MATHEMATICS 2
Instructional goals
The aim of the course is to provide working knowledge of techniques and tools of linear algebra that will be needed in the study of economics, management and finance. At the end of the course students should be able to solve problems adapting all the techniques learned and to discuss the importance of the main theoretical results presented during the course.
Intended learning outcomes
Knowledge and understanding:
The course will provide many basic mathematical tools, together with their theoretical interpretation and with examples of their possible applications to quantitative analysis in Economics, Finance, Statistics, with the help of ad hoc Gemini Gems when needed.
Applying knowledge and understanding:
Students will be able to adapt and apply a variety of techniques to solve non-standard mathematical problems, as well as to discuss the arguments that justify usage of such techniques.
Making judgements:
Students are expected to be able to choose properly the best solution strategy for each mathematical problem and to understand how to apply concepts and tools to economics and financial problems. This ability will be evaluated via exercises and exams.
Communications Skills:
Students will learn how to properly formulate and communicate mathematical concepts and logical reasoning, using the English language. They will also understand how different concrete problems can be studied using similar techniques.
Learning skills:
Students will broaden their mathematical knowledge and reasoning and they will become able to work independently with advanced mathematical concepts and tools.
Course Contents
An introduction to linear algebra, with applications to economics.
Matrices, Systems of Linear Equations, Vector Spaces, Subspaces and Bases, Eigenvalues and Eigenvectors, Orthogonality, Least-squares, linear programming.
Reference Books
Main text: R.Larson, Elementary Linear Algebra, 8th edition, Cengage
Other required readings:
D.C. Lay, S.R. Lay and J.J. McDonald, Linear Algebra and Its Applications, 5th Edition, Pearson.
S.J. Leon, Linear Algebra with Applications, 8th Edition, Pearson.
K.Sydsaeter, P.J.Hammond, A.Strom, A.Carvajal, Essential Mathematics for Economic Analysis, 6th Edition, Pearson.
W.H.Green, Econometric Analysis, 8th Edition, Pearson
Teaching Methods
Lectures on the main theoretical aspects of each topic, with examples and short exercises.
Weekly exercise sessions: written exercises + use of Gemini Gems.
Assessment Method
Student assessments in this course will be determined by individual performance, active participation in class activities, and consistent engagement with assigned tasks.
A midterm exam will be held after the sixth week of classes. The exact date of the midterm will be announced no later than the third week of the semester. It is important to note that retakes will not be permitted.
Final performance will also reflect active class participation. Further details will be provided before the course begins.
Thesis assignment criteria
Interview with the instructor.
Week 1
1)Systems of linear equations
2) Gauss and Gauss-Jordan Elimination. (Larson Chapter 1)
TA sessions on the lectures of the present week.
Week 2
Matrices.
1) Operations and Properties of matrices.
2) The Inverse of a Matrix.
3) Markov Chains. (Larson Chapter 2)
TA sessions on the lectures of the present week.
Week 3
Determinants.
1) The Determinant of a Matrix.
2) Properties of Determinants. (Larson Chapter 3)
TA sessions on the lectures of the present week.
Week 4
1) Vector Spaces and Subspaces
2) Spanning Sets and Linear Independence. (Larson First part of Chapter 4)
TA sessions on the lectures of the present week.
Week 5
1) Basis and Dimension. 2) Rank of a Matrix and Systems of Linear Equations. (Larson: Second part of Chapter 4)
TA sessions on the lectures of the present week.
Week 6
1) Linear programs, feasibility, polyhedra and polytopes, solving a linear problem;
2) Dual linear programs, weak and strong duality theorems, complementarity conditions. (Essential, Chapter 19)
TA sessions on the lectures of the present week.
Week 7
Inner Product Spaces
1) Length and Dot Product in the Euclidean space 2) Inner Product Spaces (Larson, First Part of Chapter 5)
TA sessions on the lectures of the present week.
Week 8
Inner Product Spaces
1) Orthonormal Bases: Gram-Schmidt Process 2) Mathematical Models and Least Squares Analysis (Larson, Second Part of Chapter 5)
TA sessions on the lectures of the present week.
Week 9
Eigenvalues and Eigenvectors
1) Definition of Eigenvalues and Eigenvectors 2) Diagonalization (Larson, First part of Chapter 7)
TA sessions on the lectures of the present week.
Week 10
Eigenvalues and Eigenvectors
1) Symmetric Matrices, Eigenvalues and Eigenvectors 2) Orthogonal Diagonalization (Larson, Second part of Chapter 7)
TA sessions on the lectures of the present week.
Week 11
1)Principal Component Analysis (Lay, Section 7.5)
2)Eigenvalues and Markov chains (Leon, Section 6.3)
TA sessions on the lectures of the present week.
Week 12
1)Quadratic forms of definite matrices (Green, Appendix A7)
2) Differentiation and optimization of functions of several variables using matrices (Green, Appendix A8)
TA sessions on the lectures of the present week.