Mathematics

PAOLO BRAVI
SIMONE DIVERIO

General aim:
to acquire basic knowledge on systems of linear equations, vector spaces, linear maps, affine spaces, euclidean spaces.

Specific aim:

Knowledge and comprehension: successful students will acquire basic notions and results about solvability of linear systems, matrix calculus, vector spaces, linear maps between them, affine and euclidean spaces.

Applied knowledge and comprehension: successful students will be able to solve systems of linear equations with a finite number of variables, and to
recognize mathematical problems that can be codified into vector spaces and linear maps and therefore to solve them;
he will be able to manipulate matrices and determine the solvability of a system of linear equations and the invertibility of a linear map by studying its rank and by
computing the determinant of the associated matrix; moreover, he/she will be able to compute the eigenvalues of a linear endomorphism
and to determine the associated decomposition into eigenspaces; he/she will be able to solve problems in which euclidean products appear; he/she will be able to solve problems involving affine and euclidean spaces; moreover
he/she will acquire rudimental knowledge of some basic fundamental algebraic structure, such as groups, which will be investigated in greater depth in later courses.

Critical thinking abilities: in this course the student will acquire basic knowledge that will make him able to discover analogies between the topics learnt in the course and topics in group theory (that will be taught in Algebra 1), functions of many real variables (that will be taught in Analisi 2), geometry of quadrics and of projective spaces (that will be taught in Geometria 1).

Communication skills: ability to illustrate the contents of the course in the oral exam and, eventually, in answering written theoretical questions.

Learning skills: the knowledge acquired will allow the student to approach the study (on an individual basis, or in a LM courses) of the theory of linear operators on vector spaces possibly of infinite dimensione, of family of vector spaces (vector bundles) and of the eigenspace decompositions of commutative algebras of endomorphisms, and of riemannian geometry.

General goals: basic knowledge regarding systems of linear equations, vector spaces, linear maps, affine spaces, euclidean spaces.

Specific goals:

Knowledge and comprehension: at the end of the course the student will master basic notions and results regarding solutions of systems of linear equations, matrix calculus, the category of vector spaces, affine spaces and euclidean affine spaces.

Application skills: at the end of the course the student will be able to solve systems of linear equations, he will recognize mathematical problems whose solution(s) can be found via linear algebra and he will be able to find the solution(s); he will be able to do computations with matrices, and determine the solvability of a system of linear equations and the invertibility of a linear map via arguments involving the rank and via computations of determinants; he will be able to compute eigenvalues and eigenspaces of an endomorphism of a vector space; he will be able to solve problems in which euclidean products appear; he will be able to solve problems involving affine spaces and euclidean affine spaces;

Autonomy of judgement: at the end of the course the student will be able to discover relations between Linear Algebra and Group Theory (that will be introduced in Algebra 1), Multivariable Calculus (that will be introduced in Analisi 2), the geometry of conics, quadrics in projective spaces (that will be introduced in Geometria 1).

Communication skills: at the end of the course the student will be able to explain the theoretical contents of the course.

Ability to learn: at the end of the course the student will be able to begin the study of the theory of linear maps on vector spaces of infinite dimension, of the theory of vector bundles, of the theory of eigenspace decomposition of elements of a commutative algebra of endomorphisms, and of Riemannian Geometry.

None.

Sets, functions, relations, induction. Rings, fields. Vector spaces, linear dependence, bases. Grassmann's formula. Linear maps, isomorphisms. Matrices and linear maps. The dual of a vector space. Affine spaces. Affine maps. Affine coordinates. Parametric and cartesian equations of affine subspaces. Multilinear maps. Determinants. Binet's formula, Laplace's formula. Endomorphisms, eigenvalues and eigenspaces, characteristic polynomial. Conjugate matrices, diagonalizable matrices. Euclidean scalar product on a real vector space, ON bases, orthogonal projection, Gram-Schmidt algorithm. Hermitian scalar product on a complex vector space. Orthogonal and unitary group. Affine euclidean spaces (of finite dimension). Isometries. Classification of isometries in low dimension. Discrete groups of isometries of a euclidean plane.

K. O'Grady, Algebra Lineare e Geometria.
M. Manetti, Algebra lineare, per matematici.

Lectures and exercise classes.

The exam consists of a written test (with problems of the same type as those carried out in the exercise classes) and an oral exam (with questions about the topics illustrated during the course).
In order to get the passing grade 18/30, the student must show basic knowledge of the main topics, and to be able to carry out at least the simplest exercises among the assigned ones.
To achieve a score of 30/30 cum laude, the student must show an excellent knowledge of all the topics covered during the course and to be able to connect them in a logical and consistent way.