Geometry I

The fundamental goal of the Geometry I course is to provide certain tools of Linear Algebra for the computation of eigenvectors and eigenvalues of an endomorphism between vector spaces, such as, for example, matrix properties. Some notions of Geometry in the plane and in space are also introduced, along with tools for the study of conic sections in the plane and quadric surfaces in space.

In particular, at the end of the course, students are expected to have acquired:

Knowledge and understanding:

• understand statements and proofs of fundamental theorems in algebra, analytic geometry, linear algebra, and the geometry of curves; demonstrate mathematical skills in reasoning, manipulation, and calculation;

• solve mathematical problems which, while not familiar, are of a similar nature to others already known to the students.

Applying knowledge and understanding:

• demonstrate known mathematical results using techniques other than those already studied;

• construct rigorous proofs;

• build simple examples.

The above skills will be achieved through interactive teaching: students will constantly verify their knowledge, working independently or in collaboration within small groups, on new simple problems proposed during exercises, both in class and in support sessions.

Making judgements:

• acquire an informed independence of judgement with respect to evaluating and interpreting the resolution of a geometry problem;

• be able to construct and develop logical arguments with clear identification of assumptions and conclusions;

• be able to recognize correct proofs and identify fallacious reasoning.

These objectives offer students practice activities during the course and supplementary support for the Geometry I course; these will provide opportunities for students to independently develop their decision-making and judgement skills. The above abilities will be achieved through interactive teaching: mathematics students will constantly verify their knowledge, working independently or in small groups, on new simple problems proposed during exercises and support activities.

Communication skills:

• communicate clearly and unambiguously information, ideas, problems, solutions, and their conclusions;

• present, orally or in writing, in a clear and comprehensible way, the most important theorems of linear algebra and analytic geometry;

• be able to work in groups and operate with defined degrees of autonomy.

To achieve communication skills, ample opportunities for verification and discussion of written work will be provided. The final exam will also offer students a further opportunity to deepen and test their ability to analyze, elaborate, and communicate the work carried out.

Learning skills:

• develop the competencies necessary to pursue further studies with a high degree of autonomy;

• possess learning skills and a high standard of knowledge and competence, sufficient to allow access to lectures or programs of Master’s degree courses in Mathematics;

• have a flexible mindset and be able to integrate quickly into work environments, adapting easily to new problems.

Learning skills will be acquired during the degree program thanks to the distribution of total working hours, which gives appropriate importance to those dedicated to individual study.

Should teaching be delivered in blended or remote mode, the necessary changes may be introduced with respect to what has been previously stated, in order to meet the planned program as reported in the syllabus.

NOTE: Information for students with disabilities and/or Specific Learning Disorders (SLD).
To guarantee equal opportunities and in compliance with current legislation, interested students may request a personal interview in order to plan any compensatory and/or dispensatory measures, based on educational objectives and specific needs.
It is also possible to contact the CInAP (Center for Active and Participatory Integration – Services for Disabilities and/or SLD) representative of our Department, Prof. Daniele.

Geometry 1 – 12 ECTS – 94 total hours
Teaching organization

• 300 total workload hours

• 206 hours of individual study

• 70 hours of lectures

• 24 hours of exercises

Lectures and in-class problem solving. Some lessons will be conducted entirely at the blackboard, others using handouts distributed to students. This will depend on the type of topic addressed.

Periodic meetings will be held with some first-year instructors, during which students will be guided to reflect on the connections among the three disciplines, in order to appreciate the unity and multidisciplinarity of mathematical knowledge.

Part of the program (maximum 3 ECTS) may be taught by a visiting professor, either foreign or Italian.

Should teaching be delivered in blended or remote mode, the necessary changes may be introduced with respect to what has been stated above, in order to comply with the planned syllabus.

NOTE: Information for students with disabilities and/or Specific Learning Disorders (SLD).
To guarantee equal opportunities and in compliance with current legislation, interested students may request a personal interview to arrange possible compensatory and/or dispensatory measures, based on educational objectives and specific needs.

It is also possible to contact the CInAP (Center for Active and Participatory Integration – Services for Students with Disabilities and/or SLD) representative of our Department, Prof. Daniele.

LINEAR ALGEBRA (First semester – content for the first midterm exam):

• Operations on a set.

• Matrices with entries in a field and their properties.

• Vector spaces and their properties.

• Determinant of a square matrix and its properties. Reduced matrices and the reduction method. Rank of reduced matrices. Systems of linear equations. Rouché–Capelli theorem. Cramer’s theorem. Homogeneous systems. Solving linear systems.

• Linear maps between vector spaces and their properties. Kernel and image of a linear map. Injectivity, surjectivity, isomorphisms. Rank–Nullity theorem. Study of linear maps.

• Eigenvalues, eigenvectors, and eigenspaces of an endomorphism. Diagonalizable endomorphisms and matrix diagonalization.

• Inner product in a real vector space. Gram–Schmidt orthonormalization process. Subspaces of a Euclidean space and their orthogonal complement. Orthogonal matrices.

ANALYTIC GEOMETRY (Second semester – content for the second midterm exam):

• Affine spaces – Affinities.

• Geometric vectors in ordinary space.

• Affine coordinate systems in the plane and in space. Geometry in the affine plane: lines and their equations. Orthogonality and parallelism. Geometry in 3-dimensional affine space: planes and lines and their various representations. Orthogonality and parallelism. Coplanar and skew lines. Distances.

• Conic sections in the plane and their associated matrices. Classification of irreducible conics. Pencils of conics.

• Quadrics and their associated matrices. Reducible and irreducible quadrics. Vertices of quadrics and degenerate quadrics. Intersections of quadrics with tangent planes. Reduced equations. Ellipsoids, hyperboloids, and paraboloids. Spheres. Plane sections of a quadric.

