires the ability to understand criteria, procedures and limits of application of mathematical methods to real problems.
CAPACITY TO APPLY KNOWLEDGE AND UNDERSTANDING:
The student must develop the ability to apply the tools of linear algebra and analytic geometry such as vector spaces, linear applications and their representations in terms of vectors and matrices in order to formalize, identify and solve problems in Information Engineering.
TRANSVERSAL SKILLS:
Solving problems and exercises, that will be worked out both during practice hours and at home, and the preparation for the oral part of the exam will improve judgment independence in general, communication skills, autonomous learning abilities and the capability of the student to draw conclusions.

Program
Vector spaces. Basis and dimension of a vector space, coordinates. Grassmann's theorem. Linear maps. Kernel and image of a linear map. Dimension theorem. Linear systems. Rouche'-Capelli's theorem. Ladder reduction. Operation on matrices and linear maps. Sum and composition of linear maps. Isomorphisms. Product of matrices. Invertible matrices.Change of basis. Matrix associated to a linear map with respect to two basis. Similar matrices. Determinant. Eigenvalues and eigenvectors. Diagonalizable endomorphisms. Characteristic polynomial. Algebraic and geometric multiplicity. Necessary and sufficient criterion for diagonalizability of an endomorphism. Symmetric bilinear forms and scalar products. Cauchy's inequality. Congruent matrices. Symmetric and orthogonal endomorphisms. Spectral theorem.

Development of the examination
LEARNING EVALUATION METHODS
The learning evaluation is carried out by two exams: - a practical examination, which consists of solving exercises and problems related to the topics explained in the course. The test must be completed in 3 hours; - a theoretical examination, consisting in a discussion, written and oral,of the topics of the course. In particular the knowledge and the understanding of all definitions, theorems and proofs explained in the classes will be tested. The practical exam is preliminary to the theoretical one. It is necessary to pass the practical exam in order to do the theoretical one. The two exams must be passed in the same exam session. If the student fails the theoretical exam, he/she must repeat also the practical one.

LEARNING EVALUATION CRITERIA
In order to pass the learning evaluation, the student must demonstrate that he/she has understood the basic concepts of linear algebra and geometry explained in the course. In particular in the practical test the student must show that he/she is able to apply independently the learned techniques in solving exercises and problems. In the theoretical exam the student must be able to expose the theoretical contents with the correct language and accuracy.

LEARNING MEASUREMENT CRITERIA
Each of the tests is graded on a scale from 0 to 30. The final grade will be decided starting from the two test grades.

FINAL MARK ALLOCATION CRITERIA
The final grade will be positive only if in both of the tests the students gets the passing grade (18/30). The maximal grade is reached if the student proves a knowledge and a thorough understanding of the course content. The maximal grade with honors is reserved to the students who passed both of the tests in a complet and correct way, showing special independence and excellence.

Recommended reading
The learning evaluation is carried out by two exams: - a practical examination, which consists of solving exercises and problems related to the topics explained in the course. The test must be completed in 3 hours; - a theoretical examination, consisting in a discussion, written and oral,of the topics of the course. In particular the knowledge and the understanding of all definitions, theorems and proofs explained in the classes will be tested. The practical exam is preliminary to the theoretical one. It is necessary to pass the practical exam in order to do the theoretical one. The two exams must be passed in the same exam session. If the student fails the theoretical exam, he/she must repeat also the practical one.

Courses

• Ingegneria Informatica e dell'Automazione (Corso di Laurea Triennale (DM 270/04))