Ma10209 Algebra 1A

Diary of Lectures 2020/21 (Semester 1)

(Lec 1) Mon 28 Sep

• Chap 1: Sets & Functions
• Sets: equality, subsets, empty set.
• Standard number notation, inc Z,Q,R,C.

(Lec 2) Tue 29 Sep am

• Power set. Cardinality.
• Set operations: union, intersection, difference, complement.

(Lec 3) Tue 29 Sep pm

• Ordered pairs & n-tuples.
• Cartesian products and powers

(Lec 4) Mon 5 Oct

• Functions/maps: equality, graphs.
• Composition: associativity.

(Lec 5) Tue 6 Oct am

• Identity & inclusion maps.
• Restriction, image.
• Injective, surjective, bijective.

(Lec 6) Tue 6 Oct pm

• Composition of -jections.
• Inverses: left, right & two-sided.

(Lec 7) Mon 12 Oct

• Pigeonhole principle.
• Defn of "finite" for a set and well-definedness of its size
• Cardinality revisited (defn of "equinumerous")

(Lec 8) Tue 13 Oct am

• Infinite sets
• Countably infinite and countable sets

(Lec 9) Tue 13 Oct pm

• Equivalence relations.
• Partitions.
• Each determines the other.

(Lec 10) Mon 19 Oct

• Chap 2: Numbers & Arithmetic
• Divisibility, primes, prime factorisation.
• Euclid's Theorem: there are infinitely many primes.
• gcd, lcm, coprime

(Lec 10) Tue 20 Oct am

• Bezout's Lemma
• Euclid's Algorithm.
• Finding Bezout coefficients.

(Lec 12) Tue 20 Oct pm

• Euclid's Lemma
• Fundamental Theorem of Arithmetic: uniqueness of prime factorisation
• application of FTA to divisors, lcm, gcd.

(Lec 13) Mon 26 Oct

• Modular arithmetic: congruences and Z_n
• Fermat's Little Theorem

(Lec 14) Tue 27 Oct am

• Invertible elements of Z_n
• Solving a single congruence
• Euler's phi function

(Lec 15) Tue 27 Oct pm

• Euler's theorem
• Chinese Remainder Theorem

(Lec 16) Mon 2 Nov

• Complex numbers: Argand plane
• Arithmetic and geometry
• Complex conjugation
• Cartesian form: real and imaginary parts
• Polar form: modulus and argument
• Product rule for argument.

(Lec 17) Tue 3 Nov am

• De Moivre's theorem
• Roots of unity
• Fundamental Theorem of Algebra (statement only)

(Lec 18) Tue 3 Nov pm

• Exponential notation
• Order and primitive roots of unity.

(Lec 19) Mon 9 Nov

• Chap 3: Polynomials & Matrices
• Field axioms and cancellation law.
• Polynomials as formal expressions, arithmetic, evaluation.

(Lec 20) Tue 10 Nov am

• leading term, degree, monic.
• divisibility
• gcd, lcm, coprime

(Lec 21) Tue 10 Nov pm

• Division Lemma and roots
• Bezout's Lemma
• Irreducibility and Euclid's Lemma
• Unique Factorisation Theorem

(Lec 22) Mon 16 Nov

• Matrices: addition and multiplication
• associativity of multiplication
• matrices as linear maps
• multiplication as composition

(Lec 23) Tue 17 Nov am

• invertible matrices
• 2x2 determinants

(Lec 24) Tue 17 Nov pm

• orthogonal matrices
• 2x2 orthogonal matrices (rotations and reflections)
• 3x3 determinants

(Lec 25) Mon 23 Nov

• Chap 4: Group Theory
• symmetric group S_n : permutations and crossing diagrams
• transpositions and r-cycles
• length and sign of a permutation

(Lec 26) Tue 24 Nov am

• Thm: product rule for sign
• alternating group A_n
• group axioms and examples
• uniqueness of identity and inverses, cancellation
• powers; order

(Lec 27) Tue 24 Nov pm

• Cayley tables
• subgroups: def and examples
• cyclic subgroup, cyclic group
• ord(g) = order of cyclic subgroup generated by g
• product group

(Lec 28) Mon 30 Nov

• homomorphisms; preservation of identity and inverses
• isomorphisms
• image and kernel

(Lec 29) Tue 1 Dec am

• injective hom'm iff trivial kernel
• cosets and Lagrange's Theorem
• ord(g) divides |G|; applications
• (to read: fibres of a hom'm are cosets of the kernel)

(Lec 30) Tue 1 Dec pm

• action of a group G on a set X
• equivalence to hom'm G to Sym(X)
• Cayley's Theorem (every group is isomorphic to a transformation group)
• orbits; G-orbits partition X

(Rev 1) Mon 9 Dec

• Q&A session

(Rev 2) Mon 4 Jan

• Q&A session

(Rev 3) Tue 5 Jan pm

• Q&A session

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