Methods of Assessment

To take the final exam, students must register through the SmartEdu portal. For any technical issues related to registration, students must contact the Teaching Office.

a) Assessment during the course:
Periodically, during the exercise sessions, students may be invited to participate by solving exercises proposed either by the instructor or by the students themselves at the blackboard. This helps monitor the students’ level of learning. During lectures, students will also be asked to recall definitions and results covered in previous lessons, in order to encourage a conscious learning of the subject. During supplementary activities, exercises will be carried out to support self-assessment.

b) Midterm exams:
Two midterm exams are scheduled, each lasting about two and a half hours. Both tests are written and are considered passed with a grade of at least 15/30 on at least three of the proposed exercises. Each exam corresponds to about 1/3 of the total course credits (i.e. about 4 credits each). These credits will only be awarded after passing both midterms and an oral exam (see final exam).

• The first midterm will take place during the first break in lectures and will cover about half of the syllabus.

• The second midterm will be held at the end of the course and will cover the second half of the syllabus (complementary to the first).

Generally, the dates of the first and second midterms coincide with those of the scheduled exam sessions in February and June, respectively, according to the written exam calendar.

The midterms (both passed with a grade of at least 15/30) are valid until the end of the autumn session: the oral exam must be taken no later than the scheduled exam dates from the end of the course until the autumn session.

Students who fail one of the two midterms must take the final exam (see point c). Passing both midterms grants access to the oral exam.

Note: The student will acquire the total credits only after passing both written midterms and the oral exam.

c) Final exam:
The final exam consists of a written and an oral test at the end of the academic year, according to the exam calendar.

• The written exam consists of solving several technical and theoretical exercises.

• The oral exam covers all topics in the syllabus. The instructor will announce details at the end of the course and, in any case, before the start of the first exam session.

The written exam lasts about two and a half hours and may be replaced by the two midterms (each lasting about two and a half hours), the first held halfway through the course during the first break in lectures, the second immediately at the end of the course.

If the student fails one of the two midterms, they must take the final exam.

The written exam is scheduled according to the calendar. The oral exam may be taken on a different date than the one scheduled.

To pass the written part of the final exam, the student must solve at least two Linear Algebra exercises and one Geometry exercise (or vice versa). The recommended grade on the written exam to be admitted to the oral exam is not lower than 15/30.

d) Grading criteria:
For both midterms and the final exam, the following aspects will be evaluated:

• clarity of presentation,

• completeness of knowledge,

• ability to connect different topics.

The student must demonstrate sufficient knowledge of the main topics covered in the course and the ability to solve at least the simplest assigned exercises. Consideration will be given, especially in the first sessions, to the fact that students are still in their first year and may not yet have reached the maturity expected in later years.

There is no averaging between the written and oral grades. Attendance/participation in the tutoring sessions planned for the course will also be taken into account.

Grading guidelines:

• Fail: the student has not acquired the basic concepts and is unable to solve the exercises.

• 18–23: the student demonstrates minimal mastery of the basic concepts; presentation skills and ability to link content are modest; can solve simple exercises.

• 24–27: the student demonstrates good mastery of the course content; presentation and ability to link content are good; solves exercises with few errors.

• 28–30 with honors: the student has mastered all course content, is able to present it thoroughly and critically, and solves exercises completely and without errors.

Note: Assessment may also be carried out online if required by circumstances.

Examples of frequently asked questions and/or exercises

Exercises assigned and solved are available on the course MS Teams platform or at http://studium.unict.it.

Common exam questions/exercises:

1. Definition of vector space. Injective and surjective linear maps. Examples. Theorem on the dimensions of the kernel and image of a linear map. Homomorphism theorem and its connection to Algebra. Preimage of a vector. Cramer’s and Rouché–Capelli’s theorems. Linearly independent vectors, criterion for linear independence. Bases. Examples. Independence of eigenspaces, algebraic and geometric multiplicity, simple endomorphisms. Diagonalization of a matrix. Real and/or complex inner products. Matrix associated with an inner product.

2. Lines in the plane. Lines and planes in space. Parabolas, hyperbolas, equilateral hyperbolas, ellipses, circles. Tangent lines to a conic at a point. Classification of quadrics. Hyperboloids, paraboloids, ellipsoids, spheres. Tangent planes to a quadric at a point. Hyperbolic, parabolic, and elliptic points. Vertices. Cones and cylinders. Plane sections of quadrics. Surfaces of revolution. Examples without using orthogonal invariants.

3. Properties of affine spaces.

Assigned and solved exercises are available on the course MS Teams platform (Microsoft Teams),
or at http://studium.unict.it,
and on the instructor’s website: santispadaro.weebly.com (Teaching section).

Frequently asked exam questions/exercises

1. Definition of vector space. Injective and surjective linear maps. Examples. Theorem on the dimensions of the kernel and image of a linear map. Homomorphism theorem and its connection with Algebra. Preimage of a vector. Cramer’s and Rouché–Capelli theorems. Linearly independent vectors, criterion for linear independence. Bases. Examples. Independence of eigenspaces, algebraic and geometric multiplicity, simple endomorphisms. Diagonalization of a matrix. Real and/or complex inner products. Matrix associated with an inner product.

2. Lines in the plane. Lines and planes in space. Parabolas, hyperbolas, equilateral hyperbolas, ellipses, circles. Tangent lines to a conic at a point. Classification of quadrics. Hyperboloids, paraboloids, ellipsoids, spheres. Tangent planes to a quadric at a point. Hyperbolic, parabolic, and elliptic points. Vertices. Cones and cylinders. Plane sections of quadrics. Surfaces of revolution. Examples without the use of orthogonal invariants.

3. Properties of affine spaces.