Math 55a Lecture Notes - Evan Chen

1 math 55a Lecture NotesEvan ChenFall 2014 This is Harvard College s famousMath 55a, instructed by Dennis formal name for this class is Honors Abstract and Linear Algebra butit generally goes by simply math 55a .The permanent URL ~ , along with all my other course 2, Boring stuff .. Functions .. Equivalence relations ..62 September 4, Review of equivalence relations go here .. Universal property of a quotient .. Groups .. Homomorphisms ..83 September 9, Direct products .. Commutative diagrams .. Sub-things .. Let s play Guess the BS! .. Kernels .. Normality .. Examples of normal groups .. 124 September 11, Rings .. Ring homomorphisms .. Modules, and examples of modules .. Abelian groups areZ-modules .. Homomorphisms ofR-modules .. Matrices .. Sub-modules and Ideals.

2 165 September 16, Review .. Direct Sums of Modules .. Direct Products of Modules .. 181 Evan Chen(Fall 2014) Sub-Modules .. Free Modules .. Return to the Finite .. 216 September 18, Linearly independent, basis, span .. Dimensions and bases .. Corollary Party .. Proof of Theorem .. 267 September 23, Midterm Solutions .. Endomorphisms .. Given a map we can split into invertible and nilpotent parts .. Eigen-blah .. Diagonalization .. 328 September 25, Eigenspaces .. Generalized eigenspaces .. Spectral Theorem .. Lemmata in building our proof .. Proof of spectral theorem .. Recap of Proof .. 379 September 30, Review .. Taking polynomials of an endomorphism .. Minimal Polynomials .. Spectral Projector.

3 Polynomials .. 4010 October 2, Jordan Canonical Form .. A big proposition .. Young Diagrams .. Proof of Existence .. 4411 October 7, Order of a Group .. Groups of prime powers .. Abelian groups and vector spaces are similar .. Chinese Remainder Theorem .. Not algebraically closed .. 4912 October 9, Group Actions .. How doG-sets talk to each other? .. Common group actions .. More group actions .. Transitive actions .. 532 Evan Chen(Fall 2014) Orbits .. Corollaries of Sylow s Theorem .. Proof of (b) of Sylow s Theorem assuming (a) .. 5513 October 14, Proof of the first part of Sylow s Theorem .. Abelian group structure on set of modules .. Dual Module .. Double dual .. Real and Complex Vector Spaces .. Obvious Theorems .. Inner form induces a map.

4 5814 October 16, Artificial Construction .. Orthogonal Subspace .. Orthogonal Systems .. Adjoint operators .. Spectral theory returns .. Things not mentioned in class that any sensible person should know .. Useful definitions from the homework .. 6415 October 21, Generators .. Basic Properties of Tensor Products .. Computing tensor products .. Complexification .. 6716 October 23, Tensor products gain module structure .. Universal Property .. Tensor products of vector spaces .. More tensor stuff .. Q & A .. 7217 October 28, Midterm Solutions .. Problem 1 .. Problem 2 .. Problem 3 .. The space nsub(V) .. The space nquot(V) .. The Wedge Product .. Constructing the Isomorphism .. Why do we care? .. 8018 October 30, Review .. Completing the proof that nsub(V) = nquot(V).

5 Wedging Wedges .. 833 Evan Chen(Fall 2014)Contents19 November 4, Representations .. Group Actions, and Sub-Representations .. Invariant Subspaces .. Covariant subspace .. Quotient spaces and their representations .. Tensor product of representations .. 8720 November 6, Representations become modules .. Subrepresentations .. Schur s Lemma .. Splittings .. Table of Representations .. Induced and Restricted Representations .. 9321 November 11, Review .. Homework Solutions .. A Theorem on Characters .. The Sum of the Characters .. Re-Writing the Sum .. Some things we were asked to read about .. 9822 November 13, Irreducibles .. Products of irreducibles .. Regular representation decomposes .. Function invariants .. A Concrete Example .. 10323 November 18, Review.

6 The symmetric group on five elements .. Representations ofS5/(S3 S2) finding the irreducible .. Secret of the Young Diagrams .. The General Theorem .. 10824 November 20, Reducing to some Theorem with Hom s .. Reducing to a Combinatorial Theorem .. Doing Combinatorics .. 11125 December 2, 201411326 December 4, Categories .. Functors .. Natural Transformations .. 1154 Evan Chen(Fall 2014)1 September 2, 2014 1 September 2, 2014 Boring stuffSets includeR,Z, et cetera. A subsetY Xis exactly what you think it ,{0},{1}, ,R R. X2,X1 .. Gaitsgory what are you doingFor a fixed universeX, we writeY,X\Y,X Yfor{x X|x / Y}.Lemma X,(Y)= this is being written out?x (Y) x / Y x (Y)= (X1 X2) =X1 X1 X2 x / X1 X2 x / X1 x / X2 x X1 x X2 x X1 X2=X1 But this is trivial and follows either from calculation or from applying theprevious two a setXwe can consider its power setP(X).

