1. Linear Algebra
1.1 Vectors
Definition and properties
Vector operations (addition, scalar multiplication, dot product, cross product)
Vector spaces and subspaces
Linear independence and dependence
Basis and dimension
1.2 Matrices
Matrix operations (addition, multiplication, transpose)
Special matrices (identity, diagonal, symmetric, orthogonal)
Determinants and their properties
Matrix inverse and solving linear systems
Rank and nullity
1.3 Eigenvalues and Eigenvectors
Definition and computation
Eigendecomposition
Diagonalization
1.4 Vector Calculus
Gradients and directional derivatives
Hessian matrices
2. Probability and Statistics
2.1 Probability Theory
Sample spaces and events
Probability axioms and properties
Conditional probability and independence
Bayes' theorem
2.2 Random Variables
Discrete and continuous random variables
Probability mass and density functions
Cumulative distribution functions
Expected value, variance, and standard deviation
2.3 Common Probability Distributions
Discrete: Bernoulli, Binomial, Poisson
Continuous: Uniform, Normal (Gaussian), Exponential
2.4 Descriptive Statistics
Measures of central tendency (mean, median, mode)
Measures of dispersion (variance, standard deviation, range)
Percentiles and quartiles
2.5 Inferential Statistics
Sampling distributions
Central Limit Theorem
Confidence intervals
Hypothesis testing (t-tests, chi-square tests, ANOVA)
2.6 Correlation and Regression
Correlation coefficients (Pearson, Spearman)
Simple linear regression
Multiple linear regression
3. Calculus
3.1 Differential Calculus
Limits and continuity
Derivatives and differentiation rules
Partial derivatives
Gradient, Jacobian, and Hessian
3.2 Integral Calculus
Definite and indefinite integrals
Fundamental theorem of calculus
Multiple integrals
3.3 Optimization
Maxima and minima
Constrained optimization
Lagrange multipliers
4. Information Theory
4.1 Entropy
Shannon entropy
Joint and conditional entropy
4.2 Mutual Information
Definition and properties
Relationship with entropy
4.3 Kullback-Leibler Divergence
Definition and properties
Applications in machine learning
5. Numerical Methods
5.1 Numerical Linear Algebra
Gaussian elimination
LU decomposition
QR decomposition
Singular Value Decomposition (SVD)
5.2 Optimization Algorithms
Gradient descent
Stochastic gradient descent
Newton's method
Quasi-Newton methods (e.g., BFGS)
5.3 Interpolation and Approximation
Polynomial interpolation
Spline interpolation
Least squares approximation
6. Graph Theory
6.1 Basic Concepts
Graphs, vertices, and edges
Directed and undirected graphs
Weighted graphs
6.2 Graph Properties
Connectivity
Cycles and paths
Trees and spanning trees
6.3 Graph Algorithms
Breadth-first search (BFS)
Depth-first search (DFS)
Shortest path algorithms (Dijkstra's, Bellman-Ford)
7. Additional Topics
7.1 Fourier Analysis
Fourier series
Fourier transforms
Applications in signal processing
7.2 Dimensionality Reduction
Principal Component Analysis (PCA)
Singular Value Decomposition (SVD)
7.3 Convex Optimization
Convex sets and functions
Convex optimization problems
Duality
Remember, the depth of understanding required for each topic may vary depending on your specific focus within ML, AI, and DS. As you progress, you may need to delve deeper into certain areas based on your projects and interests.