Wie man die lineare Algebra für Ökonomen versteht

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1 Springer Texts in Business and Economics Linear Algebra for Economists Bearbeitet von Fuad Aleskerov, Hasan Ersel, Dmitri Piontkovski 1. Auflage Buch. xii, 280 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 602 g Weitere Fachgebiete > Mathematik > Algebra > Lineare und multilineare Algebra, Matrizentheorie Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 1 Some Basic Concepts Introduction Linearity System of Coordinates on the Plane R Microeconomics: Market Equilibrium Equilibrium in a Single Market Multi-Market Equilibrium Macroeconomic Policy Problem A Simple Macroeconomic Policy Model with One Target A Macroeconomic Policy Model with Multiple Targets and Multiple Instruments Problems Vectors and Matrices Vectors Algebraic Properties of Vectors Geometric Interpretation of Vectors and Operations on Them Geometric Interpretation in R Dot Product of Two Vectors The Length of a Vector, and the Angle Between Two Vectors An Economic Example: Two Plants Another Economic Application: Index Numbers Matrices Operations on Matrices Matrix Multiplication Trace of a Matrix Transpose of a Matrix Rank of a Matrix Elementary Operations and Elementary Matrices Problems ix

3 x 3 Square Matrices and Determinants Transformation of Coordinates Translation Rotation Square Matrices Identity Matrix Power of a Matrix and Polynomial of a Matrix Systems of Linear Equations: The Case of Two Variables Determinant of a Matrix The Basic Properties of Determinants Determinant and Elementary Operations Problems Inverse Matrix Inverse Matrix and Matrix Division Rank and Determinants Problems Systems of Linear Equations The Case of Unique Solution: Cramer s Rule Gauss Method: Sequential Elimination of Unknown Variables Homogeneous Equations Problems Mathematical Problems Economic Problems Linear Spaces Linear Independence of Vectors Addition of Vectors and Multiplication of a Vector by a Real Number Isomorphism of Linear Spaces Subspaces Examples of Subspaces A Method of Constructing Subspaces One-Dimensional Subspaces Hyperplane Coordinate Change Economic Example: Production Technology Set Problems Euclidean Spaces General Definitions Orthogonal Bases Least Squares Method Isomorphism of Euclidean Spaces Problems

4 xi 8 Linear Transformations Addition and Multiplication of Linear Operators Inverse Transformation, Image and Kernel under a Transformation Linear Transformation Matrices with Respect to Different Bases Problems Eigenvectors and Eigenvalues Macroeconomic Example: Growth and Consumption The Model Numerical Example Self-Adjoint Operators Orthogonal Operators Quadratic Forms Problems Linear Model of Production in a Classical Setting Introduction The Leontief Model Existence of a Unique Non-Negative Solution to the Leontief System Conditions for Getting a Positive (Economically Meaningful) Solution to the Leontief Model Prices of Production in the Linear Production Model Perron Frobenius Theorem Linear Production Model (continued) Sraffa System: The Case of Basic Commodities Sraffa System: Non-Basic Commodities Added Problems Linear Programming Diet Problem Linear Production Model Convexity Transportation Problem Dual Problem Economic Interpretation of Dual Variables A Generalization of the Leontief Model: Multiple Production Techniques and Linear Programming Problems A Natural Numbers and Induction A.1 Natural Numbers: Axiomatic Definition A.2 Induction Principle A.3 Problems

5 xii B Methods of Evaluating Determinants B.1 Transformation of Determinants B.2 Methods of Evaluating Determinants of High Order B.2.1 Reducing to Triangular Form B.2.2 Method of Multipliers B.2.3 Recursive Definition of Determinant B.2.4 Representation of a Determinant as a Sum of Two Determinants B.2.5 Changing the Elements of Determinant B.2.6 Two Classical Determinants B.3 Problems C Complex Numbers C.1 Operations with Complex Numbers C.1.1 Conjugation C.1.2 Modulus C.1.3 Inverse and Division C.1.4 Argument C.1.5 Exponent C.2 Algebraic Equations C.3 Linear Spaces Over Complex Numbers C.4 Problems D Pseudoinverse D.1 Definition and Basic Properties D.1.1 The Basic Properties of Pseudoinverse D.2 Full Rank Factorization and a Formula for Pseudoinverse D.3 Pseudoinverse and Approximations D.4 Problems E Answers and Solutions References Index

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