Linear Algebra. Core Topics for the Second Course 9789811258541, 9789811258558, 9789811258565
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English Pages 332 Year 2023
Table of contents :
Contents
Preface
1 Vector Spaces
Introduction
1.1 Definitions and examples
1.2 Subspaces
1.3 Linearly independent vectors and bases
1.4 Direct sums
1.5 Dimension of a vector space
1.6 Change of basis
1.7 Exercises
1.7.1 Definitions and examples
1.7.2 Subspaces
1.7.3 Linearly independent vectors and bases
1.7.4 Direct sums
1.7.5 Dimension of a vector space
1.7.6 Change of basis
2 Linear Transformations
Introduction
2.1 Basic properties
2.1.1 The kernel and range of a linear transformation
2.1.2 Projections
2.1.3 The Rank-Nullity Theorem
2.2 Isomorphisms
2.3 Linear transformations and matrices
2.3.1 The matrix of a linear transformation
2.3.2 The isomorphism between Mn×m(K) and L(V,W)
2.4 Duality
2.4.1 The dual space
2.4.2 The bidual
2.5 Quotient spaces
2.6 Exercises
2.6.1 Basic properties
2.6.2 Isomorphisms
2.6.3 Linear transformations and matrices
2.6.4 Duality
2.6.5 Quotient spaces
3 Inner Product Spaces
Introduction
3.1 Definitions and examples
3.2 Orthogonal projections
3.2.1 Orthogonal projections on lines
3.2.2 Orthogonal projections on arbitrary subspaces
3.2.3 Calculations and applications of orthogonal projections
3.2.4 The annihilator and the orthogonal complement
3.2.5 The Gram-Schmidt orthogonalization process and orthonormal bases
3.3 The adjoint of a linear transformation
3.4 Spectral theorems
3.4.1 Spectral theorems for operators on complex inner product spaces
3.4.2 Self-adjoint operators on real inner product spaces
3.4.3 Unitary operators
3.4.4 Orthogonal operators on real inner product spaces
3.4.5 Positive operators
3.5 Singular value decomposition
3.6 Exercises
3.6.1 Definitions and examples
3.6.2 Orthogonal projections
3.6.3 The adjoint of a linear transformation
3.6.4 Spectral theorems
3.6.5 Singular value decomposition
4 Reduction of Endomorphisms
Introduction
4.1 Eigenvalues and diagonalization
4.1.1 Multilinear alternating forms and determinants
4.1.2 Diagonalization
4.2 Jordan canonical form
4.2.1 Jordan canonical form when the characteristic polynomial has one root
4.2.2 Uniqueness of the Jordan canonical form when the characteristic polynomial has one root
4.2.3 Jordan canonical form when the characteristic polynomial has several roots
4.3 The rational form
4.4 Exercises
4.4.1 Diagonalization
4.4.2 Jordan canonical form
4.4.3 Rational form
5 Appendices
Appendix A Permutations
Appendix B Complex numbers
Appendix C Polynomials
Appendix D Infinite dimensional inner product spaces
Bibliography
Index
