Author | Valenza, Robert J. author |
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Title | Linear Algebra [electronic resource] : An Introduction to Abstract Mathematics / by Robert J. Valenza |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1993 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0901-0 |
Descript | XVIII, 237 p. online resource |
1 Sets and Functions -- 1.1 Notation and Terminology -- 1.2 Composition of Functions -- 1.3 Inverse Functions -- 1.4 Digression on Cardinality -- 1.5 Permutations -- Exercises -- 2 Groups and Group Homomorphisms -- 2.1 Groups and Subgroups -- 2.2 Group Homomorphisms -- 2.3 Rings and Fields -- Exercises -- 3 Vector Spaces and Linear Transformations -- 3.1 Vector Spaces and Subspaces -- 3.2 Linear Transformations -- 3.3 Direct Products and Internal Direct Sums -- Exercises -- 4 Dimension -- 4.1 Bases and Dimension -- 4.2 Vector Spaces Are Free -- 4.3 Rank and Nullity -- Exercises -- 5 Matrices -- 5.1 Notation and Terminology -- 5.2 Introduction to Linear Systems -- 5.3 Solution Techniques -- 5.4 Multiple Systems and Matrix Inversion -- Exercises -- 6 Representation of Linear Transformations -- 6.1 The Space of Linear Transformations -- 6.2 The Representation of Hom(kn,km) -- 6.3 The Representation of Hom(V,Vโ) -- 6.4 The Dual Space -- 6.5 Change of Basis -- Exercises -- 7 Inner Product Spaces -- 7.1 Real Inner Product Spaces -- 7.2 Orthogonal Bases and Orthogonal Projection -- 7.3 Complex Inner Product Spaces -- Exercises -- 8 Determinants -- 8.1 Existence and Basic Properties -- 8.2 A Nonrecursive Formula; Uniqueness -- 8.3 The Determinant of a Product; Invertibility -- Exercises -- 9 Eigenvalues and Eigenvectors -- 9.1 Definitions and Elementary Properties -- 9.2 Hermitian and Unitary Transformations -- 9.3 Spectral Decomposition -- Exercises -- 10 Triangulation and Decomposition of Endomorphisms -- 10.1 The Cayley-Hamilton Theorem -- 10.2 Triangulation of Endomorphisms -- 10.3 Decomposition by Characteristic Subspaces -- 10.4 Nilpotent Mappings and the Jordan Normal Form -- Exercises -- Supplementary Topics -- 1 Differentiation -- 2 The Determinant Revisited -- 3 Quadratic Forms -- 4 An Introduction to Categories and Functors