 ## Linear Algebra

### byJim HefferonMathematicsSaint Michael's College Colchester, Vermont USA 05439

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.

• Chapter One: Linear Systems
1. Solving Linear Systems
1. Gauss' Method
2. Describing the Solution Set
3. General=Particular+Homogeneous
2. Linear Geometry of n-Space
1. Vectors in Space
2. Length and Angle Measure* (page 38)
3. Reduced Echelon Form
1. Gauss-Jordan Reduction (page 46)
2. 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
1. Definition of Vector Space
1. Definition and Examples (page 80)
2. Subspaces and Spanning Sets (page 91)
2. Linear Independence
1. Definition and Examples (page 102)
3. Basis and Dimension
1. Basis (page 113)
2. Dimension (page 119)
3. Vector Spaces and Linear Systems (page 124)
4. 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
1. Isomorphisms
1. Definition and Examples (page 159)
2. Dimension Characterizes Isomorphism (page 168)
2. Homomorphisms
1. Definition (page 176)
2. Range Space and Null Space (page 183)
3. Computing Linear Maps
1. Representing Linear Maps With Matrices (page 195)
2. Any Matrix Represents a Linear Map* (page 205)
4. Matrix Operations
1. Sums and Scalar Products (page 212)
2. Matrix Multiplication (page 214)
3. Mechanics of Matrix Multiplication (page 222)
4. Inverses (page 231)
5. Change of Basis
1. Changing Representations of Vectors (page 238)
2. Changing Map Representations (page 242)
6. Projection
1. Orthogonal Projection Into a Line* (page 250)
2. Gram-Schmidt Orthogonalization* (page 254)
3. 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
1. Definition
1. Exploration* (page 294)
2. Properties of Determinants (page 299)
3. The Permutation Expansion (page 303)
4. Determinants Exist* (page 312)
2. Geometry of Determinants
1. Determinants as Size Functions (page 319)
3. Other Formulas
1. 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
1. Complex Vector Spaces
1. Factoring and Complex Numbers: A Review* (page 350)
2. Complex Representations (page 351)
2. Similarity
1. Definition and Examples (page 353)
2. Diagonalizability (page 355)
3. Eigenvalues and Eigenvectors (page 359)
3. Nilpotence
1. Self-Composition* (page 367)
2. Strings* (page 370)
4. Jordan Form
1. Polynomials of Maps or Matrices* (page 381)
2. 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