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CS 601: Mathematics for Computer Science

Linear Algebra:
1. Fields.
2. Vectors spaces, examples,Rn, Cn; subspaces.
3. Linear independence, dependence and dimension.
4. Linear transformations.
5. Matrices, matrix algebra, determinants. Properties of matrices and determinants.
6. Systems of linear equations.
7. Eigenvalues, eigenvectors, eigenspaces, diagonalization and the spectral theorem.
8. Factorization and singular value decomposition.
Probability:
1. Sample spaces, events, axioms of probability.
2. Conditional probability and independence.
3. Random variables. Discrete and continuous random variables, densities and distributions.
4. Expectation and its properties.
5. Normal distribution and its properties.
6. Law of large numbers, central limit theorem.
7. Bounds on deviations: Chebyshev, Markov, Hoeffding, Chernoff.
8. Introduction to Markov chains, random walks.
Logic:
1. What is a proof? And proof methods.
2. Propositional logic syntax and semantics.
3. Tautologies, axiom system and deduction.
4. Proof of soundness and completeness.
5. First order logic syntax and semantics.
6. Converting natural language into FoL wffs.
7. Structures, models, satisfaction and validity.
8. Axiomatization, soundness and completeness.
9. Refutation and logic programming.