We now come to Fundamentals Of Linear Algebra And Analytical
Geometry by Ya. S. Bugrov, S. M. Nikolsky.

The present book is the first part of our three-part textbook
Higher-Mathematics. We deal here with the fundamentals of the theory of deteminants, the elements of theory of matrices, the theory of systems of linear equations, and vector algebra. The book is also intended to introduce its readers to the basic aspects of linear algebra: linear operators, orthogonal transformations, self-adjoint operators, the quadratic forms and reducing it to the canonical form. Elements of analytical geometry (the straight line, the plane, the straight line in space, and second order curves and surfaces)are also included.

This book was translated from the Russian by Leonid Levant and was first published by Mir Publishers in 1982.

You can get the book here.

Edit: 26 June 2023 Link updated to IA

Table of Contents

Preface 6
1. Second Order Determinants 7
2. Determinants of third and n-th order 8
3. Matrices 19
4. Systems of Linear Equations. Kronecker-Capelli theory 21
5. Three dimensional Space. Vectors. Cartesian Systems of Coordinates 39
6. n-Dimensional Euclidean Space. Scalar Product of Two Vectors 48
7. Segment of a Line. Dividing a segment in a given ratio 53
8. The straight line 56
9. The Equation of a Plane 66
10. A Straight Line in Space 74
11. Orientation of Rectangular Coordinate Systems 78
12. The Vector Product of Two Vectors 81
13. Triple Scalar Product 87
14. Linearly Independent System of Vectors 89
15. Linear Operators 96
16. Bases in R_n 102
17. Orthogonal Bases in R_n 107
18. Invariant Properties of a Scalar and a Vector Product 113
19. Transformation of Rectangular Coordinates in a Plane 116
20. Linear Subspaces in R_n 119
21.  Fredholm-type Theorems 125
22. Self-adjoint Operator. Quadratic Form 132
23. Quadratic Form in Two-dimensional Space 142
24. Second-order Curves 146
25. Second-order Surfaces in Three-dimensional Space 162
26. The General Theory of a Second-order Surface in Three-dimensional
Space  180

Subject Index 187