# GEOMETRIAModule DIDATTICA FRONTALE

Academic Year 2023/2024 - Teacher: Antonio CAUSA

## Expected Learning Outcomes

The aim of the programme is to give some preliminaries and tools for a basic introduction to Linear Algebra and Cartesian Geometry. In this course we look at properties of matrices, systems of linear equations and vector spaces useful to find eigenvalues and eigenvectors of endomorphisms.
We will learn about classification of plane conics and quadric surfaces, using their invariants and geometric intuition. At the end of the course students will have learnt and have been able to apply the notions of vector spaces, basis, linear applications and the techniques of matrix calculus and methods of resolution of linear systems. Find eigenvalues and eigenspaces of endomorphisms of a vector space.

We will also solve some problems similar to the ones assigned at the final exam.

At the conclusion of the course, the students should be able to understand the basic notions, to apply their knowledge and understanding. They also should be able to give oral and written presentation of the most important theorems of the contents of the course; to work both in collaboration with other people and by themselves. making judgements, communication skills and learning skills.

Knowledge and understanding. The student will develop the ability to: understand statements and proofs of fundamental theorems in the field of linear algebra and Cartesian geometry; apply mathematical skills in formal reasoning, and in matrix calculation.

Ability to apply knowledge and understanding.The student will develop the ability to: demonstrate Linear Algebra results, state and understand rigorous proofs. These skills will be achieved through interactive teaching: the student will constantly check their knowledge, working independently or in collaboration in small working groups, on simple new problems, proposed during exercises, both frontal and during classroom hours.

Independence of judgment. The student will develop: a conscious autonomy of judgment with reference to the evaluation and interpretation of the resolution of an algebra or geometry problem; will be able to construct and develop logical arguments with a clear identification of assumptions and conclusions.

Communication skills. The student will develop the ability to: know how to communicate information, ideas, problems, solutions and conclusions in a clear and unambiguous way; know how to present, orally or in writing, in a clear and understandable way and with formally correct language, the most important theorems of linear algebra and Cartesian geometry.

Learning skills. The student will develop: the skills necessary to undertake subsequent studies with a high degree of autonomy; learning skills and competence that allow access to lessons or programs of advanced or specialist courses.

## Course Structure

Frontal lectures and classroom exercise. The teaching approach is a traditional one. The program offers personal feedback and attention from tutors in order to help students in their studies.

The taching is composed by lectures, exercises and tutoring. In all three activities the student is encouraged to participate with suggestions and proposals.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

Learning assessment may also be carried out on line, should the conditions require it.

## Required Prerequisites

Elementary Euclidean Geometry. Elementary Cartesian Geometry. Basic knowledge of polynomial algebra. Naive set theory.

## Detailed Course Content

LINEAR ALGEBRA

• Vector spaces and their properties over R. Examples: R[x], Rn, Rm,n.
• Free vectors of euclidean space. Inner product, cross product, scalar triple product. Geometric interpretation.
• Matrices over a field. Matrices addition, scalar multiplication, matrix multiplication, ring of square matrices. Diagonal, triangular, scalar, symmetric, skew-simmetric matrices and transpose of matrix.
• Subspaces. Intersection and sum of vector spaces. Direct sum. Linear combinations. Span, Linear Independence and dependence. Finitely generated vector spaces, Base, Dimension, coordinates of a vector. Steinitz’s Lemma, Grassmann’s formulas.
• Determinants and their properties. Theorems of Binet*,Laplace I*, Laplace II, Adjunct matrix, Inverse, Rank and Reduction of a matrix. Theorem of Kronecker*. Systems of linear equations. Rouchè-Capelli‘s rule, Cramer’s rule. Solving systems of linear equations.
• Linear maps between vector spaces and their properties. Kernel and image of a linear map. Injective, surjective maps and isomorphisms. Study of linear maps. Matrices associated to linear maps. Change of base matrix. Similar matrices.
• Eigenvalues, Eigenvectors and Eigenspaces of a matrix. Characteristic polynomial. Dimension of an eigenspace. Relation between Algebraic multiplicity and geometric multiplicity. Linear Independence of the eigenvectors. Diagonalizable linear maps and diagonalization of a matrix.
• Inner product spaces, euclidean and hermitian cases. Gram-Schmidt orthogonalization.
• Spectral theorem for self-adjoint operators acting on finite dimensional spaces.

