LINEAR ALGEBRA AND GEOMETRY
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No prerequisite is required
Verification of learning will consist of two parts:
1) A 2 hours written exam where the student is asked to solve some exercises at the level of those assigned during the course. The aim of the written examination is to verify if the student is able to apply the theoretical abstract results to deduce properties of vector spaces, linear maps and geometric objects in concrete situations.
2) A traditional oral exam, during which the student will have to show that she/he has acquired the basic notions, the proofs of the main theorems, and the ability to analyze concrete examples.
The course aims at introducing the students to the basic concepts and results of the algebra of linear spaces of finite dimension on the real and complex field. Due to its abstract nature, at first glance students tend to consider linear algebra one of the hardest courses of the first year. We aim at helping the digest of the topic and guiding the intuition of the students by both illustrating crucial applications to the analytic geometry of linear and quadratic objects in the Euclidean space and by examining many concrete examples and by giving regular homework. At the end of the course we expect that:
1) the student has acquired the basic notions and results from the theory of linear spaces;
2) the student is able to apply these results to deduce properties of vector spaces, linear maps and geometric objects both of linear and quadratic nature.
3) on the base of the main proofs illustrated during the lectures, the student is able to reproduce some simple reasonings to deduce more abstract properties of generic vector spaces, linear maps between these spaces and geometric objects of linear and quadratic nature.
The course is essentially divided in three interconnected parts.
A) Algebra of linear entities.
B) Algebra of quadratic entities.
C) Elements of analytic geometry in Euclidean spaces.
More specifically, the following are the main topics of the course.
 Vector spaces and their subspaces
 Generators, linear independence and bases
 Operations on subspaces and the Grassmann formula
 Linear maps, kernel and image
 Duality in finite dimensional vector spaces
 Linear maps and matrices: the representation theorem
 Change of bases and its matrix counterpart
 Determinants and linear systems
 The canonical form of the endomorphisms
 Euclidean vector spaces and linear isometries
 Symmetric and diagonalizable endomorphisms: the real spectral theorem
 Normal endomorphisms and the complex spectral theorem
 Bilinear and quadratic forms
 Affine Euclidean spaces: subspaces, frames and transformations
 Conics in the Euclidean plane and an overview of quadric hypersurfaces in the Euclidean space
The course is essentially divided in three interconnected parts.
A) Algebra of linear entities.
B) Algebra of quadratic entities.
C) Elements of analytic geometry in Euclidean spaces.
More specifically, the following are the main topics of the course.
 Vector spaces and their subspaces
 Generators, linear independence and bases
 Operations on subspaces and the Grassmann formula
 Linear maps, kernel and image
 Duality in finite dimensional vector spaces
 Linear maps and matrices: the representation theorem
 Change of bases and its matrix counterpart
 Determinants and linear systems
 The canonical form of the endomorphisms
 Euclidean vector spaces and linear isometries
 Symmetric and diagonalizable endomorphisms: the real spectral theorem
 Outline of the complex spectral theorem
 Bilinear and quadratic forms
 Affine Euclidean spaces: subspaces, frames and transformations
 Conics in the Euclidean plane and an overview of quadric hypersurfaces in the Euclidean space
Textbooks:
1) Marco Abate, Geometria. McGrawHill Education (1996)
2)Serge Lang, Algebra Lineare. Bollati Boringhieri, 3a edizione (2014)
Further teaching support
1) Slides of the lectures
2) Lecture notes on special topics
3) Regular homeworks
The teaching method will consist of classical frontal lectures of abstract flavor. Beside them, time slots will be regularly devoted to solve exercises on concrete cases, under the guide and supervision of a teacher.
Office Hours: each Tuesday 3pm  4pm or by appointment
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Degree course in: Physics