Advanced Linear Algebra 1st edition by Hugo Woerdeman – Ebook PDF Instant Download/Delivery: 1498754057 , 9781498754057
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ISBN 10: 1498754057
ISBN 13: 9781498754057
Author: Hugo Woerdeman
Advanced Linear Algebra features a student-friendly approach to the theory of linear algebra. The author’s emphasis on vector spaces over general fields, with corresponding current applications, sets the book apart. He focuses on finite fields and complex numbers, and discusses matrix algebra over these fields. The text then proceeds to cover vector spaces in depth. Also discussed are standard topics in linear algebra including linear transformations, Jordan canonical form, inner product spaces, spectral theory, and, as supplementary topics, dual spaces, quotient spaces, and tensor products. Written in clear and concise language, the text sticks to the development of linear algebra without excessively addressing applications. A unique chapter on “How to Use Linear Algebra” is offered after the theory is presented. In addition, students are given pointers on how to start a research project. The proofs are clear and complete and the exercises are well designed. In addition, full solutions are included for almost all exercises.
Advanced Linear Algebra 1st Table of contents:
1 Fields and Matrix Algebra
1.1 The field ℤ3
1.2 The field axioms
1.3 Field examples
1.3.1 Complex numbers
Figure 1.1: The complex number z in the complex plane.
1.3.2 The finite field ℤp, with p prime
Proposition 1.3.1
Proposition 1.3.2
Algorithm 1 Euclid’s algorithm
Example 1.3.3
Theorem 1.3.4
1.4 Matrix algebra over different fields
Example 1.4.1
Example 1.4.2
Example 1.4.3
Example 1.4.4
Example 1.4.5
Example 1.4.6
Example 1.4.7
Example 1.4.8
1.4.1 Reminders about Cramer’s rule and the adjugate matrix.
Theorem 1.4.9
Example 1.4.10
Example 1.4.11
Example 1.4.12
Theorem 1.4.13
1.5 Exercises
Exercise 1.5.1
Exercise 1.5.2
Exercise 1.5.3
Exercise 1.5.4
Exercise 1.5.5
Exercise 1.5.6
Exercise 1.5.7
Exercise 1.5.8
Exercise 1.5.9
Exercise 1.5.10
Exercise 1.5.11
Exercise 1.5.12
Exercise 1.5.13
Exercise 1.5.14
Exercise 1.5.15
Exercise 1.5.16
Exercise 1.5.17
Exercise 1.5.18
Exercise 1.5.19
Exercise 1.5.20
Exercise 1.5.21
Exercise 1.5.22
Exercise 1.5.23
Exercise 1.5.24
Claim
Exercise 1.5.25
Exercise 1.5.26
Exercise 1.5.27
2 Vector Spaces
2.1 Definition of a vector space
Lemma 2.1.1
2.2 Vector spaces of functions
Proposition 2.2.1
2.2.1 The special case when X is finite
2.3 Subspaces and more examples of vector spaces
Proposition 2.3.1
Proposition 2.3.2
Proposition 2.3.3
2.3.1 Vector spaces of polynomials
Proposition 2.3.4
Example 2.3.5
Proposition 2.3.6
Proposition 2.3.7
2.3.2 Vector spaces of matrices
Proposition 2.3.8
2.4 Linear independence, span, and basis
Example 2.4.1
Example 2.4.2
Example 2.4.3
Proposition 2.4.4
Remark 2.4.5
Example 2.4.6
Example 2.4.7
Example 2.4.8
Example 2.4.9
Example 2.4.10
Example 2.4.11
2.5 Coordinate systems
Theorem 2.5.1
Example 2.5.2
Example 2.5.3
2.6 Exercises
Exercise 2.6.1
Exercise 2.6.2
Exercise 2.6.3
Exercise 2.6.4
Exercise 2.6.5
Exercise 2.6.6
Exercise 2.6.7
Exercise 2.6.8
Exercise 2.6.9
Exercise 2.6.10
Exercise 2.6.11
Exercise 2.6.12
Exercise 2.6.13
Exercise 2.6.14
Exercise 2.6.15
Exercise 2.6.16
Exercise 2.6.17
3 Linear Transformations
3.1 Definition of a linear transformation
Example 3.1.1
Example 3.1.2
Proposition 3.1.3
3.2 Range and kernel of linear transformations
Proposition 3.2.1
Example 3.1.1 continued.
Proposition 3.2.2
Lemma 3.2.3
Example 3.2.4
Theorem 3.2.5
Proposition 3.2.6
Theorem 3.2.7
Example 3.2.8
3.3 Matrix representations of linear maps
Theorem 3.3.1
Example 3.3.2
Example 3.3.3
Theorem 3.3.4
Corollary 3.3.5
Example 3.3.3 continued.
