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The present book is the first part of our three-part textbook “Higher Mathematics”. We deal here with the fundamentals of the theory of determinants, the elements of the 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 form 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.
As a rule, our arguments are accompanied by exhaustive proofs. The material is introduced so that the proofs for the general n-dimensional case may be omitted without the loss of either the formulation of the statement or a detailed explanation of how the subject matter stands in the two- and three-dimensional cases.
The canonical forms of second-order curves and surfaces are treated briefly in this book since it is assumed that they will be studied additionally by solving particular problems using the methods of mathematical analysis. The quadratic form is dealt with by the methods of mathematical, or, if more convenient, by functional analysis.
Although we have mentioned that this is the first book in our series, its material is closely interwoven with that of the second book dedicated to differential and integral calculus. The third book will consider differential equations, multiple integrals, series, and the theory of functions of a complex variable. Being complementary books of a single series entitled “Higher Mathematics”, at the same time they may be regarded as independent educational aids for students of higher engineering institutions.
Determinants of the Third and nth Order.
Matrices.
Systems of Linear Equations. The Kronecker-Capelli Theory.
Three-dimensional Space. Vectors. Cartesian System of Coordinates.
n-Dimensional Euclidean Space. Scalar Product of Two Vectors.
Segment of a Line. Dividing a Segment in a Given Ratio.
The Straight Line.
The Equation of a Plane.
A Straight Line in Space.
Orientation of Rectangular Coordinate Systems.
The Vector Product of Two Vectors.
Triple Scalar Product.
Linearly Independent System of Vectors.
Linear Operators.
Bases in Rn.
Orthogonal Bases in Rn.
Invariant Properties of a Scalar and a Vector Product.
Transformation of Rectangular Coordinates in a Plane.
Linear Subspaces in Rn.
Fredholm-type Theorems.
Self-adjoint Operator. Quadratic Form.
Quadratic Form in Two-dimensional Space.
Second-order Curves.
Second-order Surfaces in Three-dimensional Space.
The General Theory of a Second-order Surface in Three-dimensional Space.
Subject Index