## Linear Algebra And Multi Dimensional Geometry – Efimov, Rozendorn

In this post, we will see the book Linear Algebra And Multi Dimensional Geometry by
N. V. Efimov, E. R. Rozendorn

This book was conceived as a text combining the course of linear algebra and analytic geometry. It originated as a course of lectures delivered by N. V. Efimov at Moscow State University (mechanics and mathematics department) in 1964-1966. However, the material of these lectures has been completely reworked and substantially expanded. We have tried to bear in mind the requi­rements of other mathematical disciplines and also of mechanics and physics. We hope that all parts of the text will be useful. The only preparation required for this text can be given an a first- semester course of analytic geometry and algebra at the most ele­mentary level. All that is needed is a firm grasp of the elements of these subjects. For Chapter XII the student should be ac­quainted with projective transformations and the projective pro­perties of figures in the plane. Also, in Chapter X the reader may simplify his task by skipping Subsections 13 to 23 (Section 3) and Subsection 10 of Section 7. What is left of Chapter X can serve as a minimal algebraic basis for the theory of multidimensional in­tegration.
It may be noted in conclusion that the first five chapters already contain material with broad applications in mathematics, mecha­nics, and physics. These chapters, supplemented with some of the material of subsequent chapters, can be utilized in higher tech­nical schools with a more advanced mathematics curriculum.

The book was translated from the Russian by George Yankovsky and published by Mir in 1975.

Many thanks to shankar.leo for providing the scans and the pdf.

Note: I tried to optimise the file for size, but somehow the archive kept on rejecting for some error in the pdf. I could have tried a few more things, but didn’t get time for that, hence the delay in post. For now, hence this large file (~120M) was uploaded. This file is OCRed but without bookmarks and pagination.

I will try to update a smaller file in the future, or if someone can add a link to a smaller file,  it would be great.

Preface 9

Introduction 11

Chapter I. Linear Spaces

1. Axioms in linear space 15
2. Examples Of linear spaces 17
3. Elementary corrolaries to the axioms of a linear space 23
4. Linear combinations. Linear dependence.25
5. Lemma on the basis minor 27
6. Basic lemma on two systems of vectors 30
7. The rank of a matrix 32
8. Finite-dimensional and infinite-dimensional spaces. Bases 34
9. Linear operations in components 36
10. Isomorphism between linear spaces 38
11. Correspondence between complex and real spaces 40
12. Linear subspace 42
13. Linear hull 44
14. Sum of subspaces. Direct sum 47

Chapter II. Linear Transformations of Variables. Transformations of Coordinates

1. Abbreviated notation for summation 53
2. Linear transformation of variables. The product of linear
transformations of variables and matrix products 56
3. Square matrices and nonsingular transformations 60
4. The rank of a product of matrices 64
5. Transformation of coordinates in a change of basis 66

Chapter III. Systems of Linear Equations. Planes In Affine Space

1. Affine space 70
2. Affine coordinates 71
3. Planes 73
4. Systems of first-degree equations 77
5. Homogeneous systems 81
6. Nonhomogeneous systems 88
7. Mutual positions of planes 91
8. Systems of linear inequalities and convex polyhedrons 98

Chapter IV. Linear, Bilinear and Quadratic Forms

1. Linear forms 108
2. Bilinear forms 112
3. The matrix of a bilinear form 116
5. Reducing a quadratic form to canonical form by Lagrange’s method 121
6. The normal form of a quadratic form 124
7. The law of inertia of quadratic forms 125
8. Reducing a quadratic form to canonical form by Jacobi’s method 127
9. Positive definite and negative definite quadratic forms 129
10. Gram’s determinant. The Cauchy-Bunyakovsky inequality 132
11. Zero subspaces of a bilinear and a quadratic form 134
12. The zero cone of a quadratic form.137
13. Elementary examples of zero cones of quadratic forms 139

Chapter V. Tensor Algebra

1. Reciprocal bases. Contravariant and covariant vectors 142
2. Tensor product of linear spaces 149
3. Basis in a tensor product. Components of a tensor 153
4. Tensors of bilinear forms 159
5. Multiple-order tensors. Tensor product 162
6. Components of multiple-order tensors 166
7. Multilinear forms and their tensors 168
8. Symmetrization and antisymmetrization (alternation). Skewsymmetric forms 170
9. An alternative description of the tensor product of two linear spaces 174