Corollary 3.3.6
3.4 Exercises
Exercise 3.4.1
Exercise 3.4.2
Exercise 3.4.3
Exercise 3.4.4
Exercise 3.4.5
Exercise 3.4.6
Exercise 3.4.7
Exercise 3.4.8
Exercise 3.4.9
Exercise 3.4.10
Exercise 3.4.11
Exercise 3.4.12
4 The Jordan Canonical Form
4.1 The Cayley–Hamilton theorem
Theorem 4.1.1
Example 4.1.2
4.2 Jordan canonical form for nilpotent matrices
Theorem 4.2.1
Example 4.2.2
Example 4.2.2 continued.
4.3 An intermezzo about polynomials
Proposition 4.3.1
Example 4.3.2
Proposition 4.3.3
Proposition 4.3.4
Example 4.3.5
Proposition 4.3.6
4.4 The Jordan canonical form
Theorem 4.4.1
Remark 4.4.2
Proposition 4.4.3
Lemma 4.4.4
Example 4.4.5
4.5 The minimal polynomial
Example 4.5.1
Proposition 4.5.2
Theorem 4.5.3
Example 4.2.2 continued.
Example 4.5.4
Corollary 4.5.5
4.6 Commuting matrices
Proposition 4.6.1
Theorem 4.6.2
Example 4.6.3
Lemma 4.6.4
4.7 Systems of linear differential equations
Theorem 4.7.1
Example 4.7.2
Example 4.7.3
4.8 Functions of matrices
Example 4.8.1
Remark 4.8.2
Theorem 4.8.3
Theorem 4.8.4
Example 4.8.5
Proposition 4.8.6
Corollary 4.8.7
Example 4.8.8
4.9 The resolvent
Proposition 4.9.1
Theorem 4.9.2
Theorem 4.9.3
4.10 Exercises
Exercise 4.10.1
Exercise 4.10.2
Exercise 4.10.3
Exercise 4.10.4
Exercise 4.10.5
Exercise 4.10.6
Exercise 4.10.7
Exercise 4.10.8
Exercise 4.10.9
Exercise 4.10.10
Exercise 4.10.11
Exercise 4.10.12
Exercise 4.10.13
Exercise 4.10.14
Exercise 4.10.15
Exercise 4.10.16
Exercise 4.10.17
Exercise 4.10.18
Exercise 4.10.19
Exercise 4.10.20
Exercise 4.10.21
Exercise 4.10.22
Exercise 4.10.23
Exercise 4.10.24
Exercise 4.10.25
Exercise 4.10.26
Exercise 4.10.27
Exercise 4.10.28
Exercise 4.10.29
Exercise 4.10.30
Exercise 4.10.31
Exercise 4.10.32
Exercise 4.10.33
Exercise 4.10.34
Exercise 4.10.35
5 Inner Product and Normed Vector Spaces
5.1 Inner products and norms
Example 5.1.1
Example 5.1.2
Example 5.1.3
Example 5.1.4
Example 5.1.5
Example 5.1.6
Example 5.1.7
Example 5.1.8
Example 5.1.9
Proposition 5.1.10
Remark 5.1.11
Lemma 5.1.12
Example 5.1.13
Example 5.1.14
Example 5.1.15
Example 5.1.16
Example 5.1.17
Example 5.1.18
Theorem 5.1.19
Example 5.1.20
Corollary 5.1.21
Example 5.1.22
Example 5.1.23
Example 5.1.24
Theorem 5.1.25
Example 5.1.26
5.2 Orthogonal and orthonormal sets and bases
Lemma 5.2.1
Example 5.2.2
Theorem 5.2.3
Example 5.2.4
Lemma 5.2.5
Proposition 5.2.6
Corollary 5.2.7
5.3 The adjoint of a linear map
Lemma 5.3.1
Example 5.3.2
Theorem 5.3.3
5.4 Unitary matrices, QR, and Schur triangularization
Theorem 5.4.1
Example 5.4.2
Theorem 5.4.3
5.5 Normal and Hermitian matrices
Lemma 5.5.1
Theorem 5.5.2
Proposition 5.5.3
Theorem 5.5.4
Theorem 5.5.5
Lemma 5.5.6
5.6 Singular value decomposition
Theorem 5.6.1
Proposition 5.6.2
Proposition 5.6.3
Example 5.6.4
5.7 Exercises
Exercise 5.7.1
Exercise 5.7.2
Exercise 5.7.3
Exercise 5.7.4
Exercise 5.7.5
Exercise 5.7.6
Exercise 5.7.7
Exercise 5.7.8
Exercise 5.7.9
Exercise 5.7.10
Exercise 5.7.11
Exercise 5.7.12
Exercise 5.7.13
Exercise 5.7.14
Exercise 5.7.15
Exercise 5.7.16
Exercise 5.7.17
Exercise 5.7.18
Exercise 5.7.19
Exercise 5.7.20
Exercise 5.7.21
Exercise 5.7.22
Exercise 5.7.23
Exercise 5.7.24
Exercise 5.7.25
Exercise 5.7.26
Exercise 5.7.27
Exercise 5.7.28
Exercise 5.7.29
Exercise 5.7.30
Exercise 5.7.31
Exercise 5.7.32
Exercise 5.7.33
Exercise 5.7.34
Exercise 5.7.35
Exercise 5.7.36
Exercise 5.7.37
Example 5.7.38
6 Constructing New Vector Spaces from Given Ones
6.1 The Cartesian product
Proposition 6.1.1
6.2 The quotient space
Example 6.2.1
Lemma 6.2.2
Proposition 6.2.3
Proposition 6.2.4
Proposition 6.2.5
Example 6.2.6
Proposition 6.2.7
Proposition 6.2.8
Proposition 6.2.9
Example 6.2.10
Lemma 6.2.11
Proposition 6.2.12
6.3 The dual space
Proposition 6.3.1
Theorem 6.3.2
Example 6.3.3
Proposition 6.3.4
Proposition 6.3.5
Proposition 6.3.6
Proposition 6.3.7.
