Open Linear Algebra Book
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This book is suited for a standard linear algebra course for engineering students at a bachelor level. Except for some basic algebra skills generally taught in secondary education, no prior knowledge is expected.
The main concepts of linear algebra are introduced from a geometrical perspective. We start by introducing the basic concepts of vectors, lines, and planes. There follows a thorough treatment of standard subjects like systems of linear equations, matrix arithmetic, eigenvalues and eigenvectors, orthogonality etc. In the final chapters, more advanced topics like symmetric matrices and discrete dynamical systems are discussed.
Throughout the book, many interactive applets are inserted to give the student hands-on experience with linear algebra. Thanks to an ample selection of embedded exercises with individualized feedback, the book offers a stimulating learning environment for studying linear algebra!
The chapters and paragraphs of this book are:
Chapter 1. Vectors, Lines and Planes
- §1.1 Vectors
- §1.2 Dot Prodcut
- §1.3 Cross Product
- §1.4 Lines and Planes
Chapter 2. Systems of Linear Equations, Vector Equations and Matrix Equations
- §2.1 Systems of Linear Equations
- §2.2 Linear Combinations
- §2.3 The Solution Set of a System of Linear Equations
- §2.4 The Matrix-Vector Product Ax
Chapter 3. Matrix Operations
- §3.1 Linear Transformations
- §3.2 Matrix Operations
- §3.3 Some Important Classes of Linear Transformations
- §3.4 The Inverse of a Matrix
- §3.5 Injectivity, Surjectivity, and Bijectivity
Chapter 4. Subspaces
- §4.1 Subspaces of n
- §4.2 Basis and Dimension
- §4.3 Change of Basis
Chapter 5. Determinants
- §5.1 Determinants as Areas or Volumes
- §5.2 Determinants via Cofactor Expansion
- §5.3 Determinants via Row Reduction
- §5.4 Miscellaneous Applications of Determinants
Chapter 6. Eigenvalues and Eigenvectors
- §6.1 Definitions and Examples
- §6.2 The Characteristic Polynomial
- §6.3 Diagonalizability
- §6.4 Complex Eigenvalues (and Eigenvectors)
Chapter 7. Orthogonality
- §7.1 Orthogonal Complements
- §7.2 Ortogonal and Orthonormal bases
- §7.3 The Gram-Schmidt Process
- §7.4 Least Squares Solutions
Chapter 8. Symmeteric Matrices
- §8.1 Symmetric Matrices
- §8.2 Quadratic Forms
- §8.3 Singular Value Decomposition (SVD)
Chapter 9. Dynamical Systems
- §9.1 Discrete Dynamical Systems
- §9.2 Markov Chains
- §9.3 The Power Method
- §9.4 Continuous Dynamical Systems (Under construction)
Appendices
- 10 Complex numbers
- 11 The inverse matrix theorem