Chapter VI. Groups and Some Applications

1. Groups and subgroups. Distribution of bases into classes
with respect to a given subgroup of matrices. Orientation 180
2. Transformation groups. Isomorphism and homomorphism of groups 186
3. Invariants. Axial invariants. Pseudoinvariants 191
4. Tensor quantities 197
5. The oriented volume of a parallelepiped. The discriminant tensor 201

Chapter VII. Linear Transformations of Linear Spaces

1. Generalities 207
2. A linear transformation as a tensor 210
3. The geometrical meaning of the rank and determinant of a linear transformation. The group of nonsingular linear transformations.213
4. Invariant subspaces 216

5. Examples of linear transformations 218
6. Eigenvectors and the characteristic polynomial of a transformation 224
7. Basic theorems on the characteristic polynomial and eigenvectors 227
8 Nilpolent transformations. The general structure of singular
transformations 229
9. The canonical basis of a nilpotent transformation 233
10. Reducing a transformation matrix to the Jordan normal form 242
11. Transformations of a simple structure 248
12. Equivalence of matrices 250
13. The Hamilton-Cayley formula 252

Chapter VIII. Spaces with Quadratic Metric

1. Scalar products 254
2. The norm of a vector 256
3. Orthonormal bases 258
4. Orthogonal projection. Orthogonalization 259
5. Metric isomorphism 265
6. ^-orthogonal matrices and ^ orthogonal groups 266
7. The group of Euclidean rotations 270
8. The group of hyperbolic rotations 278
9. Tensor algebra in quadratic-metric spaces 287
10. The equation of a hyperplane in quadratic-metric space 295
11. Euclidean space. Orthogonal matrices. Orthogonal group 297
12. The normal equation of a hyperplane in Euclidean space 302
13. The volume of a parallelepiped in Euclidean space. The discriminant tensor. Vector product 304

Chapter IX. Linear Transformations of Euclidean Space

1. Adjoint of a transformation 308
2. Lemma on the characteristic roots of a symmetric matrix 310
4. Reducing a quadratic form to canonical form in an orthonormal basis 317
5. The joint reduction to canonical form of two quadratic forms 319
7. Isometric transformations 325
8. The canonical form of an isometric transformation 330
9. The motion of a rigid body with one fixed point 335
10. The curvature and torsion of a space curve 338
11. The decomposition of an arbitrary linear transformation into the product of a self-adjoint and an isometric transformation 340
12. Applications to the theory of elasticity. The strain tensor and the stress tensor 343

Chapter X. Multivectors and Outer Forms

1. Alternation 346
2. Multivectors. Outer product 351
3. Bivectors 357
4. Simple multivectors 366
5. Vector product 370
6. Outer forms and operations on them 376
7. Outer forms and covariant multivectors 379
8. Outer forms in three-dimensional Euclidean space 386

1. The general equation of a quadric hypersurface 391
2. Changes in the left member of the equation under translation of the origin 392
3. Changes in the left member of the equation for a change in the orthonormal basis 395
4. The centre of a quadric hypersurface 397
5. Reducing to canonical form the general equation of a quadric hypersurface in Euclidean space 399
6. Classification of quadric hypersurfaces in Euclidean space 402
7. Affine transformations. 410
8. Affine classification of quadric hypersurfaces 414
9. The intersection of a straight line with a quadric hypersurface. Asymptotic directions 415
10. Conjugate directions.418

Chapter XII. Projective Space

1. Homogeneous coordinates in affine space. Points at infinity 422
2. The concept of a projective space 425
3. A bundle of planes in affine space 435
4. Central projection 443
5. Projective equivalence of figures 446
6. Projective classification of quadric hypersurfaces 453
7. The intersection of a quadric hypersurface and a straight line. Polars 459

Appendix 1. Proof of the theorem on the classification of linear quantities 467

Appendix 2. Hermitian forms. Unitary space.471

Bibliography. 484

Index. 486

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### 4 Responses to Linear Algebra And Multi Dimensional Geometry – Efimov, Rozendorn

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