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Linear Algebra and Its Applications
Fourth Edition
Gilbert Strang
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Contents
Preface iv
1 Matrices and Gaussian Elimination 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Geometry of Linear Equations . . . . . . . . . . . . . . . . . . . . 4
1.3 An Example of Gaussian Elimination . . . . . . . . . . . . . . . . . . 13
1.4 Matrix Notation and Matrix Multiplication . . . . . . . . . . . . . . . . 21
1.5 Triangular Factors and Row Exchanges . . . . . . . . . . . . . . . . . 36
1.6 Inverses and Transposes . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.7 Special Matrices and Applications . . . . . . . . . . . . . . . . . . . . 66
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2 Vector Spaces 77
2.1 Vector Spaces and Subspaces . . . . . . . . . . . . . . . . . . . . . . . 77
2.2 Solving Ax = 0 and Ax = b ........................ 86
2.3 Linear Independence, Basis, and Dimension . . . . . . . . . . . . . . . 103
2.4 The Four Fundamental Subspaces . . . . . . . . . . . . . . . . . . . . 115
2.5 Graphs and Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
2.6 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
3 Orthogonality 159
3.1 Orthogonal Vectors and Subspaces . . . . . . . . . . . . . . . . . . . . 159
3.2 Cosines and Projections onto Lines . . . . . . . . . . . . . . . . . . . . 171
3.3 Projections and Least Squares . . . . . . . . . . . . . . . . . . . . . . 180
3.4 Orthogonal Bases and Gram-Schmidt . . . . . . . . . . . . . . . . . . 195
3.5 The Fast Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . 211
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
i
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ii
CONTENTS
4 Determinants 225
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.2 Properties of the Determinant . . . . . . . . . . . . . . . . . . . . . . . 227
4.3 Formulas for the Determinant . . . . . . . . . . . . . . . . . . . . . . . 236
4.4 Applications of Determinants . . . . . . . . . . . . . . . . . . . . . . . 247
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
5 Eigenvalues and Eigenvectors 260
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
5.2 Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 273
5.3 Difference Equations and Powers A
k
................... 283
5.4 Differential Equations and e
At
...................... 296
5.5 Complex Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
5.6 Similarity Transformations . . . . . . . . . . . . . . . . . . . . . . . . 325
Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
6 Positive Definite Matrices 345
6.1 Minima, Maxima, and Saddle Points . . . . . . . . . . . . . . . . . . . 345
6.2 Tests for Positive Definiteness . . . . . . . . . . . . . . . . . . . . . . 352
6.3 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . 367
6.4 Minimum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 376
6.5 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . 384
7 Computations with Matrices 390
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390
7.2 Matrix Norm and Condition Number . . . . . . . . . . . . . . . . . . . 391
7.3 Computation of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 399
7.4 Iterative Methods for Ax = b ....................... 407
8 Linear Programming and Game Theory 417
8.1 Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
8.2 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
8.3 The Dual Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
8.4 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
8.5 Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
A Intersection, Sum, and Product of Spaces 459
A.1 The Intersection of Two Vector Spaces . . . . . . . . . . . . . . . . . . 459
A.2 The Sum of Two Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 460
A.3 The Cartesian Product of Two Vector Spaces . . . . . . . . . . . . . . . 461
A.4 The Tensor Product of Two Vector Spaces . . . . . . . . . . . . . . . . 461
A.5 The Kronecker Product A ≠B of Two Matrices . . . . . . . . . . . . . 462
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CONTENTS
iii
B The Jordan Form 466
C Matrix Factorizations 473
D Glossary: A Dictionary for Linear Algebra 475
E MATLAB Teaching Codes 484
F Linear Algebra in a Nutshell 486
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Preface
Revising this textbook has been a special challenge, for a very nice reason. So many
people have read this book, and taught from it, and even loved it. The spirit of the book
could never change. This text was written to help our teaching of linear algebra keep up
with the enormous importance of this subject—which just continues to grow.
One step was certainly possible and desirable—to add new problems. Teaching for all
these years required hundreds of new exam questions (especially with quizzes going onto
the web). I think you will approve of the extended choice of problems. The questions are
still a mixture of explain and compute—the two complementary approaches to learning
this beautiful subject.
I personally believe that many more people need linear algebra than calculus. Isaac
Newton might not agree! But he isn’t teaching mathematics in the 21st century (and
maybe he wasn’t a great teacher, but we will give him the benefit of the doubt). Cer-
tainly the laws of physics are well expressed by differential equations. Newton needed
calculus—quite right. But the scope of science and engineering and management (and
life) is now so much wider, and linear algebra has moved into a central place.
May I say a little more, because many universities have not yet adjusted the balance
toward linear algebra. Working with curved lines and curved surfaces, the first step is
always to linearize. Replace the curve by its tangent line, fit the surface by a plane,
and the problem becomes linear. The power of this subject comes when you have ten
variables, or 1000 variables, instead of two.
You might think I am exaggerating to use the word “beautiful” for a basic course
in mathematics. Not at all. This subject begins with two vectors v and w, pointing in
different directions. The key step is to take their linear combinations. We multiply to
get 3v and 4w, and we add to get the particular combination 3v + 4w. That new vector
is in the same plane as v and w. When we take all combinations, we are filling in the
whole plane. If I draw v and w on this page, their combinations cv + dw fill the page
(and beyond), but they don’t go up from the page.
In the language of linear equations, I can solve cv + dw = b exactly when the vector
b lies in the same plane as v and w.
iv
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