Algebra I

Finite sets

Maps between sets

Injectivity, surjectivity, bijectivity

Domain, codomain

Composition

Groups

Group of symmetries, permutation group

Group formed by bijective self-maps under composition

Images, pre-images

Cartesian product

Graphs of maps

"Matrix" representations of maps (given a labelling)

Labellings

Relabellings and permutations of matrices

Equivalence Relations

Similarity of maps

Equivalence of maps from X to Y; algorithm to classify maps from X to Y upto equivalence using a normal form

Classification of self-maps from X to X

Classification of bijections from X to X using a matrix consisting of cyclic blocks

Magma

Semigroup

Monoid

Ring

Field

Vector Space

Abelian groups; the set of integers as an abelian group

Family of finite groups Z/nZ

Partitions 💀

Equipping with "multiplication"

Multiplicative inverses

Z mod p where p is prime

Polynomial rings

Z/nZ

Spaces

Field axioms

Function spaces; F^X

Vector space of polynomials with coefficients in F - F[x]

The vector space of F^n

Abelian group structure; axioms

Subspaces

Subspaces of F_p^n

Sums of subspaces

Direct sums of subspaces

Nullspaces

Images

Basis

Linear independence

Spanning lists

Dimension

Basis for the vector space of polynomials with deg <= k using Lagrange interpolation polynomials

Linear maps and Matrices

Gaussian Elimination

Linear functionals; the Dual Space V*

Group homomorphisms