CARTESIAN GEOMETRY

• Euclidean (geometric) vectors and their properties. Scalar multiplication, dot (or scalar) product, wedge (or cross) product.
• Cartesian coordinates. Points, lines , Homogeneous coordinates, Points at infinity (Improper Points),
• Parallel and orthogonal Lines. Slope of a line. Distances from a point to a line. Pencil of lines.
• Planes in the tridimensional euclidean space. Coplanar and Skew lines. Pencil of Planes.
• Angles between lines and planes. Distance from a point to a plane and from a point to a line in the space.
• Conics and their associated matrices. Orthogonal Invariants. Canonical reduction of a conic. Irreducible and degenerate conics. Rank of the associated matrix of a planar conic.
• Discriminant of a conic. Parabolas, Ellipses, Hyperbolas: equations, focus, eccentricity, directrix, semi-maior axis, center. Circumferences, Tangents, and pencils of conics.
• Polarity induced by a non degenerate conic.
• Quadrics and its associated matrix. Non degenerate, degenerate and singular quadric surfaces. Cones and cylinders. Classification.

## Textbook Information

1) S. Giuffrida, A.Ragusa, Corso di Algebra Lineare, Ed. Il Cigno G.Galilei, Roma 1998 (Linear Algebra).

2) G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005 (Geometry) available at www.giuseppepaxia.com

3) D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/

## Course Planning

SubjectsText References
1Linear spaces: definition and basic properties. Examples: R^n, R^m,n, R[X]. Intersection and sum of subspaces. Generators of a linear space.  Linear dependence. Bases of a linear space. Dimension of a linear space. Grassmann formula.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
21 Matrices with coefficients in a field. Matrix algebra: sum, scalar product, dot product. Basic properties of the ring of square matrices. Triangular and diagonal matrices, symmetric and antysimmetric matrices. Operations on matrices.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
3Determinant of a matrix. Binet and Laplace theorems on determinants. Inverse of matrix. Gaussian elimination: reduced row echelon form. Rank of a matrix. Kroneker theorem. Linear equations. Cramer theorem, Rouchè theorem. Homogeneous linear systems of equations.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
4Vector spaces homomorphisms. Null space and range space. Injectivity and surjectivity. Rank-nullity theorem. Matrix representation of a linear transformation. Matrix similarity.3) D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
5Endomorphism and their eigenvalues and eigenvectors. Characteristic polynomial of a matrix. Eigenspaces. Endomorphisms and diagonalization. Diagonalizable matrices.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
6Linear spaces with scalar product. Self-adjoint operators. Spectral theorem, corollaries.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
7Geometric vectors and basic operations with geometric vectors: dot product, cross product, triple product.D. Margalit , J. Rabinoff, Interactive linear algebra, available at https://textbooks.math.gatech.edu/ila/
8Cartesian coordinates in plane and space. Orthogonality and parallelism.Planes and lines in space. G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005
9Classification of affine conics. Quadratic forms. Canonical forms of a conic. Tangent lines to a conic.G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005
10Quadrics in affine and projective spaces. Classification of quadrics. Double points of a quadric.G. Paxia, Lezioni di Geometria, Spazio Libri, Catania, 2005

## Learning Assessment

### Learning Assessment Procedures

The exam aims to verify the achievement by students of the educational objectives described above. In particular: mastery of the methods and techniques develepod during the course, awareness of their theoretical methods, appropriateness of the language used.

To assess students' knowledge and skills, students are requested to attend a written exam and an oral exam.

During the written exam, the student maust solve some exercises in the format of open-ended question with the aim of assessing the student's ability to solve problems in linear algebra and cartesian geometry. The duration of the written exam will be proportional to the number of exercises assigned, also taking in into account the nature and complexity of exercises themselves.

The oral exam can be taken only if the written component has been successfully passed.

### Examples of frequently asked questions and / or exercises

Linear algebra

• Linear spaces: definition and basic properties.
• Linear independence, bases of a vector space.
• Coordinates of a vector.
• Subspaces, sum and intersection of subspaces.
• Steinitz's lemma and dimension of a vector space.
• Dimension of sum and intersection of subspaces: Grassmann formula.
• Determinant of a matrix, rank of a matrix. Matrices in rowechelen form.
• Systems of  linear equations. Cramer's theorem, Rouchè Theorem.
• Linear transformations: definition and basic properties. Matrix associated to a linear transformation, rank-nullity theorem.
• Eigenvalues and eigenvectors. Characteristic polynomial.
• Spaces endowed  with a scalar product. Cauchy-Schwartz inequality. The Spectral theorem.
• An hermitian matrix can be diagonalized by a similarity operation performed by a unitary matrix.

Geometry

• Euclidean coordinates frames.
• Projective plane/space and projective coordinates.
• Lines and planes in space. Orthogonality between lines and/or planes.
• Conics and quadrics: classification, tangent lines.
• Double points of a quadric. Degenerate quadrics.
• Polarity in plane/space.