First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Engineering
Course unit
IN08122537, A.A. 2019/20

Information concerning the students who enrolled in A.Y. 2019/20

Information on the course unit
Degree course First cycle degree in
IN0506, Degree course structure A.Y. 2011/12, A.Y. 2019/20
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Degree course track Common track
Number of ECTS credits allocated 9.0
Type of assessment Mark
Course unit English denomination TOPICS IN LINEAR ALGEBRA AND GEOMETRY
Department of reference Department of Industrial Engineering
E-Learning website
Mandatory attendance No
Language of instruction Italian
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge FRANCESCO ESPOSITO MAT/03

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Basic courses MAT/02 Algebra 4.0
Basic courses MAT/03 Geometry 5.0

Course unit organization
Period First semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
Hours of
Individual study
Lecture 9.0 72 153.0 No turn

Start of activities 30/09/2019
End of activities 18/01/2020
Show course schedule 2019/20 Reg.2011 course timetable

Examination board
Board From To Members of the board
25 A.A. 2018/19 canale 3 25/02/2019 30/11/2019 LARESE DE TETTO ANTONIA (Presidente)
GRAZIAN VALENTINA (Membro Effettivo)
24 A.A. 2018/19 canale 2 01/10/2018 30/11/2019 CARNOVALE GIOVANNA (Presidente)
23 A.A. 2018/19 canale 1 01/10/2018 30/11/2019 ESPOSITO FRANCESCO (Presidente)

Prerequisites: Mathematics at High school level
Target skills and knowledge: Basic notions of Linear algebra and their geometrical interpretation. The main focus is on vector spaces, linear maps and solution of systems of linear equations.
Knowledge of spectral theorem and its main applications.
Course unit contents: Introduction to linear algebra and its applications to analytic geometry.

complex numbers: definition, operations and properties
R-vector spaces. The vector space R^n, the vector space of matrices with real entries.
The vector space of polynomials in one variable with real coefficients.
Intersection and sum of subspaces.
Finitely generated spaces.
Bases of a vector space.
Existence of a basis for a finitely generated vector space.
Dimension of a vector space.
Coordinates of a vector with respect to a basis.
Direct sum of vector subspaces.
Grassmann formula and its applications.
Linear maps between vector spaces.
Construction of linear maps: existence and uniqueness conditions.
Kernel and image of linear maps, infectivity and subjectivity.
Dimension formula and its consequences.
Premiere of a vector through a linear map.
Associated matrices.
Rank of a matrix.
Systems of linear equations.
Rouche' Capelli theorem
Elementary operations on rows of a matrix.
Gauss reduction and application to the solution of systems of linear equations.
Invertible matrices and computation of the inverse of a matrix.
Basis change.
Conjugate matrices.
Determinant and its properties.
Eigenvalues and eigenvectors of an endomorphism. Eigenspaces.
Characteristic polynomial.
Algebraic multiplicity and geometric multiplicity of an eigenvalue and relations between them.
Diagonalisable matrices.
Diagonalisability of a matrix over the reals: necessary and sufficient coefficients.
Diagonalisability of a matrix depending on one or more parameters.
Inner product and its properties.
Cauchy-Schwarz inequality and triangular inequality.
Orthogonality, orthogonal complement of a subspace.
Gram-Schmidt process.
Orthogonal projection. Isometries, orthogonal matrices. Isometries of the plane.
Symmetric matrices. Positive definite matrices.
Diagonalisability over the complex numbers.
Diagonalisation of real symmetric matrices.
The affine n-space.
Affine subspaces and their reciprocal positions.
Matric properties in the plane an 3-dimansional space.
Orthogonality of affine subspaces.
Distance of affine subspaces in the plane and in the 3-dimensional space.
Planned learning activities and teaching methods: Classroom lectures. Students will be provided with exercises via the Moodle webpage.
Additional notes about suggested reading: Additional material will be uploaded on Moodle platform. Students who already own a textbook on Linear algebra and geometry may use it, contacting the teacher in case of doubt. Students who do not own any textbook are invited to get the indicated textbook.
Textbooks (and optional supplementary readings)
  • Cantarini, Nicoletta; Chiarellotto, Bruno; Fiorot Luisa, Un corso di matematica teoria ed esercizi Nicoletta Cantarini, Bruno Chiarellotto, Luisa Fiorot. Padova: Libreria progetto, 2014.