31C01100 - Taloustieteen matemaattiset menetelmät, Luento-opetus, 13.9.2021-17.12.2021

Topic outline

• Lecture Slides
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    Lecture 1 File

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    Course introduction; Systems of linear equations; Gauss-Jordan elimination method

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    Lecture 2 File

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    Systems of linear equations in matrix form; Rank of a matrix; Existence and multiplicity of solutions for systems of linear equations

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    Lecture 3 File

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    Matrix algebra; Nonsingular and invertible matrices; Determinant of a matrix; Cramer's rule

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    Exercise from Lecture 3 (updated 28.9.2021) File

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    This document contains the detailed steps on how to obtain the inverse matrix in the Exercise at the end of Lecture 3

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    Lecture 4 (updated 27.9.2021) File

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    Vectors in \( \mathbb{R}^n \); Linear combinations; Linear independence; Basis and dimension of subsets of \( \mathbb{R}^n \)

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    Lecture 5 File

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    Functions of several variables; Limits; Continuity; Partial derivatives

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    Lecture 6 File

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    Total differential and linear approximations; Open and closed sets; Continuously differentiable functions; Chain rule; Higher-order derivatives and Hessian matrix

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    Lecture 7 File

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    Implicit function theorem; Homogeneous functions

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    Lecture 8 File

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    Concave and convex functions; Definite and semidefinite matrices

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    Lecture 9 (updated 11.10.2021) File

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    Unconstrained optimization; Weierstrass theorem; Local and global extrema

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    Lecture 10 File

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    Quadratic forms; Positive monotone transformations

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    Lecture 11 (updated 19.10.2021) File

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    L'Hôpital's rule; Infinite series; Review

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    Lecture 12 File

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    Two-variable optimization problems with one equality constraint

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    Lecture 13 File

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    Optimization problems with n variables and m equality constraints

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    Lecture 14 (updated 16.11.2021) File

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    Optimization problems with inequality constraints; Kuhn-Tucker formulation

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    Lecture 15 (updated 17.11.2021) File

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    Optimization problems with mixed constraints; Envelope theorems

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    Lecture 16 File

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    Envelope theorems; First-order linear difference equations

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    Lecture 17 File

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    Systems of first-order linear difference equations; Eigenvalues and eigenvectors

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    Lecture 18 File

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    Nonhomogeneous systems of first-order linear difference equations; Repeated and complex eigenvalues

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    Exercises from Lecture 18 File

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    Lecture 19 File

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    First-order ordinary differential equations; Separable first-order differential equations; Linear first-order differential equations

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    Lecture 20 File

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    Linear second-order differential equations; Stability

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    Lecture 21 File

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    Systems of ordinary differential equations; Stability

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    Lecture 22 File

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    Review