Linear Algebra

A thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second section addresses more advanced topics such as the study of canonical forms for matrices.

The treatment can be tailored to satisfy the requirements of both introductory and ...

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A thorough first course in linear algebra, this two-part treatment begins with the basic theory of vector spaces and linear maps, including dimension, determinants, eigenvalues, and eigenvectors. The second section addresses more advanced topics such as the study of canonical forms for matrices.

The treatment can be tailored to satisfy the requirements of both introductory and advanced courses. Introductory courses that also serve as an initiation into formal mathematics will focus on the first six chapters. Students already schooled in matrices and linear mappings as well as theorem-proving will quickly proceed to selected chapters from part two. The selection can emphasize algebra or analysis/geometry, as needed. Ample examples, applications, and exercises appear throughout the text, which is supplemented by three helpful Appendixes.

PART I
1 Vector Spaces
1.1 Motivation (vectors in 3-space)
1.2 Rn and Cn
1.3 Vector spaces:the axioms,some examples
1.4 Vector spaces:first consequences of the axioms
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PART I
1 Vector Spaces
1.1 Motivation (vectors in 3-space)
1.2 Rn and Cn
1.3 Vector spaces:the axioms,some examples
1.4 Vector spaces:first consequences of the axioms
1.5 Linear combinations of vectors
1.6 Linear subspaces
2 Linear mappings
2.1 Linear mappings
2.2 Linear mappings and linear subspaces:kernel and range
2.4 Isomorphic vector spaces
2.5 Equivalence relations and quotient sets
2.6 Quotient vector spaces
2.7 The first isomorphism theorem
3 Structure of vector spaces
3.1 Linear subspace generated by a subset
3.2 Linear dependence
3.3 Linear independence
3.4 Finitely generated vector spaces
3.5 Basis,dimension
3.6 Rank + nullity = dimension
3.7 Applications of R + N = D
3.8 Dimension of L(V,W)
3.9 Duality in vector spaces
4 Matrices
4.1 Matrices
4.2 Matrices of linear mappings
4.3 Matrix multiplication
4.4 Algebra of matrices
4.5 A model for linear mappings
4.6 Transpose of a matrix
4.7 Calculating the rank
4.8 When is a linear system solvable?
4.9 An example
4.10 Change of basis,similar matrices
5 Inner product spaces
5.1 Inner product spaces,Euclidean spaces
5.2 Duality in inner product spaces
5.3 The adjoint of a linear mapping
5.4 Orthogonal mappings and matrices
6 Determinants (2×2 and 3×3)
6.1 Determinant of a 2×2 matrix
6.2 Cross product of vectors in R-3
6.3 Determinant of a 3×3 matrix
6.4 Characteristic polynomial of a matrix (2×2 or 3×3)
6.5 Diagonalizing 2×2 symmetric real matrices
6.6 Diagonalizing 3×3 symmetric real matrices
6.7 A geometric application (conic sections)
PART II
7 Determinants (n×n)
8 Similarity (Act I)
9 Euclidean spaces (Spectral Theory)
10 Equivalence of matrices over a PIR
11 Similarity (Act II)
12 Unitary spaces
13 Tensor products
Appendix A Foundations
A.1 A dab of logic
A.2 Set notations
A.3 Functions
A.4 The axioms for a field
Appendix B Integral domains,factorization theory
B.1 The field of fractions of an integral domain
B.2 Divisibility in an integral domain
B.3 Principle ideal ring
B.4 Euclidean integral domains
B.5 Factorization in overfields
Appendix C Weierstrass-Bolzano theorem
Index of notations
Index
Errata and Comments
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