###### Math for Economics

ECON 5700, Mathematics for Economics, Summer 2021

The primary objective of this course is to provide incoming graduate students with the mathematical foundations necessary for the first-year sequence of microeconomics, macroeconomics, and econometric courses. This course is designed on the presumption that students will have already been exposed to the majority of this material in previous studies. Thus, the scope of the material to be covered is much larger than one would encounter in an ordinary mathematics course. ( Syllabus )

Two optional lectures on probability and matrix calculus will be provided. The notes of matrix calculus refer to the cookbook by Kaare Brandt Petersen and Michael Syskind Pedersen.

The weight and format of the final will be the same as the practice exam (  | Solution )

Lecture Notes

◾ Lecture 1 - Calculus ()

Algebra of sets and mapping. Limit and Continuity, Differentiation, Partial Differentiation, Total Derivatives.

◾ Lecture 2 - Calculus ()

Directional Derivatives and Gradients, Extremums and Second Derivative Test, Jacobian Matrices, Hessian Matrices.

◾ Lecture 3 - Calculus/Real Analysis ()

Taylor Expansion, Mean Value Theorems, Newton-Raphson Method, Convex Functions, Infinite Series. Metric spaces, Open and closed sets.

◾ Lecture 4 - Real Analysis/Calculus ()

Closure, Interior, Completeness, Boundedness, Lipschitz Continuous, Connectedness, Compactness, Convex Set, Homogeneous Function. Techniques of Integration, Properties of Definite Integral.

◾ Lecture 5 - Calculus/Linear Algebra ()

Fundamental Theorem of Calculus. Improper integral, Tests of Integral Convergence, Multiple Integral, Change of Variable in Double Variable. Vector spaces, Lines and Planes.

◾ Lecture 6 - Linear Algebra ()

System of linear equations, Reduced Echelon, Gauss-Jordan Elimination, Rank of Matrix, Linearly Independent, Matrix Operations, Matrix Transpose, Symmetric Matrix, Matrix multiplication.

◾ Lecture 7 - Linear Algebra ()

Matrix inverse, Spanning and Bases, Dimensions of subspaces, Row Column and Null Spaces, Fundamental Theorem of Linear Algebra(Rank-Nullity Theorem). Fundamental Theorem of Inverse Matrices, Linear Transformation and its inverse, Eigenvalues and Eigenvectors.

◾ Lecture 8 - Linear Algebra ()

Properties of Eigenvalues and Eigenvectors, Determinants, Laplace Theorem, Cramer’s rule, Adjoint of Matrix, Similarity, Diagonalization, Power of Matrices,

◾ Lecture 9 - Linear Algebra ()

Orthogonal Matrices, Grant-Schmid Process, QR Factorization, Spectrum Decomposition, Elementary Matrices, LU factorization.

◾ Lecture 10 - Linear Algebra/ Multivariate Algebra ()

Positive (Semi) Definite, Least Square Solution, Generalized Inverse of Matrices, Singular Value Decomposition, 𝐿𝑈/𝐿𝐷𝑀/𝐿𝐷𝐿' Decompositions, Cholesky Decomposition, Range and Kernel, Derivatives of Matrix and Vectors, Lagrange Multiplier.

◾ Lecture 11 - Review Session; Probability (optional) (ReviewProb&Stat)

Discrete and continuous random variables, Probability distributions, Measures of central tendency, Measures of dispersion, Moments & Moment Generating Functions (MGF).

Chunyu Qu