Linear Algebra

Linear Algebra

For courses in Advanced Linear Algebra. Illustrates the power of linear algebra through practical applications This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra. It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate. Applications to such areas as differential equations, economics, geometry, and physics appear throughout, and can be included at the instructor's discretion.

1.1 Introduction

1.2 Vector Spaces

1.3 Subspaces

1.4 Linear Combinations and Systems of Linear Equations

1.5 Linear Dependence and Linear Independence

1.6 Bases and Dimension

1.7 Maximal Linearly Independent Subsets

2.1 Linear Transformations, Null Spaces, and Ranges

2.2 The Matrix Representation of a Linear Transformation

2.3 Composition of Linear Transformations and Matrix Multiplication

2.4 Invertibility and Isomorphisms

2.5 The Change of Coordinate Matrix

2.6 Dual Spaces

2.7 Homogeneous Linear Differential Equations with Constant Coeficients

3.1 Elementary Matrix Operations and Elementary Matrices

3.2 The Rank of a Matrix and Matrix Inverses

3.3 Systems of Linear Equation - Theoretical Aspects

3.4 Systems of Linear Equations - Computational Aspects

4.1 Determinants of Order Two

4.2 Determinants of Order $n$

4.3 Properties of Determinants

4.4 Summary - Important Facts about Determinants

4.5 A Characterization of the Determinant

5.1 Eigenvalues and Eigenvectors

5.2 Diagonalizability

5.3 Matrix Limits and Markov Chains

5.4 Invariant Subspace and the Cayley-Hamilton Theorem

6

6.1 Inner Products and Norms

6.10 Conditioning and the Rayleigh Quotient

6.11 The Geometry of Orthogonal Operators

6.2 The Gram-Schmidt Orthogonalization Process and Orthogonal Complements

6.3 The Adjoint of a Linear Operator

6.4 Normal and Self-Adjoint Operators

6.5 Unitary and Orthogonal Operators and Their Matrices

6.6 Orthogonal Projections and the Spectral Theorem

6.7 The Singular Value Decomposition and the Pseudoinverse

6.8 Bilinear and Quadratic Forms

6.9 Einstein's Special Theory of Relativity

7.1 The Jordan Canonical Form I

7.2 The Jordan Canonical Form II

7.3 The Minimal Polynomial

7.4 The Rational Canonical Form