Week 1: Sets and Logic
Week 2: Equivalence relations, partitions and the quotient construction on sets, functions.
Week 3: Countability, Isomorphisms of sets and the canonical decomposition of maps between sets.
Week 4: Binary operations, the definition, examples and basic properties of groups, rings and integral domains
Week 5: The natural numbers.
Week 6: Fields, the rational, real and complex numbers, polynomials and rational functions
Week 7: System of linear equations, Gaussian elimination, Matrices
Week 8: Vector spaces, examples of vector spaces, subspaces, spans, linear independence, bases and dimension.
Week 9: Linear maps, the rank—nullity theorem and applications.
Week 10: The connection between linear maps and matrices. Change of basis. Dual spaces, quotient spaces.
Week 11: The determinant as a multilinear map. Determinants and volumes, linear operators, eigenvectors and the characteristic polynomial
Week 12: Triangular and diagonal forms, the Jordan form
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