Chapter 1. Systems of Linear Equations and Matrices.

1.1 Introduction to Systems of Linear Equations.

1.2 Gaussian Elimination.

1.3 Matrices and Matrix Operations.

1.4 Inverses; Rules of Matrix Arithmetic.

1.5 Elementary Matrices and a Method for Finding A-1.

1.6 Further Results on Systems of Equations and Invertibility.

1.7 Diagonal, Triangular, and Symmetric Matrices.

Chapter 2. Determinants.

2.1 Determinants by Cofactor Expansion.

2.2 Evaluating Determinants by Row Reduction.

2.3 Properties of the Determinant Function.

2.4 A Combinatorial Approach to Determinants.

Chapter 3. Vectors in 2 Space and 3-Space.

3.1 Introduction to Vectors (Geometric).

3.2 Norm of a Vector; Vector Arithmetic.

3.3 Dot Product; Projections.

3.4 Cross Product.

3.5 Lines and Planes in 3-Space.

Chapter 4. Euclidean Vector Spaces.

4.1 Euclidean n-Space.

4.2 Linear Transformations from Rn to Rm.

4.3 Properties of Linear Transformations from Rn to Rm.

4.4 Linear Transformations and Polynomials.

Chapter 5. General Vector Spaces.

5.1 Real Vector Spaces.

5.2 Subspaces.

5.3 Linear Independence.

5.4 Basis and Dimension.

5.5 Row Space, Column Space, and Nullspace.

5.6 Rank and Nullity.

Chapter 6. Inner Product Spaces.

6.1 Inner Products.

6.2 Angle and Orthogonality in Inner Product Spaces.

6.3 Orthonormal Bases: Gram-Schmidt Prodcess; QR-Decomposition.

6.4 Best Approximation; Least Squares.

6.5 Change of Basis.

6.6 Orthogonal Matrices.

Chapter 7. Eigenvalues, Eigenvectors.

7.1 Eigenvalues and Eigenvectors.

7.2 Diagonalization.

7.3 Orthogonal Diagonalization.

Chapter 8. Linear Transformations.

8.1 General Linear Transformations.

8.2 Kernel and range.

8.3 Inverse Linear Transformations.

8.4 Matrices of General Linear Transformations.

8.5 Similarity.

8.6 Isomorphism.

Chapter 9. Additional topics.

9.1 Application to Differential Equations.

9.2 Geometry and Linear Operators on R2.

9.3 Least Squares Fitting to Data.

9.4 Approximation Problems; Fourier Series.

9.5 Quadratic Forms.

9.6 Diagonalizing Quadratic Forms; Conic Sections.

9.7 Quadric Surfaces.

9.8 Comparison of Procedures for Solving Linear Systems.

9.9 LU-Decompositions.

Chapter 10. Complex Vector Spaces.

10.1 Complex Numbers.

10.2 Division of Complex Numbers.

10.3 Polar Form of a Complex Number.

10.4 Complex Vector Spaces.

10.5 Complex Inner Product Spaces.

10.6 Unitary Normal, and Hermitian Matrices.

Chapter 11. Applications of Linear Algebra.

11.1 Constructing Curves and Surfaces through Specified Points.

11.2 Electrical Networks.

11.3 Geometric Linear Programming.

11.4 The Earliest Applications of Linear Algebra.

11.5 Cubic Spline Interpolation.

11.6 Markov Chains.

11.7 Graph Theory.

11.8 Games of Strategy.

11.9 Leontief Economic Models.

11.10 Forest Management.

11.11 Computer Graphics.

11.12 Equilibrium Temperature Distributions.

11.13 Computed Tomography.

11.14 Fractals.

11.15 Chaos.

11.16 Cryptography.

11.17 Genetics.

11.18 Age-Specific Population Growth.

11.19 Harvesting of Animal Populations.

11.20 A Least Squares Model for Human Hearing.

11.21 Warps and Morphs.

Answers to Exercises.

Index.