LINEAR ALGEBRA AND GEOMETRY

The course in Linear Algebra and Geometry is aimed at enabling students to acquire fundamental knowledge of linear algebra and analytic geometry, essential for studying mathematics applied to computer science, physics, and engineering. In particular, the course provides both theoretical and practical tools to understand and use concepts such as vector spaces, matrices, linear transformations, and geometric representations in space.

Expected Learning Outcomes

DD1 – Theoretical Knowledge
Knowledge of mathematical language and deductive methods, of basic algebraic structures (fields, vector spaces, bases, dimension), and of analytic geometry tools. Ability to understand and describe mathematical and applied problems, and to interpret geometric and algebraic relationships through coordinate and matrix representations.

DD2 – Practical Application
Ability to apply acquired knowledge to solve concrete problems, such as computing eigenvalues and eigenvectors, solving linear systems, and analyzing lines, planes, and conic sections in space. Ability to interpret data and results derived from exercises, simulations, or applied problems in linear algebra and geometry.

DD3 – Critical Evaluation
Ability to critically evaluate solutions and methods, choosing the most appropriate approach depending on the problem, and to tackle complex situations requiring the combined use of algebraic and geometric techniques.

DD4 – Communication Skills
Ability to effectively communicate knowledge, both in written and oral form, using precise mathematical language. Ability to present the results of exercises clearly and concisely through calculations, formulas, graphical representations, and diagrams. Ability to work in teams, exchanging ideas and collaborating to solve problems.

DD5 – Self-Assessment
Ability to self-assess one’s learning process, identifying strengths and weaknesses, with the aim of achieving a solid foundational knowledge of linear algebra and geometry.

The knowledge acquired in this course can be applied in various fields, particularly in computer science and engineering, and contributes to the achievement of the Agenda 2030 objectives related to innovation, quality education, and sustainable development.

Lectures. Group exercises. Classroom assessments.

Two in-course tests are scheduled during the course (duration: 2 hours for the first test and 2 hours for the second test).

Students are required to attend at least 70% of the course lectures in order to take the in-course tests. Attendance is, however, recommended to successfully complete the final exam.

• First in-course test:
  Consists of exercises aligned with the learning outcomes of the Linear Algebra units. Passing this test allows the student to earn up to 15 points (passing grade = 6).

• Second in-course test:
  Consists of exercises aligned with the learning outcomes of the Geometry units and lasts 2 hours.
  Participation in the second test is independent of participation or results in the first test. Passing this second test allows the student to earn up to 15 points (passing grade = 6).

The student must have attended at least 70% of the course lectures to be eligible to take the two in-course tests (one on Algebra and the other on Geometry).

Algebra

1. Sets and Operations
   General concepts of sets and set operations. Functions between sets, image and preimage, injectivity, surjectivity, bijective functions. Sets with operations and main algebraic structures: groups, rings, fields.

2. Matrices
   General concepts of matrices. Rank. Determinants and their properties. Laplace theorems*. Inverse of a square matrix and calculation of the inverse. Binet’s theorem*. Reduced matrices and reduction methods. Matrix multiplication. Linear systems, Rouché–Capelli theorem. Solving linear systems using the reduction (Gaussian elimination) method; free variables. Homogeneous systems. Cramer’s theorem.

3. Vector Spaces
   Vector spaces and their properties. Examples. Subspaces. Intersection, union, and sum of subspaces. Linear independence and corresponding criteria. Generators of a vector space. Basis of a vector space, successive elimination method, completion to a basis. Steinitz lemma*, dimension of a vector space. Grassmann formula*. Direct sums. Kronecker theorem. Proof of the Rouché–Capelli theorem. Homogeneous systems and solution subspaces.

4. Linear Maps and Their Properties
   Kernel and image of a linear map. Injectivity, surjectivity, isomorphisms. The space $L(V,W)$ and its isomorphism* with $K^{m,n}$. Study of linear maps. Change of basis, similar matrices.

5. Eigenvalues, Eigenvectors, and Eigenspaces
   Eigenvalues, eigenvectors, and eigenspaces of an endomorphism. Characteristic polynomial. Dimension of eigenspaces. Independence of eigenvectors. Simple endomorphisms and matrix diagonalization.

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Geometry

1. Vectors in Ordinary Space
   Vector addition, scalar multiplication. Scalar product, vector product, mixed product. Components of vectors and operations via components. Linear geometry in space. Cartesian and homogeneous coordinates. Planes and their equations. Lines and their representations. Points at infinity. Angular properties of lines and planes. Distances. Families of planes. Linear geometry in the plane. Cartesian and homogeneous coordinates. Lines and their equations. Intersections of lines. Slope. Distances. Families of lines.

