Basics of Mathematics
Code: 106801Credits: 6
| Degree programme | Type | Course |
|---|---|---|
| Nanoscience and Nanotechnology | FB | 1 |
Contact lecturer
- Name :
- Jordi Villadelprat Yague
- Email :
- jordi.villadelprat@uab.cat
Teaching staff
- Laura Rodriguez Cima
- Maria Doris Potosà Rosero
Group languages
You can consult this information at the end of the document.
Prerequisites
This subject is autonomous in the topics covered. However, it is recommended to have basic skills with algebraic calculations and basic notions of differential calculus in one variable.
Objectives
The aim of the subject is knowledge and skill in the use of the basic tools of linear algebra and their applications. It focuses on the study of linear transformations, diagonalization of endomorphisms and their applications. Included are fundamental calculus tools such as complex numbers and matrix calculus.
Learning outcomes
- CM06 (Identify the mathematical nature of certain physical and chemical phenomena, in order to abstract the essential variables that describe them.) Identify the mathematical nature of certain physical and chemical phenomena, in order to abstract the essential variables that describe them.
- CM07 (Solve real-world problems that occur in the field of science and technology using mathematical tools and methods.) Solve real-world problems that occur in the field of science and technology using mathematical tools and methods.
- KM08 (Identify the elementary mathematical models and tools used in calculus, linear algebra and differential equations.) Identify the elementary mathematical models and tools used in calculus, linear algebra and differential equations.
- SM09 (Express oneself clearly using basic mathematical language.) Express oneself clearly using basic mathematical language.
- SM10 (Solve simple problems related to matrix calculus, linear equations and first order differential equations.) Solve simple problems related to matrix calculus, linear equations and first order differential equations.
- SM12 (Use graphical and numerical methods to explore, describe and interpret data.) Use graphical and numerical methods to explore, describe and interpret data.
Contents
1. Complex numbers
Complex numbers and their properties. Trigonometric form and polar form. Operations with complex numbers. Roots of complex numbers. Fundamental theorem of algebra
2. Matrices
Solving systems of linear equations. Addition, product, and transpose of matrices. Elementary transformations. Rank of a matrix. Invertible matrices. Determinants
3. Vector spaces
Definition and examples. Linear dependence and independence. Vector subspaces and systems of generators. Basis, coordinates, and dimension. Basis of the intersection and of sum of subspaces. Change-of-basis matrix.
4. Linear transformations
Definition and examples. Matrix representation. Composition. Dependence of the matrix with respect to a base change. Kernel, image and rank. Computing basis of kernels and images.
5. Diagonalization
Eigenvectors and eigenvalues of an endomorphism. Characteristic polynomial. Diagonalization criterion. Spectral theorem
6. Applications of diagonalization
Sequences with linear recurrences. Linear differential equations and systems of first order linear differential equations.
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Self-developed study | 91 | 3.64 | |
| Exercise resolution classes | 10 | 0.4 | |
| Theory classes | 36 | 1.44 | |
| Tutorials | 2 | 0.08 | |
| Computer sessions | 6 | 0.24 |
The subject consists of three main activities.
Theory classes in which the scientific and technical concepts and knowledge specific to the subject are introduced and developed.
Problem classes, complementary to theory classes. Exercises will be solved in order to deepen the understanding of the new scientific and technical concepts and knowledge presented in the theory classes. Usually the student thinks and tries to solve the problems that are discussed in class and arrives at the final optimal solution.
Finally, there will be 3 practice sessions in the computer classroom, where specific software will be used for mathematical calculations.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Final exam | 50% | 2 | 0.08 | CM06, CM07, KM08, SM09, SM10 |
| Evaluation of practices | 15% | 1 | 0.04 | SM09, SM10, SM12 |
| Mid-term exam | 35% | 2 | 0.08 | CM06, CM07, SM10 |
Continuous assessment
It is organized into the following blocks, each of which will have a specific weight in the final grade:
Labs (PR): There will be three lab sessions that will be assessed.
First midterm (P1): Written test at the midpoint of the semester.
Second midterm (P2): Written test at the end of the semester.
If N1=0.15PR+0.35P1+0.5P2 is greater than or equal to 5, then N1 is the final grade for the course. If N1 is lower than 5, the student may take a retake exam (R). In this case, if N2=0.15SEM+0.85*R is greater than or equal to 5, then the final grade for the course is 5. Otherwise, the final grade for the course is max(N1,N2). In any case, the student must have participated in 66% of the assessed activities.
Single assessment
On the same day as the second midterm of the continuous assessment, students who have previously opted for the single assessment will take a final exam (F) covering the entire syllabus. The grade obtained will be N3=0.15PR+0.85F. If N3<5, the same retake system as for continuous assessment will apply.
NOTE: In this course, the use of Artificial Intelligence (AI) technologies is not permitted at any stage. Any work that includes AI-generated fragments will be considered a breach of academic honesty and may result in a partial or total penalty on the activity grade, or more serious sanctions in cases of severe misconduct.
Bibliography
J. Hefferon, Linear algebra, http://joshua.smcvt.edu/linearalgebra/
M. Masdeu, A. Ruiz, Apunts d'Àlgebra Lineal, https://mat.uab.cat/~albert/wp/wp-content/uploads/2020/09/Apunts_d__lgebra_Lineal.pdf
E. Nart X. Xarles, Apunts d'àlgebra lineal, Materials de la UAB, núm. 237, 1a edició.
D.C. Lay, Álgebra lineal y sus aplicaciones, Pearson Educación, 2016 (ebook)
Grossman, Stanley I., Álgebra lineal. Mc Graw Hill, 2012, 7a edició. (eBook)
Software
Python
Course groups and languages
The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE
| Type of teaching | Group | Language | Semester | Shift |
|---|---|---|---|---|
| (TE) Theory | 1 | Catalan | second semester | afternoon |
| (PAUL) Classroom practices | 1 | Catalan | second semester | afternoon |
| (PLAB) Practical laboratories | 1 | Catalan | second semester | morning-mixed |
| (PAUL) Classroom practices | 2 | Catalan | second semester | afternoon |
| (PLAB) Practical laboratories | 2 | Catalan | second semester | morning-mixed |
| (PLAB) Practical laboratories | 3 | Catalan | second semester | morning-mixed |
| (PLAB) Practical laboratories | 4 | Catalan | second semester | morning-mixed |