# 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.

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.

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. The third part of the course studies vector spaces, linear transformations and the diagonalization problem for linear operators and matrices. The course ends with a study of scalar and hermitian products.

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

Elements of analytic geometry in 3-dimensional space. Parametric and Cartesian equations of a straight line. Mutual position of two lines. Equation of a plane. Scalar product and distance. Wedge product and its fundamental properties. Analytic geometry of line and planes in 3-dimensional space. The algebra of matrices, matrix multiplication. Inverse of a matrix. Determinant of square matrices: definition using the Laplace formulas and fundamental properties. The Binet theorem. Elementary row operations on a matrix, row-echelon form of a matrix. Gaussian elimination and row-echelon reduction. Use of the row-echelon form. Rank of a matrix. Computation of the inverse matrix. Systems of linear equations: computation of solutions using Gaussian elimination. The Theorem of Rouche-Capelli. Real and complex vector spaces. Linear combinations of vectors, subspaces and generators. Linear dependence and independence. The notions of basis and dimension of a vector space. The Grassmann formula. Coordinates relative to a basis. Change of basis, transition matrix, change of coordinates under change of basis. Linear transformations. Operations on linear transformations: sum and composition. Isomorphisms. Definition of image and kernel, the rank-nullity theorem. Matrix associated to a linear application and change of matrix for linear transformation under change of bases. Isomorphisms. Operators (endomorphisms) of a vector space: eigenvalues, eigenvectors and eigenspaces. The characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Diagonaliztion of matrices and endomorphisms: necessary and sufficient conditions. Scalar products. Orthogonal complement of a subspace. The Gram-Schmidt orthogonalization process. The orthogonal group. Diagonalization of symmetric matrices: the spectral theorem. Positivity criterion for scalar products. Outline of the complex case.

Elements of analytic geometry of the 3-dimensional space. Parametric and cartesian equations Parametric and Cartesian of a straight line. Mutual position of two lines. Equation of a plane. Scalar product and distance. Wedge product and its fundamental properties. Real and complex vector spaces. Subspaces: sum and intersection. Linear combination of vectors: linear dependence/independence. Generators, bases and dimension of a vector space. Grassmann formula. Determinants: definition using the formulas of Laplace and fundamental properties. Binet theorem. Elementary operations of the row and column of a matrice. Calculation of the inverse matrix. Rank of a matrix. System of linear equations: Gauss-Jordan's theorem and Theorem of Rouche-Capelli. Linear applications. Definition of the kernel and of image, Dimension's theorem, matrix associated to a linear application and rule base change. Isomorphisms. Endomorphisms of a vector space: eigenvalues, eigenvectors and eigenspaces. Characteristic polynomial. Algebraic multiplicity and geometry of an eigenvalue. Diagonalizable endomorphisms. Scalar products. Orthogonal complement of a subspace. Process of Gram-Schmidt orthogonalization. The orthogonal group. Diagonalization of symmetric matrices: the spectral theorem. Positivity criterion for scalar products. Outline of the complex case.

## Recommended readings

L. Alessandrini, L. Nicolodi, GEOMETRIA E ALGEBRA LINEARE con esercizi svolti, Casa Editrice UNI.NOVA, Parma, 2012.

L. Alessandrini, L. Nicolodi, GEOMETRIA E ALGEBRA LINEARE con esercizi svolti, Casa Editrice UNI.NOVA, Parma, 2012.

## 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 “Progetto IDEA”.

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.

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.