Lecture notes
Linear Algebra Notes
- Module
- Institution
- Book
Lecture notes study book Lecture Notes for Linear Algebra of Gilbert Strang - ISBN: 9781733146647 (ITS LECTURE NOTES)
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
Seller
Follow
joyalsebastian
Content preview
Linear Algebra NotesDavid A. SANTOS
dsantos@ccp.edu
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 dsantos@ccp.edu
iv
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller joyalsebastian. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £5.79. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
72841 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now