Book:Richard Kaye/Linear Algebra
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Richard Kaye and Robert Wilson: Linear Algebra
Published $\text {1998}$, Oxford Science Publications
- ISBN 0-19-850237-0
Subject Matter
Contents
- Preface
- PART I MATRICES AND VECTOR SPACES
- 1 Matrices
- 1.1 Matrices
- 1.2 Addition and multiplication of matrices
- 1.3 The inverse of a matrix
- 1.4 The transpose of a matrix
- 1.5 Row and column operations
- 1.6 Determinant and trace
- 1.7 Minors and cofactors
- 2 Vector spaces
- 2.1 Examples and axioms
- 2.2 Subspaces
- 2.3 Linear independence
- 2.4 Bases
- 2.5 Coordinates
- 2.6 Vector spaces over other fields
- PART II BILINEAR AND SESQUILINEAR FORMS
- 3 Inner product spaces
- 3.1 The standard inner product
- 3.2 Inner products
- 3.3 Inner products over $\C$
- 4 Bilinear and sesquilinear forms
- 4.1 Bilinear forms
- 4.2 Representation by matrices
- 4.3 The base-change formula
- 4.4 Sesquilinear forms over $\C$
- 5 Orthogonal bases
- 5.1 Orthonormal bases
- 5.2 The Gram-Schmidt process
- 5.3 Properties of orthonormal bases
- 5.4 Orthogonal complements
- 6 When is a form definite?
- 6.1 The Gram-Schmidt process revisited
- 6.2 The leading minor test
- 7 Quadratic forms and Sylvester's law of inertia
- 7.1 Quadratic forms
- 7.2 Sylvester's law of inertia
- 7.3 Examples
- 7.4 Applications to surfaces
- 7.5 Sesquilinear and Hermitian forms
- PART III LINEAR TRANSFORMATIONS
- 8 Linear transformations
- 8.1 Basics
- 8.2 Arithmetic operations on linear transformations
- 8.3 Representation by matrices
- 9 Polynomials
- 9.1 Polynomials
- 9.2 Evaluating polynomials
- 9.3 Roots of polynomials over $\C$
- 9.4 Roots of polynomials over other fields
- 10 Eigenvalues and eigenvectors
- 10.1 An example
- 10.2 Eigenvalues and eigenvectors
- 10.3 Upper triangular matrices
- 11 The minimum polynomial
- 11.1 The minimum polynomial
- 11.2 The characteristic polynomial
- 11.3 The Cayley-Hamilton theorem
- 12 Diagonalization
- 12.1 Diagonal matrices
- 12.2 A criterion for diagonalizability
- 12.3 Examples
- 13 Self-adjoint transformations
- 13.1 Orthogonal and unitary transformations
- 13.2 From forms to transformations
- 13.3 Eigenvalues and diagonalization
- 13.4 Applications
- 14 The Jordan normal form
- 14.1 Jordan normal form
- 14.2 Obtaining the Jordan normal form
- 14.3 Applications
- 14.4 Proof of the primary decomposition theorem
- Appendix A A Theorem of Analysis
- Appendix B Applications to quantum mechanics
- Index
Errata
Simultaneous Linear Equations: Arbitrary System $6$
Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.5$ Row and column operations
To solve:
- $\begin {array} {rcrcrcr}
x & + & y & + & 2 z & = & -1 \\
-x & + & & & z & = & -1 \\ -x & + & y & + & 4 z & = & 3 \\ \end {array}$,
first put the equation in matrix form
- $\paren {\begin {array} {rrr} 1 & 1 & 2 \\ -1 & 0 & 1 \\ -1 & 1 & 4 \end {array} } \begin {pmatrix} x \\ y \\ z \end {pmatrix} = \paren {\begin {array} {r} -1 \\ -1 \\ 3 \end {array} }$
and then put the augmented matrix formed from the matrix on the left with the column vector on the right into echelon form:
- $\paren {\begin {array} {rrr|r}
1 & 1 & 2 & -1 \\
-1 & 0 & 1 & -1 \\ -1 & 1 & 4 & 3 \end {array} } \to \paren {\begin {array} {rrr|r}
1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 2 & 6 & -4 \end {array} } \to \paren {\begin {array} {rrr|r} 1 & 1 & 2 & -1 \\ 0 & 1 & 3 & -2 \\ 0 & 0 & 0 & 0 \end {array} }$.
Source work progress
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: Exercises: $1.15 \ \text {(a)}$