Linearity and the mathematics of several variables. With Michael N. Sinyakov and Sergei V. Tischchenko (Q2734838)

This is a textbook which is intended for a graduate course in linear algebra, presenting a different point of view in treating modern concepts of linear algebra, showing their usefulness particularly in multivariable calculus. The general concepts of vector space and linear operator are developed emphasizing especially the interpretation of abstract algebraic notions in terms of solvability of homogeneous and nonhomogeneous linear equations, algebraic or differential. Reading this book the student succeeds in familiarizing with the main concepts of linear algebra such as linearity, span, linear independence, subspace, kernel and range, eigenvectors, inner product and applying them in concrete cases, in mastering the calculational techniques of linear algebraic equations, matrix inversion, determinants, Gram-Schmidt orthogonalization, diagonalizing matrices, in studying thorougly multivariable calculus and differential equations chapters by applying the concepts of linear algebras, as in differentials of nonlinear functions, tangent lines and planes, volumes, line and surface integrals, the chain rule, implicit and inverse functions, local calculations in curvilinear coordinates, solution of systems of ordinary differential equations with constant coefficients, the proper treatment of homogeneous and nonhomogeneous differential equations and boundary conditions by linear superposition. The authors' approach enables finally students to appreciate linear algebra as an important and indispensable tool in modern applied analysis. The reading of this book assumes only a little elementary information concerning linear algebraic equations, partial derivatives, geometrical and physical vectors and elementary ordinary differential equations.NEWLINENEWLINE Contents: I) Vectors: 1) Vectors that you know; 2) Lines and Planes; 3) Points: A Deeper Look; 4) Curves and Tangent Vectors. II) Matrices: 1) Linear Systems and Matrices; 2) Matrix Algebra; 3) Inverses; 4) Functions and Gradient Vectors; 5) Elementary determinants. III) Vector Spaces and Linear Functions: 1) The Definition of a Vector Space; 2) Linear Functions; 3) Nonlinear Functions; 4) Differentials; 5) The Chain Rule. IV) Bases: 1) The Basis Concept: Independence and Span; 2) Local Bases Associated with a Coordinate System; 3) Dimension; 4) Coordinates with respect to a Basis; 5) Changes of Basis. V) Subspaces and Linear Equations: 1) Subspaces; 2) Subspaces Associated with a Linear Function: Kernel and Range; 3) Linear Equations: The Superposition Principles; 4) Rank; 5) Implicit and Inverse Function. VI) Inner Products and Differential Vector Calculus: 1) Inner Products, Norms and Metrics; 2) Orthogonality; 3) The Geometry of Curves; 4) Nabla: The Vector Differential Operations. VII) Determinants and Integral Vector Calculus: 1) Properties of Determinants; 2) Volume, Rotations and all that: The Geometrical Significance of Determinants and Antisymmetry; 3) Jacobi's Theorem: Changing Variables in Multiple Integrals; 4) Surface Integrals - Definition; 5) The Integral Theorems of Vector Analysis; 6) Direct Methods of Evaluating Line and Surface Integrals. VIII) Eigenvectors and Diagonalization: 1) Eigenvalues and Eigenvectors; 2) Diagonalization of Real, Symmetric Matrices; 3) A Note on History Index.