A Concise Introduction to Linear Algebra

336 Pages·2012·2.026 MB·English

by  Géza Schay (auth.)

Géza Schay
A Concise Introduction
to Linear Algebra
Géza Schay
Department of Mathematics
University of Massachusetts
Boston, MA, USA
ISBN 978-0-8176-8324-5 978-0-8176-8325-2 (eBook)
DOI 10.1007/978-0-8176-8325-2
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2012934162

Mathematics Subject Classification (2010): 15Axx
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This book is based on the author’s Introduction to Linear Algebra, published by Jones & Bartlett in 1996.
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Contents
Preface ....................................................... vii
1. Analytic Geometry of Euclidean Spaces................... 1
1.1 Vectors................................................ 1
1.2 Length and Dot Product of Vectors in Rn ................. 15
1.3 Lines and Planes ....................................... 28
2. Systems of Linear Equations, Matrices ................... 41
2.1 Gaussian Elimination ................................... 41
2.2 The Theory of Gaussian Elimination...................... 53
2.3 Homogeneous and Inhomogeneous Systems, Gauss–Jordan
Elimination............................................ 57
2.4 The Algebra of Matrices................................. 67
2.5 The Inverse and the Transpose of a Matrix ................ 86
3. Vector Spaces and Subspaces ............................. 99
3.1 General Vector Spaces .................................. 99
3.2 Subspaces ............................................. 105
3.3 Span and Independence of Vectors........................ 109
3.4 Bases ................................................. 117
3.5 Dimension, Orthogonal Complements ..................... 131
3.6 Change of Basis ........................................ 148
4. Linear Transformations .................................. 163
4.1 Representation of Linear Transformations by Matrices ...... 163
4.2 Properties of Linear Transformations...................... 174
4.3 Applications of Linear Transformations in Computer
Graphics .............................................. 188
5. Orthogonal Projections and Bases ........................ 199
5.1 Orthogonal Projections and Least-Squares Approximations .. 199
5.2 Orthogonal Bases....................................... 210
v
vi Contents
6. Determinants............................................. 221
6.1 Determinants: Definition and Basic Properties ............. 221
6.2 Further Properties of Determinants ....................... 235
6.3 The Cross Product of Vectors in R3....................... 245
7. Eigenvalues and Eigenvectors............................. 253
7.1 Eigenvalues and Eigenvectors, Basic Properties............. 253
7.2 Diagonalization of Matrices .............................. 263
7.3 Principal Axes ......................................... 273
7.4 Complex Matrices ...................................... 279
8. Numerical Methods....................................... 291
8.1 LU Factorization ....................................... 291
8.2 Scaled Partial Pivoting.................................. 300
8.3 The Computation of Eigenvalues and Eigenvectors.......... 305
A. Appendices ............................................... 315
A.1 Implication and Equivalence ............................. 315
A.2 Complex Numbers...................................... 318
Further Reading.............................................. 325
Index......................................................... 327
Preface
Thisbookisdesignedforaone-semester,post-calculuslinearalgebracourse,
primarily intended for mathematics, physics, and computer science majors.
While basic calculus is a prerequisite for such a course, very little of it is
used in the book. Certainly, multivariable calculus is not required. Vectors
are treated fully in Chapter 1, but for classes familiar with them, this chap-
ter may be skipped or just reviewed briefly. Complex numbers, series, and
exponentials are presented briefly in an appendix, but they are needed only
in Section 7.4, which may not be covered in some courses.
Theselectionoftopicsconformstoalargeextenttotherecommendations
of the Linear Algebra Curriculum Study Group.1 The main differences are
that the book begins with a chapter on Euclidean vector geometry, mostly
in three dimensions; determinants are treated more fully and are placed just
before eigenvalues, which is where they are needed; the LU factorization is
relegated to Chapter 8 on numerical methods; and the facts about linear
transformations are collected in one chapter and are treated in more detail.
This book is considerably shorter than the 400 to 800 pages of most
introductory linear algebra books, which are more suitable for two- or three-
semester courses.
While many applications are presented, they are mostly taken from
physics, and several new ones have been added in the second edition. How-
ever, these examples give only a glimpse of how the subject is used in other
fields, and further details are left to texts in those fields. There is, though,
a section on computer graphics and a chapter on numerical methods. Also,
mostsectionscontainMATLAB(cid:2) exercises.Ontheotherhand,wehopethat
the student’s interest will be aroused not only by the possible applications,
but also by the geometrical background and the beautiful structure of linear
algebra. Nevertheless, for readers especially interested in applications, a list
of the ones discussed follows this preface.
Themoredifficultexercisesandtheoremsaremarkedbyanasterisk.Some
exercises are used to develop new topics, whose inclusion in the main text
would have disrupted the flow of ideas. The symbols (cid:2) and (cid:3) are used to
indicate the end of proofs and examples, respectively.
1 David Carlson, Charles R. Johnson, David C. Lay, A. Duane Porter. The Lin-
