Sharipov R. - Linear Algebra and Multidimensional Geometry.pdf

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Shr2b.dvi
RUSSIAN FEDERAL COMMITTEE
FOR HIGHER EDUCATION
BASHKIR STATE UNIVERSITY
SHARIPOV R. A.
COURSE OF LINEAR ALGEBRA
AND MULTIDIMENSIONAL GEOMETRY
The Textbook
Ufa 1996
 
2
MSC 97U20
PACS 01.30.Pp
UDC 512.64
Sharipov R. A. Course of Linear Algebra and Multidimensional Geom-
etry: the textbook / Publ. of Bashkir State University | Ufa, 1996. | pp. 143.
ISBN 5-7477-0099-5.
This book is written as a textbook for the course of multidimensional geometry
and linear algebra. At Mathematical Department of Bashkir State University this
course is taught to the rst year students in the Spring semester. It is a part of
the basic mathematical education. Therefore, this course is taught at Physical and
Mathematical Departments in all Universities of Russia.
In preparing Russian edition of this book I used the computer typesetting on
the base of theA M S-T E X package and I used the Cyrillic fonts of Lh-family
distributed by the CyrTUG association of Cyrillic T E X users. English edition of
this book is also typeset by means of theA M S-T E X package.
Referees: Computational Mathematics and Cybernetics group of Ufa
State University for Aircraft and Technology (UGATU);
Prof. S. I. Pinchuk, Chelyabinsk State University for Technol-
ogy (QGTU) and Indiana University.
Contacts to author.
Oce: Mathematics Department, Bashkir State University,
32 Frunze street, 450074 Ufa, Russia
Phone: 7-(3472)-23-67-18
Fax:
7-(3472)-23-67-74
Home: 5 Rabochaya street, 450003 Ufa, Russia
Phone: 7-(917)-75-55-786
ISBN 5-7477-0099-5
c Sharipov R.A., 1996
c Bashkir State University, 1996
English translation
c Sharipov R.A., 2004
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CONTENTS.
CONTENTS. ............................................................................................... 3.
PREFACE. .................................................................................................. 5.
CHAPTER I. LINEAR VECTOR SPACES AND LINEAR MAPPINGS. ........ 6.
x1. The sets and mappings. ......................................................................... 6.
x2. Linear vector spaces. ........................................................................... 10.
x3. Linear dependence and linear independence. ......................................... 14.
x4. Spanning systems and bases. ................................................................ 18.
x5. Coordinates. Transformation of the coordinates of a vector
under a change of basis. ....................................................................... 22.
x6. Intersections and sums of subspaces. ..................................................... 27.
x7. Cosets of a subspace. The concept of factorspace. ................................. 31.
x8. Linear mappings. ................................................................................ 36.
x9. The matrix of a linear mapping. ........................................................... 39.
x10. Algebraic operations with mappings.
The space of homomorphisms Hom(V;W). ........................................... 45.
CHAPTER II. LINEAR OPERATORS. ...................................................... 50.
x1. Linear operators. The algebra of endomorphisms End(V )
and the group of automorphisms Aut(V ). ............................................. 50.
x2. Projection operators. ........................................................................... 56.
x3. Invariant subspaces. Restriction and factorization of operators. .............. 61.
x4. Eigenvalues and eigenvectors. ............................................................... 66.
x5. Nilpotent operators. ............................................................................ 72.
x6. Root subspaces. Two theorems on the sum of root subspaces. ................ 79.
x7. Jordan basis of a linear operator. Hamilton-Cayley theorem. ................. 83.
CHAPTER III. DUAL SPACE. .................................................................. 87.
x1. Linear functionals. Vectors and covectors. Dual space. .......................... 87.
x2. Transformation of the coordinates of a covector
under a change of basis. ....................................................................... 92.
x3. Orthogonal complements in a dual spaces. ............................................ 94.
x4. Conjugate mapping. ............................................................................ 97.
CHAPTER IV. BILINEAR AND QUADRATIC FORMS. ......................... 100.
x1. Symmetric bilinear forms and quadratic forms. Recovery formula. ....... 100.
x2. Orthogonal complements with respect to a quadratic form. .................. 103.
4
CONTENTS.
x3. Transformation of a quadratic form to its canonic form.
Inertia indices and signature. ............................................................. 108.
x4. Positive quadratic forms. Silvester's criterion. ..................................... 114.
CHAPTER V. EUCLIDEAN SPACES. ..................................................... 119.
x1. The norm and the scalar product. The angle between vectors.
Orthonormal bases. ........................................................................... 119.
x2. Quadratic forms in a Euclidean space. Diagonalization of a pair
of quadratic forms. ............................................................................ 123.
x3. Selfadjoint operators. Theorem on the spectrum and the basis
of eigenvectors for a selfadjoint operator. ............................................ 127.
x4. Isometries and orthogonal operators. .................................................. 132.
CHAPTER VI. AFFINE SPACES. ........................................................... 136.
x1. Points and parallel translations. Ane spaces. .................................... 136.
x2. Euclidean point spaces. Quadrics in a Euclidean space. ...................... 139.
REFERENCES. ....................................................................................... 143.
PREFACE.
There are two approaches to stating the linear algebra and the multidimensional
geometry. The rst approach can be characterized as the <coordinates and
matrices approach>. The second one is the <invariant geometric approach>.
In most of textbooks the coordinates and matrices approach is used. It starts
with considering the systems of linear algebraic equations. Then the theory of
determinants is developed, the matrix algebra and the geometry of the space R n
are considered. This approach is convenient for initial introduction to the subject
since it is based on very simple concepts: the numbers, the sets of numbers, the
numeric matrices, linear functions, and linear equations. The proofs within this
approach are conceptually simple and mostly are based on calculations. However,
in further statement of the subject the coordinates and matrices approach is not so
advantageous. Computational proofs become huge, while the intension to consider
only numeric objects prevents us from introducing and using new concepts.
The invariant geometric approach, which is used in this book, starts with the
denition of abstract linear vector space. Thereby the coordinate representation
of vectors is not of crucial importance; the set-theoretic methods commonly used
in modern algebra become more important. Linear vector space is the very object
to which these methods apply in a most simple and eective way: proofs of many
facts can be shortened and made more elegant.
The invariant geometric approach lets the reader to get prepared to the study
of more advanced branches of mathematics such as dierential geometry, commu-
tative algebra, algebraic geometry, and algebraic topology. I prefer a self-sucient
way of explanation. The reader is assumed to have only minimal preliminary
knowledge in matrix algebra and in theory of determinants. This material is
usually given in courses of general algebra and analytic geometry.
Under the term <numeric eld> in this book we assume one of the following
three elds: the eld of rational numbers Q, the eld of real numbers R, or the
eld of complex numbers C. Therefore the reader should not know the general
theory of numeric elds.
I am grateful to E. B. Rudenko for reading and correcting the manuscript of
Russian edition of this book.
May, 1996;
May, 2004.
R. A. Sharipov.

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