文档介绍：ELEMENTARY
LINEAR ALGEBRA
K. R. MATTHEWS
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF QUEENSLAND
Second Online Version, December 1998
Comments to the author at ******@
All contents copyright
c 1991 Keith R. Matthews
Department of Mathematics
University of Queensland
All rights reserved
Contents
1 LINEAR EQUATIONS 1
Introduction to linear equations . . . . . . . . . . . . . . . . . 1
Solving linear equations . . . . . . . . . . . . . . . . . . . . . 6
The Gauss–Jordan algorithm . . . . . . . . . . . . . . . . . . 8
Systematic solution of linear systems. . . . . . . . . . . . . . 9
Homogeneous systems . . . . . . . . . . . . . . . . . . . . . . 16
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 MATRICES 23
Matrix arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . 23
Linear transformations . . . . . . . . . . . . . . . . . . . . . . 27
Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 31
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Non–singular matrices . . . . . . . . . . . . . . . . . . . . . . 36
Least squares solution of equations . . . . . . . . . . . . . . . 47
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 SUBSPACES 55
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Subspaces of F n . . . . . . . . . . . . . . . . . . . . . . . . . 55
Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . 58
Basis of a subspace . . . . . . . . . . . . . . . . . . . . . . . . 61
Rank and nullity of a matrix . . . . . . . . . . . . . . . . . . 64
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 DETERMINANTS 71
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
i
PLEX NUMBERS 89
Constructing plex numbers . . . . . . . . . . . . . . . 89
Calculating plex numbers .