Fundamentals of Linear Algebra
Introduction to Vectors: Vector Operations, Properties, and
Geometric Interpretation
Matrices and Matrix Operations: Addition, Subtraction,
Scalar Multiplication, and Matrix Multiplication
Systems of Linear Equations: Gaussian Elimination, Matrix
Formulation, and Row-Reduced Echelon Form
Vector Spaces and Subspaces: Basis, Dimension, and Linear
Independence
Linear Transformations and Eigenvalues/Eigenvectors:
Definitions and Applications
Matrix Operations and Applications
Matrix Determinants: Properties, Cramer's Rule, and Inverse
Matrices
Matrix Factorizations: LU Decomposition, QR Decomposition,
and Singular Value Decomposition
Orthogonal Matrices and Orthogonal Projections
Matrix Rank, Null Space, and Column Space
Applications of Linear Algebra in Engineering, Computer
Science, Physics, and Economics
Vector Spaces and Linear Transformations
Vector Spaces: Definitions, Examples, and Properties
Subspaces: Span, Basis, and Dimension
Linear Independence and Basis Expansion Theorem
Inner Product Spaces: Dot Product, Orthogonality, and
Orthonormal Bases
Linear Transformations: Definitions, Properties, and
Examples
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors: Definitions and Properties
Diagonalization of Matrices: Diagonalizable and Defective
Matrices
Applications of Eigenvalues and Eigenvectors in Markov
Chains, Differential Equations, and Image Processing
Jordan Canonical Form: Jordan Blocks, Diagonalizability, and
Applications
Spectral Decomposition: Eigenvalue Decomposition and
Singular Value Decomposition
Advanced Topics in Linear Algebra
Orthogonalization: Gram-Schmidt Process and Orthogonal
Complement
Orthogonal Projections and Least Squares Approximation
Linear Independence and Linear Dependence: Wronskian, Linear
Dependence Test, and Cofactor Expansion
Determinants and Eigenvalues of Special Matrices: Diagonal
Matrices, Triangular Matrices, and Hermitian Matrices
Applications of Linear Algebra in Machine Learning, Data Science, and Cryptography