Lectures in course MATH-110(a): Advanced linear algebra I - vector spaces

Introduces the basics of polynomials on a field, focusing on definitions, operations, and properties.

Covers abbreviated calculation, structures, operations, and unique set properties.

Explores the properties and applications of polynomials on a field, including formal derivation and uniqueness.

Explores binary relations, Cartesian products, and applications between sets, emphasizing their properties and significance.

Introduces irreducible polynomials and the Gaussian lemma for polynomial factorization.

Covers applications between sets, functions, and graphs, exploring injectivity, surjectivity, and bijectivity.

Covers the construction of polynomials on a field and the concept of minimal polynomials.

Explores the relationships between functions and sets, focusing on composite functions properties.

Explores cardinality in sets and properties of groups, including commutativity and examples of abstract groups.

Covers properties and notation of the trivial group and discusses the uniqueness of elements.

Introduces the concept of a group and its properties through illustrative examples.

Covers powers, multiples, subgroups, and stabilizers in group theory.

Covers subgroups, generators, cyclic groups, and group morphisms in group theory.

Covers the concept of morphism of groups, actions on sets, and automorphisms.

Explores group morphisms, injectivity criteria, and conjugation concepts within groups.

Covers the ring structure, focusing on polynomials and coefficients, including associativity, distributivity, and the product of rings.

Covers the ring of polynomials, focusing on coefficients and multiplication.

Introduces sub-rings and morphisms in ring theory, emphasizing stability by multiplication.

Covers commutative ring morphisms, subrings, and injective ring morphisms.

Explains submodules, generated modules, and ring ideals in module structure.