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Linear Algebra

Georgi E. Shilov

chapter 1
		DETERMINANTS
		1.1.	Number Fields
		1.2.	Problems of the Theory of Systems of Linear Equations
		1.3.	Determinants of Order n
		1.4.	Properties of Determinants
		1.5.	Cofactors and Minors
		1.6.	Practical Evaluation of Determinants
		1.7.	Cramer's Rule
		1.8.	Minors of Arbitrary Order. Laplace's Theorem
		1.9.	Linear Dependence between Columns
			Problems
	chapter 2
		LINEAR SPACES
		2.1.	Definitions
		2.2.	Linear Dependence
		2.3.	"Bases, Components, Dimension"
		2.4.	Subspaces
		2.5.	Linear Manifolds
		2.6.	Hyperplanes
		2.7.	Morphisms of Linear Spaces
			Problems
	chapter 3
		SYSTEMS OF LINEAR EQUATIONS
		3.1.	More on the Rank of a Matrix
		3.2.	Nontrivial Compatibility of a Homogeneous Linear System
		3.3.	The Compatability Condition for a General Linear System
		3.4.	The General Solution of a Linear System
		3.5.	Geometric Properties of the Solution Space
		3.6.	Methods for Calculating the Rank of a Matrix
			Problems
	chapter 4
		LINEAR FUNCTIONS OF A VECTOR ARGUMENT
		4.1.	Linear Forms
		4.2.	Linear Operators
		4.3.	Sums and Products of Linear Operators
		4.4.	Corresponding Operations on Matrices
		4.5.	Further Properties of Matrix Multiplication
		4.6.	The Range and Null Space of a Linear Operator
		4.7.	Linear Operators Mapping a Space Kn into Itself
		4.8.	Invariant Subspaces
		4.9.	Eigenvectors and Eigenvalues
			Problems
	chapter 5
		COORDINATE TRANSFORMATIONS
		5.1.	Transformation to a New Basis
		5.2.	Consecutive Transformations
		5.3.	Transformation of the Components of a Vector
		5.4.	Transformation of the Coefficients of a Linear Form
		5.5.	Transformation of the Matrix of a Linear Operator
		*5.6.	Tensors
			Problems
	chapter 6
		THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR
		6.1.	Canonical Form of the Matrix of a Nilpotent Operator
		6.2.	Algebras. The Algebra of Polynomials
		6.3.	Canonical Form of the Matrix of an Arbitrary Operator
		6.4.	Elementary Divisors
		6.5.	Further Implications
		6.6.	The Real Jordan Canonical Form
		*6.7.	"Spectra, Jets and Polynomials"
		*6.8.	Operator Functions and Their Matrices
			Problems
	chapter 7
		BILINEAR AND QUADRATIC FORMS
		7.1.	Bilinear Forms
		7.2.	Quadratic Forms
		7.3.	Reduction of a Quadratic Form to Canonical Form
		7.4.	The Canonical Basis of a Bilinear Form
		7.5.	Construction of a Canonical Basis by Jacobi's Method
		7.6.	Adjoint Linear Operators
		7.7.	Isomorphism of Spaces Equipped with a Bilinear Form
		*7.8.	Multilinear Forms
		7.9.	Bilinear and Quadratic Forms in a Real Space
			Problems
	chapter 8
		EUCLIDEAN SPACES
		8.1.	Introduction
		8.2.	Definition of a Euclidean Space
		8.3.	Basic Metric Concepts
		8.4.	Orthogonal Bases
		8.5.	Perpendiculars
		8.6.	The Orthogonalization Theorem
		8.7.	The Gram Determinant
		8.8.	Incompatible Systems and the Method of Least Squares
		8.9.	Adjoint Operators and Isometry
			Problems
	chapter 9
		UNITARY SPACES
		9.1.	Hermitian Forms
		9.2.	The Scalar Product in a Complex Space
		9.3.	Normal Operators
		9.4.	Applications to Operator Theory in Euclidean Space
			Problems
	chapter 10
		QUADRATIC FORMS IN EUCLIDEAN AND UNITARY SPACES
		10.1.	Basic Theorem on Quadratic Forms in a Euclidean Space
		10.2.	Extremal Properties of a Quadratic Form
		10.3	Simultaneous Reduction of Two Quadratic Forms
		10.4.	Reduction of the General Equation of a Quadratic Surface
		10.5.	Geometric Properties of a Quadratic Surface
		*10.6.	Analysis of a Quadric Surface from Its Genearl Equation
		10.7.	Hermitian Quadratic Forms
			Problems
	chapter 11
		FINITE-DIMENSIONAL ALGEBRAS AND THEIR REPRESENTATIONS
		11.1.	More on Algebras
		11.2.	Representations of Abstract Algebras
		11.3.	Irreducible Representations and Schur's Lemma
		11.4.	Basic Types of Finite-Dimensional Algebras
		11.5.	The Left Regular Representation of a Simple Algebra
		11.6.	Structure of Simple Algebras
		11.7.	Structure of Semisimple Algebras
		11.8.	Representations of Simple and Semisimple Algebras
		11.9.	Some Further Results
			Problems
	*Appendix
		CATEGORIES OF FINITE-DIMENSIONAL SPACES
		A.1.	Introduction
		A.2.	The Case of Complete Algebras
		A.3.	The Case of One-Dimensional Algebras
		A.4.	The Case of Simple Algebras
		A.5.	The Case of Complete Algebras of Diagonal Matrices
		A.6.	Categories and Direct Sums
	HINTS AND ANSWERS
	BIBLIOGRAPHY
	INDEX

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Linear Algebra

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