Linear Algebra
zyBooks 2018

Table of Contents

1. Systems of Linear Equations
1.1 Systems of linear equations
1.2 Matrices and linear systems
1.3 Elementary row operations
1.4 Echelon forms of a matrix
1.5 Solution set of a system of linear equations
1.6 Gaussian elimination
1.7 Gauss-Jordan elimination
1.8 Application: Electric circuits

2. Determinants
2.1 Cofactor expansions
2.2 Properties of determinants
2.3 Application: Area and volume
2.4 Cramer’s rule

3. Euclidean Vector Spaces
3.1 Introduction to vectors
3.2 Vector operations
3.3 Dot product
3.4 Cross product
3.5 Application: 3D coordinate geometry
3.6 Application: Vectors in physics
3.7 Vector spaces and subspaces
3.8 Spanning sets
3.9 Linear independence and dependence
3.10 Basis and dimension

4. Matrix Algebra
4.1 Matrix addition and scalar multiplication
4.2 Matrix multiplication
4.3 Inverse of a matrix
4.4 Solving a system using an inverse matrix
4.5 Elementary matrices
4.6 Block matrices
4.7 LU decomposition
4.8 Application: Markov chains

5. General Vector Spaces
5.1 General vector spaces
5.2 Subspaces
5.3 Coordinatization
5.4 Four fundamental subspaces
5.5 Rank and nullity

6. Linear Transformations
6.1 Linear transformations between Euclidean spaces
6.2 General linear transformations
6.3 Isomorphisms
6.4 Rank and nullity of a linear transformation
6.5 Composition of linear transformations
6.6 Change of bases and matrices of linear transformations
6.7 Application: Transformations in 2D coordinate geometry

7. Eigenvalues and Eigenvectors
7.1 Eigenvalues and eigenvectors
7.2 Eigenspaces
7.3 Similarity and diagonalization
7.4 Complex eigenvalues and eigenvectors
7.5 Application: Inertia tensors
7.6 Application: Systems of first order differential equations

8. Inner Product Spaces and Orthogonality
8.1 Inner product spaces
8.2 Norms and distances
8.3 Orthogonal bases
8.4 Orthogonal complements
8.5 Orthogonal matrices
8.6 Singular value decomposition
8.7 Pseudoinverses
8.8 Complex inner product spaces
8.9 Application: Least-squares approximation
8.10 Application: Principal component analysis

9. Appendix
9.1 Notation