# GEOMETRY

## Learning outcomes of the course unit

The principal aim of this course is to provide students with an understanding of the basic mathematical concepts and techniques of linear algebra and Euclidean geometry; at the end of this course, students should be able to: a) solve systems of linear equations; b) solve easy problems of analytic geometry; c) operate on vectors and matrices; d) diagonalize operators and matrices.

## Prerequisites

No specific prerequisites are needed.

## Course contents summary

The course is an introduction to the basic notions of linear algebra and geometry. The first part studies Euclidean geometry in 3-space (vectors, lines, planes), while the second part is devoted to the study of vectors, matrices, and linear systems. In the third part of the course we study vector spaces, linear maps and the diagonalization problem for linear operators and matrices. The course ends with a study of scalar and hermitian products.

## Course contents

1. Real and complex vector spaces. Linear subspaces: sum and intersection. Linear combinations of vectors: linear dependence and independence. Generators, bases and dimension of a vector spaces. Grassmann formula for subspaces. 2. Determinants: Laplace expansion and basic properties. Binet theorem. Row and column elementary operations on matrices. Computation of the inverse matrix. Rank of a matrix. 3. Linear systems: Gauss-Jordan method and Rouché Capelli theorem. 4. Linear maps. Definition of kernel and image; fundamental theorem on linear maps. Matrix representation of a linear map and change of bases. Isomorphisms and inverse matrix. 5. Endomorphisms of a vector space: eigenvalues, eigenvector and eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicity. Diagonalizable endomorphisms. 6. Scalar products. Orthogonal complement of a linear subspace. Gram-Schmidt orthogonalization process. Representation of isometries by orthogonal matrices. The orthogonal group. Diagonalization of symmetric matrices: spectral theorem. Positivity criterion for scalar product. A brief discussion on the complex case. 7. Three dimensional analytic geometry. Parametric and Cartesian equations of a line. Mutual position of two lines in the space; skew lines. Equation of a plane. Canonical scalar product and distance. Vector product and its fundamental properties. Distance of a point from a line and a plane. 8. Topics in algebra and/or geometry.

## Recommended readings

L. Alessandrini, L. Nicolodi,

Geomeria e Algebra Lineare con esercizi svolti,

Editrice UNI.NOVA, Parma, 2012.

S. Lang, Linear Algebra, 3rd ed.,

Undergraduate Texts in Mathematics, Springer-Verlag, 2004.

## Teaching methods

During lectures, the material of the course is presented using formal

definitions and proofs; abstract concepts are illustrated through

significant examples, applications, and exercises. The discussion of

examples and exercises is of fundamental importance for grasping the

meaning of the abstract mathematical concepts; for this reason, besides

lectures, guided recitation sessions to discuss and solve exercises and

assignments will be provided within the the “Progetto IDEA”.

## Assessment methods and criteria

Course grades will be based on a final exam which consists of a

preliminary multiple-choice test, a written exam and an oral interview.

There will be the possibility of two intermediate written exams and tests

to avoid the final written exam and test. The written exam, through tests

and exercises, should establish that students have learned the course

materials to a sufficient level. In the colloquium, students should be able

to repeat definitions, theorems and proofs given in the lectures using a

proper mathematical language and formalism.