Geometry

At the end of the course, the student will be able to:

1. Understand and use the formal language of Linear Algebra and Analytic Geometry, identifying fundamental structures such as vector spaces, linear maps, matrices, and geometric transformations.

2. Solve algebraic and geometric problems, including linear systems, change of basis, matrix diagonalization, and the study of linear and affine transformations.

3. Compute and interpret algebraic and geometric invariants, such as rank, determinant, eigenvalues, eigenvectors, area, and volume.

4. Analyze lines, planes, conics, and quadrics, determining their equations, relative positions, and classification using algebraic and matrix-based methods.

5. Study isometries and orthogonal transformations, describing their geometric nature and representing them through matrices.

6. Apply fundamental algorithmic procedures, such as Gaussian elimination and the Gram–Schmidt process, with rigor and independence.

7. Connect analytic representations with geometric interpretation, developing visualization and modeling skills.

Assessment of learning outcomes is carried out through a final examination, consisting of a written test and an oral examination.

The written test is designed to assess the student’s ability to correctly and independently apply the tools of Linear Algebra and Analytic Geometry to the solution of exercises and problems. It may include computational questions, applied exercises, and requests to justify the methods used.

The oral examination aims to verify the student’s theoretical understanding of the topics covered in the course, their command of mathematical language, and their ability to connect and interpret the various concepts presented. During the oral exam, students may be asked to provide proofs, theoretical explanations, and geometric interpretations.

Passing the written test is a necessary condition for admission to the oral examination. The final grade is determined by the overall evaluation of both components.

Attendance of the course is strongly recommended, as lectures include theoretical explanations, worked examples, and exercises that significantly support the learning of the topics covered.

Classes are conducted in a traditional face-to-face format, with guided exercises and classroom discussions. Additional teaching materials, exercises, and in-depth content may be made available through the university’s e-learning platform.

Non-attending students are still expected to be familiar with the entire course program and may prepare using the reference textbooks and the materials provided online.

The course consists of 30 sessions of 2 hours each and is divided into two main parts: Linear Algebra and Analytic Geometry.

Part I – Linear Algebra

• Vector spaces

  • Fields and vector spaces

  • Fundamental examples

  • Subspaces, sum and direct sum

• Linear combinations, bases, and dimension

  • Linear dependence and independence

  • Bases and coordinates

  • Steinitz lemma and Grassmann formula

• Linear maps

  • Kernel and image

  • Rank–nullity theorem

  • Matrix associated with a linear map

  • Change of basis

• Matrices and linear systems

  • Matrix operations

  • Gaussian elimination

  • Rank and compatibility of linear systems

• Determinants

  • Definition and properties

  • Laplace expansion

  • Binet’s theorem

  • Invertible matrices and Cramer’s rule

  • Geometric interpretation of determinants

• Eigenvalues and diagonalization

  • Eigenvalues and eigenvectors

  • Characteristic polynomial

  • Diagonalizability criteria

• Inner product and orthogonality

  • Orthogonal and orthonormal bases

  • Gram–Schmidt process

  • Orthogonal matrices and isometries

Part II – Analytic Geometry

• Affine and vector geometry

  • Affine spaces

  • Lines and planes

  • Cartesian, parametric, and vector equations

  • Distances and angles

• Isometries

  • Plane and space isometries

  • Rotations, reflections, and translations

  • Matrix representation

• Conics

  • Definition and matrix representation

  • Classification of conic sections

  • Geometric study: tangents, foci, directrices, eccentricity

  • Pencils of lines and conics

• Quadrics

  • Equations and classification

  • Ellipsoids, paraboloids, hyperboloids

  • Cones and cylinders

• S. Giuffrida, A.Ragusa, Corso di Algebra Lineare.
• G. Paxia, Lezioni di Geometria.
• S. Greco, P. Valabrega. Lezioni di Geometria, Volume I: Algebra Lineare.
• Grossman, Stanley, Elementary linear algebra

The final examination consists of a written test and an oral examination, which take place at the end of the course according to the examination schedule.

The written test is scheduled according to the official calendar. The oral examination will take place in the days following the written test, subject to availability.