earAlgebraCurriculumStudyGroupRecommendationsfortheFirstCoursein
Linear Algebra. College Mathematics Journal, 24:1 (1993) 41–46.
vii
viii Preface
In this second edition, in response to the concerns of some users of the
firstedition,manyoftheearlierproofsandexplanationshavebeenexpanded
and a few new ones added. Also, exercises involving laborious computations
have been replaced by simpler ones, and some new ones have been added.
Foreword to Instructors
• The brevity mentioned above makes the book easier to use. Important
points are not drowned in a sea of detail, and instructors and students do
nothavetosearchforwhattokeepandwhattoomit.Inaminimalcourse,
however, the following sections may be omitted entirely: Section 4.3 on
computergraphics,Section5.1onorthogonalprojectionsandleastsquares,
Section 6.2 on cofactor expansions of determinants, Cramer’s rule, etc.,
Section6.3onthecrossproduct,Sections7.3and7.4onprincipalaxesand
complex matrices, and Chapter 8 on numerical methods. Theorem 3.4.8
(The Exchange Theorem) may also be omitted, since an alternative direct
proof of the dimension theorem is provided in the new edition.
• The geometric content is heavily emphasized. In fact, as mentioned above,
the book begins with a chapter on Euclidean vector geometry, mostly in
three dimensions. Most other similar textbooks start with the solution of
linear systems. We believe that this early introduction of the geometri-
cal background helps students to visualize the concepts of linear algebra
and provides easy concrete examples. Additionally, many students in this
course, e.g., computer science majors, are not required to take multivari-
able calculus, and do not see this important material anywhere else.
• In the first chapter, the equations of planes are given in both parametric
andnonparametricform,incontrasttomostcalculusbooks,whichpresent
only the nonparametric form. Many examples and exercises illustrate the
transition from one form to the other. However, we avoid using the cross
productatthisstage,becauseitisonlyavailableinR3.Weusethemethod
ofsolvingsimultaneousequationstoobtainanormalvectortoaplane,and
thistopicisrevisitedasanexampletoGaussianelimination.Ontheother
hand, Section 6.3 is devoted to the cross product as an illustration of the
use of determinants, and it is only at that point that it is used to obtain a
normal vector to a plane.
• The“backandforth”processbetweenparametricandnonparametricequa-
tions for lines and planes lays the groundwork for the same transition be-
tween describing a subspace of Rn as a set of linear combinations or as
the solution set of a homogeneous system of linear equations, that is, as
the column space of a matrix or the null space of another matrix. Another
generalization of thisissueisfinding orthogonal complementsof subspaces
of Rn given in either form.
• Many books use the notation (cid:2)p(cid:2) for the length of a vector p in Rn, but
we prefer |p|, because in R1 length is the absolute value, and there is no
Preface ix
reason to change notation for higher dimensions, just as there was none in
using + for addition of both scalars and of vectors. The notation (cid:2)p(cid:2) is
left for other norms.
• Important concepts are presented as definitions and theorems. Students
are advised to memorize them. It is not enough just to understand the
material; the main concepts must be remembered well to be able to build
on them.
• Except for the Spectral Theorem in the complex case and theorems from
other fields of mathematics, all theorems are proved. It is thus left to the
instructor to adjust the level of the course from the computational to the
fairly theoretical by omitting as many or as few proofs as desired.
• Great care has been taken to motivate every new concept, even those that
many books do not, such as dot product, matrix operations, linear inde-
pendence (not just in two or three dimensions), determinants, eigenvalues,
and eigenvectors.
• The letter symbols are selected to reflect the connections between related
quantities, a principle often ignored in other linear algebra books. Vectors
and their components, matrices and their column and row vectors and
entries are denoted by the same letters with different fonts, like v, vi and
A, ai, aj, aij. The main exception is the unit matrix, which is, bowing to
tradition, denoted by I, its columns by ei, and its entries by δij.
• Only standard notation is used, so that students who study further, will
have no difficulty in reading applied or more advanced texts. Nonstandard
notation, such as the use of a list in parentheses for column vectors and in
−→
bracketsforrowvectors,or ai orAi forarowvectorofamatrix,foundin
someotherintroductorylinearalgebrabooks,isavoided.Weuseai forthe
column vectors of a matrix A and ai for its row vectors. This is standard
notation in more advanced books. (See, e.g., Introduction to Linear and
Nonlinear Programming by David G. Luenberger, Addison-Wesley, 1973.)
T
WealsousexA =(xA1,xA2,...,xAn) forthecoordinatevectorofavector
x relative to an ordered basis or basis matrix A. (Compare this, e.g., with
T
the notation [x]B =(c1,c2,...cn) of Linear Algebra and Its Applications
by David Lay, Addison-Wesley, 1993, where the brackets on the left are
superfluous, the coordinates of x are denoted by the unrelated letter c,
and the basis B is not indicated on the right, not to mention that we need
anorderedbasisorbasismatrixhere.)Ournotationmakesthenotoriously
messy topic of change of basis much simpler.
• Similarity of matrices is introduced in the context of changing bases.
• Most introductory linear algebra books introduce determinants by unmo-
tivated formulas. This book introduces them by three simple properties,
expanding on the approach in Strang.2
2 GilbertStrang,LinearAlgebraanditsApplications,3rded.HarcourtBrace,San
Diego, 1988.