7 It has 2nelements. FunctionsGiven two setsXandYa map (or function)Xf Yis an assignment x Xto anelementfx :X={55 students},Y=Z. Thenf(x) =$in cents(which can benegative).Definition functionfis injective (or a monomorphism) ifx6=y= fx6= functionfis surjective (or an epimorphism) if y Y x X:fx= ;Xf Yg Chen(Fall 2014)1 September 2, 2014 Equivalence relationsAn equivalence relation must be symmetric, reflexive, and transitive. A relation willpartition its setXinto cosets or equivalence classes. (The empty set is not a coset.)Lemma a set and an equivalence relation. Then for anyx Xthere exists aunique cosetxwithx tedious manual we can take quotientsX/ , and we have projections :X X/ .6 Evan Chen(Fall 2014)2 September 4, 2014 2 September 4, 2014 Review of equivalence relations go heremeow Universal property of a quotientXf-YX/ ? f-Proposition sets and an equivalence relation onX.

8 Letf:X Ybe afunction which preserves , and let denote the projectionX X/ . Prove thatthere exists a unique function fsuch thatf= f .The uniqueness follows from the following obvious g?f2-f1-Lemma the above commutative diagram, ifgis surjective thenf1= usegto get everything equal. Yay. GroupsDefinition a setGendowed with an associative1binary operation :G2 of groups (Starfish)LetGbe an arbitrary set and fix ag0 G. Then letab=g0for anya,b G. Thisis a baby starfish has|G|= (ab)c=a(bc)7 Evan Chen(Fall 2014)2 September 4, 2014 Definition semi-groupGis amonoidif there exists an identity 1 Gsuch that g G,g 1 = 1 g= identity of any semi-groupGis 1, 1 be identities. Then1 = 1 1 = 1 .Definition a monoidGwith inverses: for anyg Gthere existsg 1withgg 1=g 1g= are ,x2are both inverses ofg. Thenx1=x1gx2= group isabelianif it is commutative. HomomorphismsDefinition groups.

9 Agroup homomorphismis a mapf:G Hthat preserves Chen(Fall 2014)3 September 9, 2014 3 September 9, 2014 Direct productsGiven two setsXandYwe can define the direct productX Y={(x,y)|x X,y Y}.For example,R2is the Euclidean the operation of a semigroup should be thought of asG Gmult :Y1 Y2we define idf f:X Y1 X Y2by(x,y1)7 (x,fy1). Commutative diagramsWe can then rephrase associativity using the following commutative G GidG -G GG G idG? -G ?We can also rephrase homomorphisms as follows: given :G Hwe require thefollowing diagram to G -H HG G? -H H? Sub-thingsDefinition a semigroup / monoid / group. We sayH Gis asub-semigroupifh1,h2 H= h1h2 H. Moreover, ifGis a monoid and 1 HthenHis asub-monoid. Finally, ifHis closed under inverses as well thenHis the additive group of integers. ThenNis a sub-semigroup,Z 0is asub-monoid, and 2 Zis a GandH2 Gare subgroups, thenH1 H2is a Chen(Fall 2014)3 September 9, 2014 Let s play Guess the BS!

10 In what follows I ll state some false statements. You will be asked to prove them at yourown risk. Lemma :G Ha homomorphism, (G) is a subgroup any (a), (b) in (G) we have (a) (b) = (ab) (G).Then use (1) = 1 to get the rest of the :G HandH H, then 1(H ) is a subgroup one turns out to be ,H2subgroups ofG,H1 H2need not be a subgroup ,H1= 100Z,H2= 101Z. KernelsDefinition a homomorphism :G H, thekernelker is defined byker = 1({1}).Proposition :G Hbe a homomorphism. Then is injective as a map of sets if andonly if ker ={1}. all cases 1 ker . If|ker |6= 1 then clearly is not injective. On the otherhand, supposeker ={1}. If a= bwe get (ab 1) = 1, so if we must haveab 1= 1ora= a group and letH Gbe a subgroup. We define the rightequivalence relation onGwith respect toH ras follows:g1 rg2if h Hsuch thatg2= ` check this is actually an equivalence relation, note that1 H= g rgandg1 g2= g1=g2h= g1h 1=g2= g2 , ifg1=g2h andg2=g3h theng1=g3(h h ), so transitivity works as thatg1 rg2 g 11g2 the set of equivalence classes ofGwith respect to Chen(Fall 2014)3 September 9, 2014 NormalityDefinition a subgroup ofG.