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Table of Contents

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ONE. Linear Systems (pdf)

1. Solving Linear Systems
   1. Gauss’ Method
      • Exercises One I 1
   2. Describing the Solution Set
      • Exercises One I 2
   3. General = Particular + Homogeneous
      • Exercises One I 3

2. Linear Geometry of n-Space
   1. Vectors in Space
      • Exercises One II 1
   2. Length and Angle Measures
      • Exercises One II 2

3. Reduced Echelon Form
   1. Gauss-Jordan Reduction
      • Exercises One III 1
   2. Row Equivalence
      • Exercises One III 2

4. Topics
   1. Computer Algebra Systems
      • Exercises One IV 1
   2. Input-Output Analysis
      • Exercises One IV 2
   3. Accuracy of Computations
      • Exercises One IV 3
   4. Analyzing Networks
      • Exercises One IV 4

TWO. Vector Spaces (pdf)

1. Definition of Vector Space
   1. Definition and Examples
      • Exercises Two I 1
   2. Subspaces and Spanning Sets
      • Exercises Two I 2

2. Linear Independence
   1. Definition and Examples
      • Exercises Two II 1

3. Basis and Dimension
   1. Basis
      • Exercises Two III 1
   2. Dimension
      • Exercises Two III 2
   3. Vector Spaces and Linear Systems
      • Exercises Two III 3
   4. Combining Subspaces
      • Exercises Two III 4

4. Topics
   1. Fields
      • Exercises Two IV 1
   2. Crystals
      • Exercises Two IV 2
   3. Dimensional Analysis
      • Exercises Two IV 3

THREE. Maps Between Spaces (pdf)

1. Isomorphisms
   1. DefinitionAndExamples
      • Exercises Three I 1
   2. DimensionCharacterizesIsomorphism
      • Exercises Three I 2
2. Homomorphisms
   1. Definition
      • Exercises Three II 1
   2. RangespaceAndNullspace
      • Exercises Three II 2
3. ComputingLinearMaps
   1. RepresentingLinearMapsWithMatrices
      • Exercises Three III 1
   2. AnyMatrixRepresentsALinearMap
      • Exercises Three III 2
4. MatrixOperations
   1. SumsAndScalarProducts
      • Exercises Three IV 1
   2. MatrixMultiplication
      • Exercises Three IV 2
   3. MechanicsOfMatrixMultiplication
      • Exercises Three IV 3
   4. Inverses
      • Exercises Three IV 4
5. ChangeOfBasis
   1. ChangingRepresentationsOfVectors
      • Exercises Three V 1
   2. ChangingMapRepresentations
      • Exercises Three V 2
6. Projection
   1. OrthogonalProjectionIntoALine
      • Exercises Three VI 1
   2. GramSchmidtOrthogonalization
      • Exercises Three VI 2
   3. ProjectionIntoASubspace
      • Exercises Three VI 3
7. Topics
   1. LineOfBestFit
      • Exercises Three VII 1
   2. GeometryOfLinearMaps
      • Exercises Three VII 2
   3. MarkovChains
      • Exercises Three VII 3
   4. OrthonormalMatrices
      • Exercises Three VII 4

FOUR. Determinants (pdf)

1. Definition
   • Exercises Four I 1
   1. Properties of Determinants
      • Exercises Four I 2
   2. The Permutation Expansion
      • Exercises Four I 3
   3. Determinants Exist
      • Exercises Four I 4
2. Geometry of Determinants
   1. Determinants as Size Functions
      • Exercises Four II 1
3. Other Formulas
   1. Laplace’s Expansion
      • Exercises Four III 1
4. Topics
   1. Cramer’s Rule
      • Exercises Four IV 1
   2. Speed of Calculating Determinants
      • Exercises Four IV 2
   3. Projective Geometry
      • Exercises Four IV 3

FIVE. Similarity (pdf)

Similarity

1. Complex Vector Spaces
   1. FactoringAndComplexNumbers

   2. ComplexRepresentations

2. Similarity
   1. DefinitionAndExamples
      • Exercises Five II 1
   2. Diagonalizability
      • Exercises Five II 2
   3. Eigenvalues and Eigenvectors
      • Exercises Five II 3

3. Nilpotence
   1. Self-Composition
      • Exercises Five III 1
   2. Strings
      • Exercises Five III 2

4. Jordan Form
   1. Polynomials of Maps and Matrices
      • Exercises Five IV 1
   2. Jordan Canonical Form
      • Exercises Five IV 2

5. Topics
   1. Method of Powers
      • Exercises Five V 1
   2. Stable Populations
      • Exercises Five V 2
   3. Linear Recurrences
      • Exercises Five V 3