# Abstract Algebra in 3 Hours

Ashwin RaoVice President, Data Science & Optimization at Target à Target
1 sur 10

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### Abstract Algebra in 3 Hours

• 1. ABSTRACT ALGEBRA IN 3 HOURS! Ashwin Rao Meant to be a quick preparation for learning Category Theory
• 2. Overview of Preliminaries • Set: unordered and unique elements • Cartesian Product of Sets • Relation: A subset of a cartesian product • Reflexive, Symmetric, Transitive Relation on a set ó Equivalence Classes (Partition) • Partially Ordered Set: Reflexive, Anti-symmetric and Transitive • Function: Just a relation on A x B with every a in A mapped to a single b in B • Domain, Codomain, Range, Injective, Surjective, Bijective functions • Inverse and Composition of functions
• 3. Semigroup • A set with an operation (*) under which the set is closed, along with associativity. • Associativity: a * (b * c) = (a * b) * c • Commutativity a * b = b * a is fairly common, but not part of semigroup definition. • Canonical Example: Positive Integers Z+ with operation as + or * • Funky Example: Integers with Min or Max operation. • Example: Free semigroup of an alphabet (List[T] except empty list, with concat) • Or, List[T] of length n, for any n in Z+ • Eg: Set of Functions f : X -> X with composition (think “shrinking” functions) • Sub-Semigroup example: nZ+, for n in Z+ • Semigroup homomorphism (structure-preserving) f: G -> H : f(a *G b) = f(a) *H f(b)
• 4. Monoid • Semigroup together with an identity element (call it “1”) • Canonical Example: Natural numbers N with + as * , 0 as 1 • or, Z+ with * as * and 1 as 1 • Example: Free monoid of an alphabet (List[T] with concat) • Or, List[T] of length n, for any n in N. • Eg: {True, False} with AND as *, True as 1 (or with OR as *, False as 1) • Eg: All subsets of a set S with Union as *, Empty as 1 (or Intersect as *, S as 1) • Note: Cartesian product of monoids is a monoid • Note: All functions from a set to a monoid form a monoid (pointwise operation) • Eg: All Functions f: X -> X for any set X, with composition as * and identity function as 1
• 5. Monoid (continued) • Submonoid example: nN • Monoid homomorphism f: G -> H : f(a *G b) = f(a) *H f(b) and f(1G) = 1H • Example: f(x) = 2x from (N,+,0) to (N,*,1) • Isomorphism is when we have homomorphisms f: G -> H and g: H -> G such that g . f = idG and f . g = idH • Isomorphism means the two monoids are “basically the same” • Kernel(f) = {a in G | f(a) = 1H} is a monoid • Isomorphism can also be defined as a homomorphism f with Kernel(f) = {1G} • Note: The f(x) = 2x example is an isomorphism
• 6. Group • Monoid together with an inverse a-1 for every a such that a * a-1 = a-1 * a = 1 • Canonical Example: Z • Eg: Bijective functions f : X -> X for any set X with {func composition, identity func, inverse func} • Great Example: All Permutations of a finite set of size n (refered to as Sn) • Eg: n-th complex root of unity zn and its powers (zn is called the generator of the group) • Example of subgroup: nZ for any n in Z+ • Homomorphism f: G -> H: f(a *G b) = f(a) *H f(b), f(1G) = f(1H), f(a-1) = f(a)-1, eg: Z -> nZ • Coseta,H for any a in G and any subgroup H if defined as: {a + h: h in H} • Quotient Group: G/H is a group consisting of all the cosets of H (H becomes identity element) • Canonical Example of Quotient Group: Z / nZ = Zn (Integers modulo n for any n in Z+) • Isomorphism is same as defined for a monoid (isomorphism means “basically the same group“) • First Isomorphism Theorem: Homomorphism f: G -> H, Kernel(f) is a subgroup of G, Range(f) is a subgroup of H, G/Kernel(f) is isomorphic to Range(f)
• 7. Semiring and Ring • Semiring has two monoid operations (*,1) and (+,0) with a * (b + c) = (a * b) + (a * c), (a + b) * c = (a * c) + (b * c), and 0 * a = a * 0 = 0. Moreover, + is commutative. • Canonical Example: N • Ring is a semiring with + operation having an inverse (i.e., a group under +) • Ring Homomorphism means homomorphism under both + and * • Canonical Example: Z • Another Canonical Example: Polynomials over R • Ideal I is a subset of Ring R s.t. for any x, y in I and r in R, x + y and r * x are in I • Canonical Example of Ideal: nZ • R / I is a ring (Quotient Ring) consisting of all the cosets of I s.t. (a+I)+(b+I) = (a+b)+I and (a+I)*(b+I) = (ab)+I
• 8. Field • Field is a ring with an inverse for *, and * commutative. • Canonical Example: Rational Numbers Q or Real Numbers R • Finite Field Example: Zp for any prime p • Every finite field is isomorphic to the set of polynomials over the finite field Zp modulo an irreducible polynomial (over Zp) • Hence, finite fields are of size pr (r is the degree of the irreducible polynomial)
• 9. Vector Space and Linear Map • Vector Space V (associated with scalar Field F) is a commutative group under vector addition, together with scalar multiplication, and the following properties: o a(bv) = (ab)v o 1(v) = v o a(u+v) = au + av o (a+b)v = av + bv • Canonical Example: Rn • Eg: Complex numbers and other field extensions • Eg: Functions from a set X to a field F (pointwise addition and pointwise scalar multiplication) • Linear Map f: V -> W has property f(v+w) = f(v) + f(w) and f(a.x) = a.f(x) • Canonical Example: m by n Matrix M: Rn -> Rm • Linear maps V -> W forms a vector space L(V,W) • Linear maps V -> F (F the scalar Field) is called the Dual Vector Space V*
• 10. Fundamental Theorem of Linear Algebra • Consider a linear map expressed as a m x n matrix M : Rn -> Rm • Column Space (Range): Subspace of Rm consisting of all Mx (over all x in Rn) • Row Space (CoRange): Subspace of Rn consisting of all MTy (over all y in Rm) • Kernel Space: Subspace of Rn mapping (through M) to 0 in Rm • CoKernel Space: Subspace of Rm mapping (through MT) to 0 in Rn • Rank r is defined as the dimension of Column Space(= Dimension of Row Space) • Kernel Space is orthogonal to Row Space and has rank n – r (a.k.a. Nullity) • CoKernel Space is orthogonal to Column Space and has rank m – r (a.k.a. CoRank) • More generally, we know from the First Isomorphism Theorem (on Groups) that the Kernel Quotient (i.e., Row Space) and Range (i.e., Column Space) are isomorphic.
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