Corso di laurea in Ottica e Optometria

Coherently with the didactic goals of the Course of Study planned by the SUA, CdS frames, it is expected that the student acquires familiarity with the basic notions of analytic geometry in the plane and 3D space.

In particular, the student is asked to understand the concept of frame and to be able to use coordinates to describe the basic geometric entities. Furthermore, the student is expected to be able to solve linear systems and to interpret their solutions in the light of the problem at hand.

Knowledge and understanding: knowing the fundamental aspects of linear algebra and the algorithms that underlie these results.

Ability to apply knowledge and understanding: Recognize the geometric problems associated with the use of the most advanced instrumentation in the field of data collection and processing in optics and optometry, and know how to connect them with the theoretical results of linear algebra learned in the course.

• Euclidean geometry.
  Concept of axiom and theorem. Use of coordinates and references.
  Natural, integer, rational and real numbers, and their relationships with points on a straight line. References on line S1, plane S2 and space S3.
  
  
• Geometry in the plane.
  Orthogonal reference and Cartesian coordinates of the points. Vector by two points. Parametric and non-parametric representation of a straight line, parallelism and perpendicularity between 2 straight lines. Line through two points, midpoint of a segment, axis of a segment. Distances between points and lines.
  
  
• Geometry in space.
  Orthogonal reference. Parametric and non-parametric equation of lines and planes in space. Intersections of three planes and solutions of a system. Intersections between straight lines and planes. Intersections between straight lines.
  Point-line and point-plane distance.
  Line-plane and line-line distance. Angles between straight lines and incident planes.
  
  
• Linear equations and systems.
  Solutions of a system. Matrix of coefficients. Gauss-Jordan method
  Rescaling of a matrix: pivot elements and rank of a matrix. Rouche'-Capelli theorem. Homogeneous systems and structure of the set of solutions.
  
  
• Matrices.
  Sum, products and their properties. Transposed matrix and inverse matrix. Matrix equations and calculation of the inverse with Gauss-Jordan. Invertibility and rank. Determinants of 2x2 and 3x3 matrices. Geometric meaning and calculation. Link with linear systems.
  
  
• Vector spaces and their properties.
  The vector space of S1, S2 and S3. Vector subspaces. Space Rn. Linear dependence and independence, free sets and generators, bases. Theorems on the basis. Dimension of a vector space. Space of rows and columns of a matrix, nullity plus rank theorem.
  
  
• Basis changes.
  Relation between coordinates with respect to two different bases.
  Scalar products and their properties, angle between two vectors. Sets and orthonormal bases, components, lengths of vectors.
  Orthonormal base changes. Orthogonal matrices.
  
  
• Conics.
  Circle, center and radius. Tangent lines to a circle and passing through a given point.
  Circle passing through three non-aligned points.
  Ellipse, hyperbola and parabola in canonical form: semiaxes, foci and focal property. Equilateral hyperbola referred to the axes.
  Parabola in canonical form: focus, directrix and focal property.
  
  
• Eigenvalues and eigenvectors.
  Notes on eigenvectors and eigenvalues of a matrix, orthogonality of eigenvectors in different eigenspaces. Calculation of eigenvalues/eigenvectors of a 2x2 matrix and determination of the principal axes of the conic.

The course is articulated in 48 hours. A substantial part of the lessons will be dedicated to examples and exercises. The courses will take place exclusively in presencand streaming will not be allowed, unless otherwise specified.

The exam consists of a written test followed by an oral test.
The written test consists of 4-6 exercises to be carried out in 2 hours (120 minutes).
A score out of 30 is assigned to the test.
The topics of the written paper concern the exercises performed in class and the course programme.
The oral test is compulsory and mainly consists of a comment on the written test, followed by a few questions about the course programme.
The use of a calculator is permitted, and no notes or textbooks may be kept.
The exam consists of a written test and an oral one, which concern the entire programme.

The exams will take place exclusively in presence, unless otherwise specified.

Texts

Linear Algebra: A First Course in Linear Algebra

Geometry:  E.Abbena, A.M.Fino, G.M.Gianella, Algebra lineare e Geometria Analitica, Vol I, Aracne Ed, 2012

Applications and exercises: Linear Algebra with Applications.

Take notes during the lectures!

Office hours: by appointment.