2. Coordinate Transformations, Conics
   Coordinate changes in the plane, rotations, and translations. Conics and associated matrices, orthogonal invariants. Reduced equations, reduction of a conic to canonical form. Classification of irreducible conics. Study of conics in canonical form. Circles. Tangent lines. Families of conics and their use to determine specific conics.

3. Quadrics in Space
   Quadrics and associated matrices. Irreducible quadrics. Vertices and degenerate quadrics. Cones and cylinders, their sections. Reduced equations, reduction of a quadric to canonical form. Classification of non-degenerate quadrics. Sections of quadrics with lines and planes. Tangent lines and planes.

Note: Proofs of the theorems marked with * can be omitted.

The assessment consists of passing a written exam and an oral exam, both of which are mandatory.

During the course, in-course tests will be administered exclusively to attending students for self-assessment (DD5) and evaluation purposes. Students who achieve a positive average in these tests may be exempted from taking the written exam.

Assessment may also be conducted remotely, if circumstances require.

Information for Students with Disabilities and/or Specific Learning Disorders (DSA)

To ensure equal opportunities and comply with current regulations, students with disabilities and/or DSA may request a personal meeting with the instructor to plan any compensatory or dispensatory measures, according to the learning objectives and their specific needs.

Students can also contact the CInAP Coordinator (Centro per l’Integrazione Attiva e Partecipata – Services for Disabilities and/or DSA) at the Department.

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Registration

Registration for each exam session is mandatory and must be done online through the University student portal. Exams are published on the University portal, the degree program page, and the homepage:  http://www.dmi.unict.it/lmarino/

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Preliminaries

The topics of the first week of the course will cover “Preliminaries.”

During the course, two in-course tests will be held, each lasting two hours.

The Preliminaries Test is open to all students (1st, 2nd, and 3rd year, repeating students, and FC). Students who pass the test will receive a bonus of 3 points.

Validity of the bonus:

• The 3 points are valid for all written exams in the current academic year.

• Students who have the bonus do not need to complete the “Preliminaries question” included in each written exam.

Students enrolled in the 1st, 2nd, or 3rd year, as well as repeating students or FC, who do not have the bonus, must complete the “Preliminaries question” to pass the written exam in Linear Algebra and Geometry.

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In-course Tests

1. First in-course test: Covers the Algebra portion of the course.

   • Passing grade: 6/15

   • Maximum grade: 15/15

2. Second in-course test: Covers the Geometry portion of the course and will be held at the end of the course.

   • Passing grade: 6/15

   • Maximum grade: 15/15

Participation in the second in-course test is independent of participation or results in the first test.

Passing both in-course tests means that the final exam will consist only of the oral exam, to be taken within the first semester (March of the same year).

Students who pass only one of the two in-course tests may take the remaining test by March of the same year.

Students who do not pass any in-course tests or who have never attended must take both a written and an oral exam.

• The written exam, a prerequisite for the oral exam, lasts three hours and covers the full program of Algebra and Geometry.

• Passing grade: 12/30

• Maximum grade: 30/30

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Evaluation Criteria

The final grade will be based on:

• Level of knowledge of the required topics

• Expressive ability and correctness of mathematical language

• Ability to apply knowledge to simple case studies

• Ability to connect different topics of the course program

Grading Criteria:

• NOT PASSED: the student demonstrates a poor and fragmented knowledge of the subject, shows serious misunderstandings, and is unable to present the subject matter acceptably.

• 18-21: the student demonstrates limited knowledge and a basic understanding of the subject, presenting in an unclear and imprecise manner.

• 22-24: the student demonstrates acceptable knowledge and an essential understanding of the subject, presenting correctly but not fully structured.

• 25-27: the student demonstrates broad knowledge and an adequate understanding of the subject, presenting correctly but not completely.

• 28-29: the student demonstrates in-depth knowledge and a solid understanding of the subject, presenting in a clear and structured manner.

• 30-30 cum laude: the student demonstrates complete and detailed knowledge and an excellent understanding of the subject, presenting in a clear and structured manner.

Linear Algebra Exercises

1. Study of a linear map as a function of a parameter, determining its kernel and image.

2. Study of the simplicity of an endomorphism as a function of a parameter, determining, when possible, a basis of eigenvectors.

3. Computation of the preimage of a vector, followed by the solution of a linear system depending on a real parameter; computation of the preimage of a vector space and the image of a vector space.

4. Direct sums, operations with linear maps, induced linear maps, restrictions, and extensions.

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Geometry Exercises

1. Linear geometry in space: parallelism and perpendicularity, distances, orthogonal projections, angles.

2. Study of a pencil of conics, either given or to be determined. Complete study of a conic. Conics under given conditions.

3. Study of quadrics as a function of a parameter. Quadrics under given conditions.

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