Linear Algebra
Table of Contents
The subsections are usually a day's lecture.
The ones that are *-ed are optional; I anticipate that some
instructors will pass over these in favor of spending time on other
sections (all of the Topics are optional).
Every subsection includes many exercises; perhaps
ten straightforward computations, ten middling-hard verifications or proofs,
and five reasonably-hard proofs.
To download the book, or the
worked answers to all of the exercises, including the
proofs, visit the text's home page.
- Chapter One: Linear Systems
- Solving Linear Systems
- Gauss' Method
- Describing the Solution Set
- General=Particular+Homogeneous
- Linear Geometry of n-Space
- Vectors in Space
- Length and Angle Measure* (page 38)
- Reduced Echelon Form
- Gauss-Jordan Reduction (page 46)
- Row Equivalence (page 52)
- Topic: Computer Algebra Systems (page 62)
- Topic: Input-Output Analysis (page 64)
- Topic: Accuracy of Computations (page 68)
- Topic: Analyzing Networks (page 72)
- Chapter Two: Vector Spaces
- Definition of Vector Space
- Definition and Examples (page 80)
- Subspaces and Spanning Sets (page 91)
- Linear Independence
- Definition and Examples (page 102)
- Basis and Dimension
- Basis (page 113)
- Dimension (page 119)
- Vector Spaces and Linear Systems (page 124)
- Combining Subspaces* (page 131)
- Topic: Fields (page 141)
- Topic: Crystals (page 143)
- Topic: Voting Paradoxes (page 147)
- Topic: Dimensional Analysis (page 152)
- Chapter Three: Maps Between Spaces
- Isomorphisms
- Definition and Examples (page 159)
- Dimension Characterizes Isomorphism (page 168)
- Homomorphisms
- Definition (page 176)
- Range Space and Null Space (page 183)
- Computing Linear Maps
- Representing Linear Maps With Matrices (page 195)
- Any Matrix Represents a Linear Map* (page 205)
- Matrix Operations
- Sums and Scalar Products (page 212)
- Matrix Multiplication (page 214)
- Mechanics of Matrix Multiplication (page 222)
- Inverses (page 231)
- Change of Basis
- Changing Representations of Vectors (page 238)
- Changing Map Representations (page 242)
- Projection
- Orthogonal Projection Into a Line* (page 250)
- Gram-Schmidt Orthogonalization* (page 254)
- Projection Into a Subspace* (page 260)
- Topic: Line of Best Fit (page 269)
- Topic: Geometry of Linear Maps (page 274)
- Topic: Markov Chains (page 281)
- Topic: Orthonormal Matrices (page 287)
- Chapter Four: Determinants
- Definition
- Exploration* (page 294)
- Properties of Determinants (page 299)
- The Permutation Expansion (page 303)
- Determinants Exist* (page 312)
- Geometry of Determinants
- Determinants as Size Functions (page 319)
- Other Formulas
- Laplace's Expansion* (page 326)
- Topic: Cramer's Rule (page 331)
- Topic: Speed of Calculating Determinants (page 334)
- Topic: Projective Geometry (page 337)
- Chapter Five: Similarity
- Complex Vector Spaces
- Factoring and Complex Numbers: A Review* (page 350)
- Complex Representations (page 351)
- Similarity
- Definition and Examples (page 353)
- Diagonalizability (page 355)
- Eigenvalues and Eigenvectors (page 359)
- Nilpotence
- Self-Composition* (page 367)
- Strings* (page 370)
- Jordan Form
- Polynomials of Maps or Matrices* (page 381)
- Jordan Canonical Form* (page 388)
- Topic: Computing Eigenvalues---the Method of Powers (page 401)
- Topic: Stable Populations (page 405)
- Topic: Linear Recurrences (page 407)
- Appendix
- Introduction
- Propositions
- Quantifiers
- Techniques of Proof
- Sets, Functions, and Relations
- Bibliography
- Index
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Math Department, Saint Michael's College, Colchester, VT USA 05439
This page was last revised on 2001-Oct-09.