Linear Algebra Notes
David A. SANTOS
January 2, 2010 REVISION
,ii
Contents
Preface iv 6 Determinants 111
6.1 Permutations . . . . . . . . . . . . . . . . 111
To the Student v 6.2 Cycle Notation . . . . . . . . . . . . . . . . 114
1 Preliminaries 1 6.3 Determinants . . . . . . . . . . . . . . . . 119
1.1 Sets and Notation . . . . . . . . . . . . . . 1 6.4 Laplace Expansion . . . . . . . . . . . . . 129
1.2 Partitions and Equivalence Relations . . . . 4 6.5 Determinants and Linear Systems . . . . . 136
1.3 Binary Operations . . . . . . . . . . . . . . 6
1.4 Zn . . . . . . . . . . . . . . . . . . . . . . 9 7 Eigenvalues and Eigenvectors 138
1.5 Fields . . . . . . . . . . . . . . . . . . . . 13 7.1 Similar Matrices . . . . . . . . . . . . . . . 138
1.6 Functions . . . . . . . . . . . . . . . . . . 15 7.2 Eigenvalues and Eigenvectors . . . . . . . 139
7.3 Diagonalisability . . . . . . . . . . . . . . 143
2 Matrices and Matrix Operations 18 7.4 Theorem of Cayley and Hamilton . . . . . 147
2.1 The Algebra of Matrices . . . . . . . . . . . 18
2.2 Matrix Multiplication . . . . . . . . . . . . 22 8 Linear Algebra and Geometry 149
2.3 Trace and Transpose . . . . . . . . . . . . 28 8.1 Points and Bi-points in R2 . . . . . . . . . 149
2.4 Special Matrices . . . . . . . . . . . . . . . 31 8.2 Vectors in R2 . . . . . . . . . . . . . . . . 152
2.5 Matrix Inversion . . . . . . . . . . . . . . . 36 8.3 Dot Product in R2 . . . . . . . . . . . . . . 158
2.6 Block Matrices . . . . . . . . . . . . . . . 44
8.4 Lines on the Plane . . . . . . . . . . . . . 164
2.7 Rank of a Matrix . . . . . . . . . . . . . . 45
8.5 Vectors in R3 . . . . . . . . . . . . . . . . 169
2.8 Rank and Invertibility . . . . . . . . . . . . 55
8.6 Planes and Lines in R3 . . . . . . . . . . . 174
3 Linear Equations 65 8.7 Rn . . . . . . . . . . . . . . . . . . . . . . 178
3.1 Definitions . . . . . . . . . . . . . . . . . 65
3.2 Existence of Solutions . . . . . . . . . . . 70 A Answers and Hints 182
3.3 Examples of Linear Systems . . . . . . . . 71 Answers and Hints . . . . . . . . . . . . . . . . 182
4 Vector Spaces 76
GNU Free Documentation License 262
4.1 Vector Spaces . . . . . . . . . . . . . . . . 76
1. APPLICABILITY AND DEFINITIONS . . . . . . 262
4.2 Vector Subspaces . . . . . . . . . . . . . . 79
2. VERBATIM COPYING . . . . . . . . . . . . . 262
4.3 Linear Independence . . . . . . . . . . . . 81
3. COPYING IN QUANTITY . . . . . . . . . . . . 262
4.4 Spanning Sets . . . . . . . . . . . . . . . . 84
4. MODIFICATIONS . . . . . . . . . . . . . . . 262
4.5 Bases . . . . . . . . . . . . . . . . . . . . 87
5. COMBINING DOCUMENTS . . . . . . . . . . 263
4.6 Coordinates . . . . . . . . . . . . . . . . . 91
6. COLLECTIONS OF DOCUMENTS . . . . . . . 263
5 Linear Transformations 97 7. AGGREGATION WITH INDEPENDENT WORKS 263
5.1 Linear Transformations . . . . . . . . . . . 97 8. TRANSLATION . . . . . . . . . . . . . . . . 263
5.2 Kernel and Image of a Linear Transformation 99 9. TERMINATION . . . . . . . . . . . . . . . . 263
5.3 Matrix Representation . . . . . . . . . . . 104 10. FUTURE REVISIONS OF THIS LICENSE . . 263
, iii
Copyright c 2007 David Anthony SANTOS. Permission is granted to copy, distribute and/or
modify this document under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation; with no Invariant Sections, no
Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section
entitled “GNU Free Documentation License”.
, Preface
These notes started during the Spring of 2002, when John MAJEWICZ and I each taught a section of
Linear Algebra. I would like to thank him for numerous suggestions on the written notes.
The students of my class were: Craig BARIBAULT, Chun CAO, Jacky CHAN, Pho DO, Keith HAR-
MON, Nicholas SELVAGGI, Sanda SHWE, and Huong VU. I must also thank my former student William
CARROLL for some comments and for supplying the proofs of a few results.
John’s students were David HERNÁNDEZ, Adel JAILILI, Andrew KIM, Jong KIM, Abdelmounaim
LAAYOUNI, Aju MATHEW, Nikita MORIN, Thomas NEGRÓN, Latoya ROBINSON, and Saem SOEURN.
Linear Algebra is often a student’s first introduction to abstract mathematics. Linear Algebra is well
suited for this, as it has a number of beautiful but elementary and easy to prove theorems. My purpose
with these notes is to introduce students to the concept of proof in a gentle manner.
