
Course unit
TOPICS IN LINEAR ALGEBRA AND GEOMETRY
IN08122537, A.A. 2017/18
Information concerning the students who enrolled in A.Y. 2017/18
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 
5 A.A. 2018/19 
01/10/2018 
30/11/2019 
LONGO
MATTEO
(Presidente)
CANDILERA
MAURIZIO
(Membro Effettivo)
CAILOTTO
MAURIZIO
(Supplente)

4 A.A. 2016/17 
01/10/2016 
30/11/2017 
LONGO
MATTEO
(Presidente)
CANDILERA
MAURIZIO
(Membro Effettivo)

Prerequisites:

Nothing required in advance 
Target skills and knowledge:

Basic notions in linear algebra and their fundamental applications to geometry, with particular attention
devoted to the study of vector spaces, linear functions and linear systems. Spectral theorem and its applications. 
Assessment criteria:

Written exam 
Course unit contents:

Introduction to linear algebra with a view to its main applications to analytic geometry. 
Planned learning activities and teaching methods:

Real vector spaces and subspaces. The real vector space R^n; the vector space of mxn matrices with real entries.
The vector space of polynomials in one variable with real coefficients.
Finetely generated vector spaces.
Intersection, union and sum of subspaces.
Bases and their existence. Dimension of a vector space.
Coordinates of a vector with respect to a basis.
Direct sums. Grassmann formula and its applications.
Linear functions. Definition of a linear function: existence and uniqueness conditions.
Kernel and image. Injective and surjective functions. Ranknullity Theorem.
Preimage. Matrices associated to a linear function. Rank of a matrix.
Linear systems. Rouche'Capelli Theorem.
Elementary row operations. Reduction to row echelon form. Systems of linear equations.
Systems of linear equations depending on a parameter.
Product of matrices, composition of linear maps. Invertible square matrices and
the computation of the inverse of an invertible matrix. Changes of bases.
Conjugated matrices.
The determinant of a matrix and its properties.
Eigenvalues and eigenvectors of an endomorphism. Eigenspaces.
Characteristic polynomial. Algebraic and geometric multiplicities.
Spectral Theorem.
Inner product. The standard inner product.
CauchySchwarz inequality and triangular inequality.
Orthogonal vectors and orthogonal subspaces.
GramSchmidt orthogonalization.
Orthogonal projection.
Isometries, orthogonal matrices.
Symmetric matrices.
Complex numbers. Spectral Theorem for symmetric matrices.
ndimensional affine space. Description and properties of linear manifolds.
Euclidean space. Orthogonal submanifolds.
Vector product. Distances between two manifolds.
Affine and euclidean reference systems. 
Additional notes about suggested reading:

ChiarellottoCantariniFiorot, Un Corso di Matematica, Libreria Progetto, Padova.
F. Bottacin, Algebra Lineare e Geometria, Ed. Escupapio, Bologna 
Textbooks (and optional supplementary readings) 

Nicoletta Cantarini, Bruno Chiarellotto, Luisa Fiorot, Un corso di Matematica. Padova: Progetto, .

F. Bottacin, Algebra Lineare e Geometria. Bologna: Ed. Escupapio, .


