Introduction To Linear Algebra Book

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