David A. Santos
iv
David A. SANTOS
January 2, 2010 REVISION
,ii
Contents
Preface iv 6 Determinants 111
6.1 Permutations . . . . . . . . . . . . . . . . 111
To the Student v 6.2 Cycle Notation . . . . . . . . . . . . . . . . 114
1 Preliminaries 1 6.3 Determinants . . . . . . . . . . . . . . . . 119
1.1 Sets and Notation . . . . . . . . . . . . . . 1 6.4 Laplace Expansion . . . . . . . . . . . . . 129
1.2 Partitions and Equivalence Relations . . . . 4 6.5 Determinants and Linear Systems . . . . . 136
1.3 Binary Operations . . . . . . . . . . . . . . 6
1.4 Zn . . . . . . . . . . . . . . . . . . . . . . 9 7 Eigenvalues and Eigenvectors 138
1.5 Fields . . . . . . . . . . . . . . . . . . . . 13 7.1 Similar Matrices . . . . . . . . . . . . . . . 138
1.6 Functions . . . . . . . . . . . . . . . . . . 15 7.2 Eigenvalues and Eigenvectors . . . . . . . 139
7.3 Diagonalisability . . . . . . . . . . . . . . 143
2 Matrices and Matrix Operations 18 7.4 Theorem of Cayley and Hamilton . . . . . 147
2.1 The Algebra of Matrices . . . . . . . . . . . 18
2.2 Matrix Multiplication . . . . . . . . . . . . 22 8 Linear Algebra and Geometry 149
2.3 Trace and Transpose . . . . . . . . . . . . 28 8.1 Points and Bi-points in R2 . . . . . . . . . 149
2.4 Special Matrices . . . . . . . . . . . . . . . 31 8.2 Vectors in R2 . . . . . . . . . . . . . . . . 152
2.5 Matrix Inversion . . . . . . . . . . . . . . . 36 8.3 Dot Product in R2 . . . . . . . . . . . . . . 158
2.6 Block Matrices . . . . . . . . . . . . . . . 44
8.4 Lines on the Plane . . . . . . . . . . . . . 164
2.7 Rank of a Matrix . . . . . . . . . . . . . . 45
8.5 Vectors in R3 . . . . . . . . . . . . . . . . 169
2.8 Rank and Invertibility . . . . . . . . . . . . 55
8.6 Planes and Lines in R3 . . . . . . . . . . . 174
3 Linear Equations 65 8.7 Rn . . . . . . . . . . . . . . . . . . . . . . 178
3.1 Definitions . . . . . . . . . . . . . . . . . 65
3.2 Existence of Solutions . . . . . . . . . . . 70 A Answers and Hints 182
3.3 Examples of Linear Systems . . . . . . . . 71 Answers and Hints . . . . . . . . . . . . . . . . 182
4 Vector Spaces 76
GNU Free Documentation License 262
4.1 Vector Spaces . . . . . . . . . . . . . . . . 76
1. APPLICABILITY AND DEFINITIONS . . . . . . 262
4.2 Vector Subspaces . . . . . . . . . . . . . . 79
2. VERBATIM COPYING . . . . . . . . . . . . . 262
4.3 Linear Independence . . . . . . . . . . . . 81
3. COPYING IN QUANTITY . . . . . . . . . . . . 262
4.4 Spanning Sets . . . . . . . . . . . . . . . . 84
4. MODIFICATIONS . . . . . . . . . . . . . . . 262
4.5 Bases . . . . . . . . . . . . . . . . . . . . 87
5. COMBINING DOCUMENTS . . . . . . . . . . 263
4.6 Coordinates . . . . . . . . . . . . . . . . . 91
6. COLLECTIONS OF DOCUMENTS . . . . . . . 263
5 Linear Transformations 97 7. AGGREGATION WITH INDEPENDENT WORKS 263
5.1 Linear Transformations . . . . . . . . . . . 97 8. TRANSLATION . . . . . . . . . . . . . . . . 263
5.2 Kernel and Image of a Linear Transformation 99 9. TERMINATION . . . . . . . . . . . . . . . . 263
5.3 Matrix Representation . . . . . . . . . . . 104 10. FUTURE REVISIONS OF THIS LICENSE . . 263
, iii
Copyright c 2007 David Anthony SANTOS. Permission is granted to copy, distribute and/or
modify this document under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation; with no Invariant Sections, no
Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section
entitled “GNU Free Documentation License”.
, Preface
These notes started during the Spring of 2002, when John MAJEWICZ and I each taught a section of
Linear Algebra. I would like to thank him for numerous suggestions on the written notes.
The students of my class were: Craig BARIBAULT, Chun CAO, Jacky CHAN, Pho DO, Keith HAR-
MON, Nicholas SELVAGGI, Sanda SHWE, and Huong VU. I must also thank my former student William
CARROLL for some comments and for supplying the proofs of a few results.
John’s students were David HERNÁNDEZ, Adel JAILILI, Andrew KIM, Jong KIM, Abdelmounaim
LAAYOUNI, Aju MATHEW, Nikita MORIN, Thomas NEGRÓN, Latoya ROBINSON, and Saem SOEURN.
Linear Algebra is often a student’s first introduction to abstract mathematics. Linear Algebra is well
suited for this, as it has a number of beautiful but elementary and easy to prove theorems. My purpose
with these notes is to introduce students to the concept of proof in a gentle manner.
David A. Santos
iv
