
Course unit
TOPICS IN LINEAR ALGEBRA AND GEOMETRY (Numerosita' canale 2)
IN08122537, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2018/19
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Basic courses 
MAT/02 
Algebra 
4.0 
Basic courses 
MAT/03 
Geometry 
5.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Lecture 
9.0 
72 
153.0 
No turn 
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)
ESPOSITO
FRANCESCO
(Membro Effettivo)
COLPI
RICCARDO
(Supplente)

23 A.A. 2018/19 canale 1 
01/10/2018 
30/11/2019 
ESPOSITO
FRANCESCO
(Presidente)
CARNOVALE
GIOVANNA
(Membro Effettivo)

22 A.A. 2017/18 
01/10/2017 
30/11/2018 
CARNOVALE
GIOVANNA
(Presidente)
ESPOSITO
FRANCESCO
(Membro Effettivo)
COLPI
RICCARDO
(Supplente)

21 A.A. 2017/18 matricole dispari 
01/10/2017 
30/11/2018 
ESPOSITO
FRANCESCO
(Presidente)
CARNOVALE
GIOVANNA
(Membro Effettivo)
COLPI
RICCARDO
(Supplente)

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. 
Examination methods:

Written exam 
Assessment criteria:

Correctness of solutions. Satisfactory explanations of the proposed solutions. Clarity. 
Course unit contents:

Introduction to linear algebra and its applications to analytic geometry.
Program:
complex numbers: definition, operations and properties
Rvector 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.
Subspaces.
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.
CauchySchwarz inequality and triangular inequality.
Orthogonality, orthogonal complement of a subspace.
GramSchmidt 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 nspace.
Affine subspaces and their reciprocal positions.
Matric properties in the plane an 3dimansional space.
Orthogonality of affine subspaces.
Distance of affine subspaces in the plane and in the 3dimensional 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) 

Abate, Marco; De_Fabritiis, Chiara, Geometria analitica con elementi di algebra lineareMarco Abate, Chiara De Fabritiis. Milano [etc.]: McGrawHill, 2015.

Innovative teaching methods: Teaching and learning strategies
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)
 Latex

