29228 - Geometry and Algebra T

Academic Year 2022/2023

  • Docente: Luca Marchese
  • Credits: 6
  • SSD: MAT/03
  • Language: Italian
  • Moduli: Luca Marchese (Modulo 1) Marta Morigi (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: First cycle degree programme (L) in Chemical and Biochemical Engineering (cod. 8887)

Learning outcomes

Provide the basic notions in linear algebra and vector calculus, with applications to the study of systems of linear equations and to the geometry in the euclidean space

Course contents

1) Real and complex numbers, polynomials, vector in linear space of dimension n. 

2) Systems of linear equations and matrices (I): sum and scalar multiplication in the euclidean space. Linear equations. Elementary operations on the rows of a system of linear equations. Relation with matrices. 

3) Systems of linear equations and matrices (II): parametric form of the solutions of a linear system.

4) Matrices: sum of matrices and scalar multiplication. Product of vectors by matrices. Product of matrices. Invertible matrices. Inversion and solution of linear systems. 

5) Vector spaces: basic definitions, linear combinations. Vectors subspaces. Sum and direct sum of vector subspaces. 

6) Families of vectors, linearly independent families and generating families. Basis of a vector space. Dimension and coordinates. 

7) Linear maps between vector spaces. Relation between linear maps and matrices. Image and Kernel. Injective and surjective linear maps. 

8) Matrix of a linear map in a fixed basis. Change of basis. 

9) Determinants: definitions, basic properties and computation. 

10) Eigenvalues and eigenvector. Definition of diagonalisable matrix. Eigenspaces. 

11) Diagonalization of symmetric matrices. Scalar products. 

12) Orthonormal basis.

 

 

 

Readings/Bibliography

  • Rita Fioresi, Marta Morigi, Introduzione all'algebra lineare. Casa editrice Ambrosiana
  • Francesco Bottacin, Algebra Lineare e Geometria. Società editrice Esculapio.

 

Teaching methods

There are 60 hours in total, devoted both to the theory, with proofs of relevant theorems, and to the complete solution of various exercises of different levels

Assessment methods

Written exam, where the student is asked to solve 4 exercises and to answer to 1 question on the general theory. 

Office hours

See the website of Luca Marchese

See the website of Marta Morigi