(NOTE: Each chapter concludes with Review Exercises.)
1. The Geometry of the Plane and 3 Space.
Vectors. Length and Direction. Lines, Planes, Cross Product. Projections. Euclidean n-space.
2. Matrices and Linear Equations.
The Algebra of Matrices. The Inverse and Transpose. Systems of Linear Equations. Homogeneous Systems and Linear Independence. The LU Factorization of a Matrix. LDU Factorizations. Finding the Inverse of a Matrix.
3. Determinants and Eigenvalues.
The Determinant of a Matrix. Properties of Determinants. The Eigenvalues and Eigenvectors of a Matrix. Similarity and Diagonalization.
4. Vector Spaces.
The Theory of Linear Equations. Basic Terminology and Concepts (Mostly Review). Basis and Dimensions; Rank and Nullity. One-sided Inverses.
5. Linear Transformations.
Fundamentals. Matrix Multiplication Revisited. The Matrices of a Linear Transformation. Changing Coordinates.
6. Orthogonality.
Projection Matrices and Least Square Approximation. The Gram-Schmidt Algorithm and QR Factorization. Orthogonal Subspaces and Complements. The Pseudoinverse of a Matrix.
7. The Spectral Theorem.
Complex Numbers and Vectors. Complex Matrices. Unitary Diagonalization. The Orthogonal Diagonalization of Real Symmetric Matrices. The Singular Value Decomposition.
8. Applications.
Data Fitting. Linear Recurrence Relations. Markov Processes. Quadratic Forms and Conic Sections. Graphs.
Show and Prove.
Things I Must Remember.
Solutions to Selected Exercises.
Solutions to True/False Questions.
Glossary.
Index.
