Bachelor's Degree in Electrical Engineering and Information Technology

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Geometry and Linear Algebra

COURSE LANGUAGE: English

YEAR OF THE DEGREE PROGRAMME (I, II, III): I

SEMESTER (I, II, ANNUAL): I

CFU: 6

REQUIRED PRELIMINARY COURSES (IF MENTIONED IN THE COURSE STRUCTURE “REGOLAMENTO”)
None.

PREREQUISITES (IF APPLICABLE)
The mathematical content of school syllabus.

LEARNING GOALS
Students should learn all the basic tools of linear algebra and geometry. The goal of this course is, on one hand, to get used to afford formal problems by using adequate tools and correct language, on the other hand to solve specific problems of algebraic or geometric type, applying the methods of linear algebra.

EXPECTED LEARNING OUTCOMES (DUBLIN DESCRIPTORS)

Knowledge and understanding
Students are requested to prove to be acquainted with notions (definitions, statements, proofs, if specifically indicated in the syllabus) related to the known algebraic and geometric structures (vector spaces, the plane and the space of the elementary geometry, matrix spaces) and the calculation tools developed during the course. They should be able to understand topics like those learned, by reworking known notions. 

Applying knowledge and understanding
Students should be able to apply what they learned in solving suitable exercises elaborated by the teacher, mainly related with lines and planes, matrices, equations, vectors, and they should know the basic problematics related to algebraic and geometric structures.

COURSE CONTENT/SYLLABUS

  • Recalls on set theory and algebraic structures:
    Union, intersection, complement, cartesian product; correspondences and relations, maps, restrictions: injective, surjective, and bijective maps. Composition of maps. A characterization of bijective maps. Equivalence relations (the congruence of applied vectors as an example). Internal operations: associativity, neutral element, symmetric elements, commutativity (the sum on numers and on vectors an example). Abelian and non-abelian groups (with some examples). Fields. The example of real numbers and the prime field of order 2. External operations (multiplication on vectors as an example). 
  •  Vector spaces and Euclidean spaces:
    Definition and elementary properties of a vector space. Examples: numerical vector spaces, polynomials, matrices, vectors of the elementary geometry. Linear combinations. Linear dependence and independence, and their characterizations. Generators. Subspaces and their characterizations. The linear spam of a subset of vectors. Bases of a vector space, and related components. Extraction of a basis from a set of generators. The Steinitz lemma and its consequences: the dimension of a vector space, the construction of a basis starting from a linearly independent set of vectors. Intersection, sum, direct sum of subspaces. The Grassmann formula. The notion of Euclidean vector spaces: inner products in real vector spaces, the length of a vector, angles between vectors, orthonormal bases. The Gram-Schmidt process. Orthogonal complements. The standard inner product on numerical vector spaces. The inner product between geometric vectors. The vector product in 3-dimensional spaces.
  • Matrices and determinants:
    Elementary operations on the rows of a matrix and stepped matrices. The rank of a matrix and the number of pivots of a stepped matrix. Triangular and diagonal matrices. The row-by-column product. The notion of determinant of a square matrix: classical definition and elementary properties (without proofs). Characterization of the maximum rank of a square matrix by the non-vanishing of their determinants. Computation of determinants: first and second Laplace theorems, Kronecker theorem (without proof). Invertible matrices and the determination of the inverse. Similarity of matrices.
  • Linear systems of equations: 
    Solutions, compatibility and Rouchè-Capelli theorem. Cramer theorem and Gauss elimination algorithm. Resolution of a linear system. Determination of a basis of the vector space of solutions of a homogeneous linear system. Each subspace of a numerical vector space is the space of solutions of a homogeneous linear system. Cartesian and parametric representation of subspaces of numerical vector spaces.
  • Linear maps: 
    The notion of linear map: elementary properties. Linear maps preserve linear dependence. Kernel and image of a linear map. A characterization of injective and surjective linear maps. The fundamental theorem of linear maps. Endomorphisms and isomorphisms. The isomorphism associated to given bases. The transition matrices associated to a change of bases. The rank-nullity theorem (without proof). Similarity of matrices associated to a given endomorphism in different bases.
  • Diagonalization of endomorphisms and matrices: 
    Eigenvalues, eigenvectors, and eigenspaces of endomorphisms (and square matrices). The characteristic polynomial. Algebraic and geometric multiplicity of an eigenvalue. Characterization of the diagonalizability of endomorphisms and square matrices through the existence of a basis of eigenvectors. Determination of the eigenvalues and of a basis of eigenvectors of a diagonalizable endomorphism or square matrix.
  • Affine Euclidean spaces: 
    Basic definitions. Affine frames and coordinates of a point. Affine Euclidean subspaces. Parallelism among subspaces. Skew lines. Cartesian and parametric representation of affine (Euclidean) subspaces. The study of incidence and parallelism of subspaces. Orthogonality conditions among subspaces in dimension 2 or 3. Distance among subsets of points. Distance among a point and a hyperplane. The distance between Euclidean subspaces in dimension 2 or 3. The common perpendicular theorem. Proper and improper pencils of straight lines (in dimension 2) and planes (in dimension 3).

READINGS/BIBLIOGRAPHY
Check the teacher's website.

TEACHING METHODS 
Plenary lectures. Approximately one third of the lectures will be based on exercises.

EXAMINATION/EVALUATION CRITERIA 

Exam type:

  • Written and oral.

In case of a written exam, questions refer to:

  • Numerical exercises.
  • Open answers.
  • Multiple choice answers.

Evaluation pattern: 
The grade is formulated by the Examination Commission on the basis of the outcome of written test, in particular on the basis of the consistency and accuracy of the exercises performed and the adequacy of the answers provided by the student to the theory questions. The final grade is also suitably motivated to the student.