Example 6.3.8
Example 6.3.9
Example 6.3.10
Proposition 6.3.11
6.4 Multilinear maps and functionals
Example 6.4.1
Example 6.4.2
Example 6.4.3
Example 6.4.4
Proposition 6.4.5
6.5 The tensor product
Example 6.5.1
Proposition 6.5.2
Example 6.5.3
Proposition 6.5.4
Example 6.5.5
Proposition 6.5.6
Example 6.5.7
Proposition 6.5.8
Remark 6.5.9
Lemma 6.5.10
Proposition 6.5.11
6.6 Anti-symmetric and symmetric tensors
Example 6.6.1
Lemma 6.6.2
Proposition 6.6.3
Corollary 6.6.4
Proposition 6.6.5
Remark 6.6.6
Proposition 6.6.7
Example 6.6.8
Lemma 6.6.9
Proposition 6.6.10
Example 6.6.11
Lemma 6.6.12
Proposition 6.6.13
Proposition 6.6.14
Remark 6.6.15
Proposition 6.6.16
Example 6.6.17
Lemma 6.6.18
Proposition 6.6.19
Remark 6.6.20
6.7 Exercises
Exercise 6.7.1
Exercise 6.7.2
Exercise 6.7.3
Exercise 6.7.4
Exercise 6.7.5
Exercise 6.7.6
Exercise 6.7.7
Exercise 6.7.8
Exercise 6.7.9
Exercise 6.7.10
Exercise 6.7.11
Exercise 6.7.12
Exercise 6.7.13
Exercise 6.7.14
Exercise 6.7.15
Exercise 6.7.16
Exercise 6.7.17
Exercise 6.7.18
Exercise 6.7.19
Exercise 6.7.20
Exercise 6.7.21
Exercise 6.7.22
Exercise 6.7.23
Exercise 6.7.24
Exercise 6.7.25
7 How to Use Linear Algebra
7.1 Matrices you can’t write down, but would still like to use
7.2 Algorithms based on matrix vector products
Theorem 7.2.1
Example 7.2.2
Lemma 7.2.3
Corollary 7.2.4
Proof of Theorem 7.2.1.
7.3 Why use matrices when computing roots of polynomials?
Example 7.3.1
Theorem 7.3.2
Lemma 7.3.3
Proposition 7.3.4
Figure 7.1: These are the roots of the polynomial where pk(n) is the number of partitions of n in k parts, which is the number of ways n can be written as the sum of k positive integers.
Corollary 7.3.5
Proposition 7.3.6
7.4 How to find functions with linear algebra?
Figure 7.2: A Meyer wavelet.
Example 7.4.1
Proposition 7.4.2
Proposition 7.4.3
Example 7.4.4
Example 7.4.5
Example 7.4.6
Example 7.4.7
Figure 7.3: Blurring function.
Figure 7.4: The original image (of size 3000 × 4000 × 3).
7.5 How to deal with incomplete matrices
Example 7.5.1
Theorem 7.5.2
7.6 Solving millennium prize problems with linear algebra
7.6.1 The Riemann hypothesis
Figure 7.5: The Redheffer matrix of size 500 × 500.
7.6.2 P vs. NP
Figure 7.6: A sample graph.
Lemma 7.6.1
Proposition 7.6.2
Corollary 7.6.3
7.7 How secure is RSA encryption?
Example 7.7.1
Example 7.7.2
7.8 Quantum computation and positive maps
Proposition 7.8.1
Proposition 7.8.2
Example 7.8.3
Example 7.8.4
Proposition 7.8.5
Theorem 7.8.6
Example 7.8.7
7.9 Exercises
Exercise 7.9.1
Exercise 7.9.2
Exercise 7.9.3
Exercise 7.9.4
Exercise 7.9.5
Exercise 7.9.6
Exercise 7.9.7
Exercise 7.9.8
Theorem 7.9.9
Exercise 7.9.10
Exercise 7.9.11
Exercise 7.9.12
Exercise 7.9.13
Exercise 7.9.14
Exercise 7.9.15
Exercise 7.9.16
Exercise 7.9.17
Exercise 7.9.18
Bibliography for Chapter 7
Back Matter
How to Start Your Own Research Project
Answers to Exercises
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Tags: Hugo Woerdeman, Advanced Linear