Introduction to Linear Algebra: A Primer for Social Scientists

by Mills,Gordon

• ISBN13: 9780202361598
• ISBN10: 0202361594
• eBook ISBN(s): 9781351511568
• Edition: Reprint
• Format: Nonspecific Binding
• Copyright: 2007-08-30
• Publisher: Routledge
• More Book Details

Summary

This is the first book on linear algebra written specifically for social scientists. It deals only with those aspects of the subject applicable in the social sciences and provides a thorough understanding of linear algebra for those who wish to use it as a tool in the design, execution, and interpretation of research. Linear mathematical models play an important role in all of the social sciences. This book provides a step-by-step introduction to those parts of linear algebra which are useful in such model building. It illustrates some of the applications of linear analysis and helps the reader learn how to convert his formulation of a social science problem into algebraic terms.

Author Biography

Gordon Mills is an honorary professor in the department of economics at the University of Sydney

Table of Contents

Preface	p. ix
Introduction	p. 1
Linear and non-linear systems	p. 1
Simultaneous linear equations: an economic example	p. 2
Simultaneous linear equations: general systems	p. 5
Summation notation	p. 6
Exercises	p. 8
Inequalities	p. 10
Linear inequalities: graphical representation	p. 12
Linear inequalities: an economic example	p. 14
Exercises	p. 16
Some concepts from set theory	p. 17
Necessary and sufficient conditions	p. 20
Exercises	p. 22
Vectors	p. 25
Vectors - the basic concepts	p. 25
Vector operations	p. 27
Exercises	p. 30
Vector representation of simultaneous equations	p. 31
Euclidean space and linear dependence	p. 31
Exercises	p. 36
Bases	p. 37
Exercises	p. 40
Matrices	p. 42
The concept of a matrix	p. 42
The first matrix operations	p. 43
Exercises	p. 45
Some particular matrix types	p. 46
Exercises	p. 51
Matrix multiplication - definition	p. 51
Exercises	p. 54
Some properties of matrix multiplication	p. 55
Exercises	p. 59
Summation notation again	p. 60
Matrix multiplication and simultaneous equations	p. 63
Exercises	p. 68
Submatrices and partitioning	p. 69
Transposition; symmetric and skew-symmetric matrices	p. 72
Exercises	p. 76
Elementary Operations and the Rank of a Matrix	p. 80
Introduction	p. 80
Elementary operations	p. 81
Exercises	p. 84
Echelon matrices	p. 84
Exercises	p. 89
Elementary operations and linear dependence	p. 89
Exercises	p. 91
The rank of a matrix	p. 91
Product matrices and rank	p. 94
Computing the rank of a matrix	p. 98
Exercises	p. 101
The Inverse of a Square Matrix	p. 102
The concept of an inverse matrix	p. 102
An approach to calculating the inverse matrix	p. 103
Exercises	p. 106
The existence of the inverse matrix	p. 106
Some further properties of inverse matrices	p. 112
Some computational aspects of finding the inverse	p. 114
Exercises	p. 117
Computation of the inverse by using partitioning	p. 117
Exercises	p. 122
The Solution of Simultaneous Linear Equations	p. 125
Introduction	p. 125
Exercises	p. 128
A formal computational method	p. 128
Exercises	p. 132
Non-homogeneous equations: some general considerations	p. 133
Exercises	p. 138
Non-homogeneous equations: further discussion	p. 139
Exercises	p. 147
Homogeneous equations	p. 147
Exercises	p. 151
A summary of the results	p. 152
A variant of the computational approach	p. 153
Exercises	p. 155
Some applied examples	p. 156
Exercises	p. 162
Integer Variables and Other Topics	p. 165
Introduction	p. 165
Basic solutions for linear equations	p. 165
Exercises	p. 169
An appreciation of linear programming	p. 170
Exercises	p. 175
Quadratic forms	p. 175
Exercises	p. 180
The need for integer-variable analysis	p. 180
The single linear equation in one or two integer variables	p. 182
Exercises	p. 187
Larger systems of linear equations in integers	p. 188
Exercises	p. 189
General integer systems and computational approaches	p. 190
Problems where the integer requirement is automatically satisfied	p. 193
Exercises	p. 195
Determinants	p. 196
Solutions and Hints for Some of the Exercises	p. 206
Suggestions for Further Reading	p. 221
Index	p. 223
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