### Teachings

Select Academic Year:     2017/2018 2018/2019 2019/2020 2020/2021 2021/2022 2022/2023
Professor
MARIA PAOLA PIU (Tit.)
Period
First Semester
Teaching style
Convenzionale
Lingua Insegnamento
ITALIANO

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/60]  PHYSICS [60/00 - Ord. 2012]  PERCORSO COMUNE 8 64

### Objectives

The student will gain the ability to address and resolve a problem of linear algebra and a problem of analytical geometry (in plane and space), using the tools provided by linear algebra.

More specifically: he should acquire the ability to calculate the rank, the determinant of a matrix, find a basis of a vector space, components with respect to a basis, matrix of a change of basis and of linear map. Moreover the student has to gain the ability to solve problems concerning the search for eigenvalues and eigenvectors of an endomorphism.
He should acquire the ability to recognize and represent the lines (both in plane and in space) and the conics. He must know how to solve problems in plane geometry by using the methods of linear algebra and vector calculus. He must be able to recognize and represent planes and quadrics in space, and to determine the geometric transformations of the plane and space.

### Prerequisites

Ability to manipulate and simplify rational expressions even in more variables. Knowing how to solve equations and inequalities in one variable of 1° and 2° degree. Familiarity with the geometric properties of elementary plane figures. Knowing how to calculate the length of a circle, the area of the circle, the volumes of cube, cuboid, pyramid, cone, cylinder, and sphere. Familiarity with the theorems of Thales, Pythagoras and Euclid and ability to use them to solve problems of elementary geometry. Familiarity with the geometric meaning of sine, cosine and tangent, and the main trigonometric formulas.

### Contents

Linear systems, matrices, determinants Linear equations. Linear systems. Matrices, determinats and rank. Determinants and linear systems. Matrices operations. The linear group.
Vectors in space Definition of vector. Sum of vectors, product by a real number. Linear dependent vectors. Basis and components. Scalar product, orthonormal basis. Vector product. Mixed product.
Vector spaces Definition and examples of vector spaces. Subspaces. Linear combinations and liner independency. Basis. Dimension. Infinite dimensional vector spaces.
Linear maps, eigenvalues and eigenvectors Maps between vector spaces. Linear maps and associated matrices. Linear systems. Operations between maps and associated matrices. Dual space and transpose. Eigenvalues and eigenvectors.
Bilinear forms Symmetric and skew-symmetric forms. Associated matrices. Rank of a bilinear forms. Non degeneracy. Quadratic forms. Positive definite and indefinite forms. Canonical form of a quadratic form.
Euclidean spaces. Euclidean vector spaces. Modulus of a vector, Orthogonal vectors. Schwarz and triangular inequality. Angle of two non vanishing vectors. Gram-Schmidt method. Orthonormal bases. Decomposition of a vector with respect to an orthonormal basis. Orthogonal complement of a subspace. Orthogonal projection. Minimum distance properties of the orthogonal projection. Applications of orthogonal projections. Cartesian coordinates. Changes of Cartesian coordinates. Euclidean transformations. Linear isometries. Isometries. The Group of isometries.
Geometry of the plane and space. Cartesian coordinates. Distance of two points. Midpoint of a segment. Parametric equation of the affine line. Affine line in the plane. Mutual position of two lines. Straight line for two points. Proper and improper pencil of lines. Distance from a point to a line. Parametric equation of the affine plane. Cartesian equation of the plane. Normal to the plane. Plane for three points. Cartesian equation of the straight line in space. Pencil of planes. Mutual position of a line and a plane. Mutual position of two lines in space. Distance from a point to a plane, distance of a point from a straight line. Distance of two points on a straight line. Symmetric of a point relative to a line or a plane.
Isometries. Classification of orthogonal transformation in dimension two. Classification of isometries of the plane. Classification of orthogonal transformations in three dimension. Classification of isometries of space.
Circles, spheres, cones and cylinders. The circle and the sphere: geometric definition; Cartesian equation: parametric equation. Cones and cylinders: Cartesian and parametric equations.
Geometry of conics and quadrics. Ellipses, hyperbolas and parabolas as geometric loci. Eccentricity. Quadrics of revolution: Ellipsoids, hyperboloids, paraboloids. Non-degenerate quadrics in canonical form: Ellipsoids, hyperboloids, elliptic and hyperbolic paraboloids. General equation of a conic in the plane. Matrix expression. The intersection of a line with a conic: definition of asymptotic direction and asymptote. Tangency of a straight line to a conic in a point. Center of symmetry of a conic. Definition of a conjugate diameter to a direction. Conjugate directions. Principal directions. Proof that a symmetric endomorphism is always diagonalizable. Definition of symmetry axis and plane of symmetry. Orthogonal classification of conics. Invariants of a quadric with respect to a change of Cartesian coordinates. Recognition of a Conic through the invariants. The method of invariants to determine the Euclidean canonical form of a conic. Recognition of a Hyperbola. Affine Canonical equation of a conic. Reduction of a quadric in canonical form. Quadrics of revolution.

### Teaching Methods

Blackboard and slides during the lectures. For the preparation of the student at home there is a website dedicated (unica2.unica.it/~piu/) where students can find the slides of the course, the instructor's notes and the weekly exercises to perform at home. The weekly exercises will be corrected in class by the tutor.
By conducting a large number of exercises on all topics, the course tries to teach a general approach to troubleshooting. It is also structured in such a way as to make the student study of the discipline independent.

### Verification of learning

A written test. Students who have passed the test are admitted to the oral test. The oral examination is considered to be exceeded if the student answer correctly at least three questions on different topics of the syllabus. In any case a reply too insufficient can impair the entire oral test. The final grade is determined by the vote in the written test and by the evaluation of the oral test.

### Texts

A. Sanini, Lezioni di geometria, ed Levrotto e Bella Torino 1984.

M.R. Casali, C. Gagliardi, L. Grasselli, Geometria, Esculapio – Bologna