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LINEAR ALGEBRA

Thamarai Selvi

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Contents

Dependencies among sections3
Chapter 1.Euclidean lines and hyperplanes5
Definition5
Euclidean plane and Euclidean space6
The standard scalar product9
Angles, orthogonality, and normal vectors14
Orthogonal projections and normality19
Projecting onto lines and hyperplanes containing zero19
Projecting onto arbitrary lines and hyperplanes24
Distances26
Reflections30
Reflecting in lines and hyperplanes containing zero31
Reflecting in arbitrary lines and hyperplanes33
Cauchy-Schwarz35
What is next?37
Chapter 2.Vector spaces41
Definition of a vector space42
Examples43
Basic properties50
Chapter 3.Subspaces53
Definition and examples53
The standard scalar product (again)56
Intersections58
Linear hulls, linear combinations, and generators60
Sums of subspaces65
Chapter 4.Linear maps71
Definition and examples71
Linear maps form a vector space76
Linear equations81
Characterising linear maps84
Isomorphisms86
Chapter 5.Matrices89
Definition of matrices90
Matrix associated to a linear map91
The product of a matrix and a vector92
Linear maps associated to matrices94
Addition and multiplication of matrices96
Row space, column space, and transpose of a matrix102
1
2CONTENTS
Chapter 6.Computations with matrices105
Elementary row and column operations105
Row echelon form110
Generators for the kernel116
Reduced row echelon form119
Chapter 7.Linear independence and dimension123
Linear independence123
Bases129
The basis extension theorem and dimension135
Dimensions of subspaces143
Chapter 8.Ranks149
The rank of a linear map149
The rank of a matrix152
Computing intersections156
Inverses of matrices159
Solving linear equations164
Chapter 9.Linear maps and matrices167
The matrix associated to a linear map167
The matrix associated to the composition of linear maps171
Changing bases174
Endomorphisms175
Similar matrices and the trace176
Classifying matrices178
Similar matrices178
Equivalent matrices179
Chapter 10.Determinants183
Determinants of matrices183
Some properties of the determinant190
Cramer’s rule193
Determinants of endomorphisms194
Linear equations with parameters195
Chapter 11.Eigenvalues and Eigenvectors197
Eigenvalues and eigenvectors197
The characteristic polynomial199
Diagonalization203
Appendix A.Review of maps213
Appendix B.Fields215
Definition of fields215
The field of complex numbers.217
Appendix C.Labeled collections219
Appendix D.Infinite-dimensional vector spaces and Zorn’s Lemma22

692 pages, Kindle Edition

Published December 16, 2020

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