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Introduction to
Linear Algebra
Fourth Edition
Gilbert Strang
INTRODUCTION
TO
LINEAR ALGEBRA
Fourth Edition
GILBERT STRANG
Massachusetts Institute
of
Technology
WELLESLEY - CAMBRIDGE PRESS
Box 812060 Wellesley MA 02482
Introduction
to
Linear
Algebra,
4th
Edition
Copyright ©2009 by Gilbert Strang
ISBN
978-0-9802327-1-4
Fourth
International
Edition
Copyright ©2009 by Gilbert Strang
ISBN
978-0-9802327-2-1
All rights reserved. No part
of
this work may be reproduced or stored or transmitted
by any means, including photocopying, without written permission from
Wellesley - Cambridge Press. Translation
in
any language
is
strictly prohibited -
authorized translations are arranged by the publisher.
Typeset by www.valutone.co.in Printed
in
the United States
of
America
987654
QAI84.S78 2009 512'.5 93-14092
Other
texts from Wellesley -
Cambridge
Press
Computational
Science
and
Engineering,
Gilbert Strang
ISBN 978-0-9614088-1-7 ISBN 0-9614088-1-2
Wavelets
and
Filter
Banks,
Gilbert Strang and Truong Nguyen
ISBN 978-0-9614088-7-9 ISBN 0-9614088-7-1
Introduction
to Applied
Mathematics,
Gilbert Strang
ISBN 978-0-9614088-0-0 ISBN 0-9614088-0-4
An Analysis
of
the
Finite
Element
Method,
2008 edition, Gilbert Strang and George Fix
ISBN 978-0-9802327-0-7 ISBN 0-9802327-0-8
Calculus
Second
edition
(2010), Gilbert Strang
ISBN 978-0-9802327-4-5 ISBN 0-9802327-4-0
Wellesley -
Cambridge
Press
Box 812060
Wellesley
MA
02482 USA
www.wellesleycambridge.com
gs@math.mit.edu
math.mit.ed
u;''"
.....
gs
phone (781) 431-8488
fax (617) 253-4358
The website for this book
is
math.mit.edu/linearalgebra.
A Solutions Manual
is
available to instructors by email from the publisher.
Course material including syllabus and Teaching Codes and exams and also
videotaped lectures are available on the teaching website: web.mit.edu/lS.06
Linear Algebra is included in MIT's OpenCourseWare site oew.mit.edu.
This provides video lectures
of
the full linear algebra course 18.06.
MATLAB®
is
a registered trademark
of
The Math Works, Inc.
The
front
cover captures a central idea
of
linear algebra.
Ax
= b is solvable when b is in the (orange) column space
of
A.
One particular solution y is in the (red) row space:
Ay
= b.
Add any vector z from the (green) nullspace
of
A:
Az
=
O.
The complete solution is x = y + z. Then
Ax
=
Ay
+
Az
= b.
The cover design was the inspiration
of
a creative collaboration:
Lois Sellers (birchdesignassociates.com) and Gail Corbett.
Table
of
Contents
1
Introduction to Vectors
1
1.1
Vectors and Linear Combinations.
2
1.2
Lengths and Dot Products.
11
1.3
Matrices
.........
22
2
Solving Linear Equations
31
2.1
Vectors and Linear Equations .
31
2.2
The Idea
of
Elimination . . .
45
2.3
Elimination Using Matrices .
57
2.4
Rules for Matrix Operations
68
2.5
Inverse
Matrices.
. . . . . .
82
2.6
Elimination = Factorization: A = L U
96
2.7
Transposes and Permutations .
.....
108
3 Vector Spaces and Subspaces
121
3.1
Spaces
of
Vectors . . . . . . .
....
121
3.2
The Nullspace
of
A: Solving
Ax
= 0
133
3.3
The Rank and the Row Reduced Form
145
3.4
The Complete Solution to
Ax
= b .
156
\
3.5
Independence, Basis and Dimension
169
3.6
Dimensions
of
the Four Subspaces
185
4 Orthogonality
196
4.1
Orthogonality
of
the Four Subspaces .
196
4.2
Projections . . . . . . . . . . . . . .
207
4.3
Least Squares Approximations . . . .
219
4.4
Orthogonal Bases and Gram-Schmidt
231
5 Determinants
245
5.1
The Properties
of
Determinants .
245
5.2
Permutations and Cofactors . . .
256
5.3
Cramer's Rule, Inverses, and Volumes
270
iii
iv
6 Eigenvalues and Eigenvectors
6.1
Introduction
to
Eigenvalues . . . . . .
6.2 Diagonalizing a Matrix . . . . . . . .
6.3 Applications
to
Differential Equations
6.4 Symmetric Matrices. . . .
6.5 Positive Definite Matrices. . . . . . .
6.6 Similar Matrices
...........
.
6.7 Singular Value Decomposition (SVD)
7 Linear Transformations
7.1
The Idea of a Linear Transformation
7.2 The Matrix
of
a Linear Transformation .
7.3 Diagonalization and the Pseudoinverse .
8 Applications
8.1
Matrices
in
Engineering
...........
.
8.2 Graphs and Networks . . . . . . . . . . . . .
8.3
Markov Matrices, Population, and Economics
8.4 Linear Programming
............
.
8.5
Fourier Series: Linear Algebra for Functions.
8.6 Linear Algebra for Statistics and Probability .
8.7 Computer Graphics
.............
.
9 Numerical Linear Algebra
9.1
Gaussian Elimination in Practice
9.2 Norms and Condition Numbers.
9.3
Iterative Methods and Preconditioners
10 Complex Vectors and Matrices
10.1
Complex Numbers
.....
.
10.2 Hermitian and Unitary Matrices
10.3
The Fast Fourier Transform .
Solutions to Selected Exercises
Conceptual Questions for Review
Glossary: A Dictionary for Linear Algebra
Matrix Factorizations
Teaching Codes
Index
Linear Algebra
in
a Nutshell
Table
of
Contents
284
284
299
313
331
343
356
364
376
376
385
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441
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460
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