Chapter 1: Matrices and Gaussian Elimination
1.1: Introduction to Vectors, Matrices, and Linear Equations and Combinations
1.2: Applications – Probability, Graphs and Networks, Economics, Engineering, Computer Graphics, Cryptography, etc
1.3: Lengths and Dot Products
1.4: Matrix Notation and Matrix Multiplication/Rules for Matrix Operations
1.6: Eliminations = Factorization; A = LU
1.7: Inverse Matrices; Transposes and Permutations
1.8: Triangular Factors and Row Exchanges
Chapter 2: Vector Spaces
2.1: Vector Spaces and Subspaces
2.2: Nullspace of A – Solving Ax=0 and Rx=0; Complete Solution to Ax=b
2.3: Linear Independence, Basis and Dimension
2.4: Dimensions of the Four Fundamental Subspaces
2.5: Graphs and Networks
2.6: Linear Transformations
Chapter 3: Orthogonality
3.1: Vectors and Subspaces; Orthogonality of the Four Subspaces
3.2: Projections (Cosines and Projections onto Lines)
3.3: Projections and Least Squares Approximations
3.4: Orthogonal Bases and Gram-Schmidt
Chapter 4: Determinants
4.1: What are Determinants? Properties, Formulas and Applications
4.2: Permutations and Cofactors
4.3: Cramer’s Rule, Inverses, and Volumes
Chapter 5: Eigenvalues and Eigenvectors
5.1: What are Eigenvalues and Eigenvectors
5.2: Diagonalization of a Matrix
5.3: Systems of Differential Equations
5.4: Symmetric Matrices
5.5: Positive Definite Matrices
Chapter 6: Singular Value Decomposition (SVD)
6.1: Image Processing by Linear Algebra
6.2: Bases and Matrices in the SVD
6.3: Principal Component Analysis (PCA by SVD)
6.4: Geometry of the SVD
Chapter 7: Linear Transformations
7.1: The Idea of a Linear Transformation
7.2: Matrix of a Linear Transformation
7.3: Search for a Good Basis
Chapter 8: Complex Vectors and Matrices
8.1: Complex Numbers
8.2: Hermitian and Unitary Matrices
8.3: Fast Fourier Transform
Chapter 9: Numerical Linear Algebra
9.1: Gaussian Elimination in Practice
9.2: Norms and Condition Numbers
9.3: Iterative Methods and Preconditioners
Extra Practice Problems
Link to Textbook: Linear Algebra & Its Applications by Gilbert Strang