Watched lecture 10 of Abstract Algebra E-222

A lecture by Peter on Linear Algebra. I scanned parts of the video, didn't really like it. Maybe its because I think I am done with this part of Linear Algebra,( have to go through it once more for M208, I suppose ) or its simply because I am interested mainly in groups at the moment.

Anyway, a very abstract and fast talk starting with the definition of a vector space all leading up to the change of basis problem. The change of basis problem is in itself not easy and takes a lot of practive to get comfortable with. A lot of material to go through in just one lecture. It took me months to fully understand these topics. Just to compare: Strang took about 10 lectures to cover this material.
See also: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/index.htm

Improving the study process

Studying is a process which can be improved continuously. There are tons of books written with advice on how to do this. I intend to try to improve my own way of studying. I have found several books on the subject, will browse them and select one or two which I will read through. But before I do that I want to define some key statistics which I can monitor so that I can measure the effects of changes I might implement. - In order to facilitate the monitoring I started to use Personal Brain as a study tool. - ( More later. )

P.S.

I had my life to live over again, I would have made a rule to read some poetry and listen to some music at least once a week; for perhaps the parts of my brain now atrophied would have thus been kept active through use. The loss of these tastes is a loss of happiness, and may possibly be injurious to the intellect, and more probably to the moral character, by enfeebling the emotional part of our nature.
—Charles Darwin

Watched lecture 9 of Abstract Algebra E-222

( Gross is Jewish. )

Nothing much really in this lecture.
Vector spaces.
Spans.
Linear combinations.
Dimensions.

Maybe a hint to a very important notion in mathematics: all vectorspaces of dimension n are isomorphic.

Peter on Monday.

Watched lecture 8 of Abstract Algebra E-222

The following topics are discussed in this lecture:

Isomorphism Theorem

Vector spaces over an arbitrary field
- Definition field
- Examples of finite fields

Proof that Z/pZ is a field
- Added to what we know from Z/nZ as an additive subgroup of Z we must prove that each a in Z/pZ has a multiplicative inverse, so we must show that if a is not a multiple of p then there is an integer b such that a*b congruent 1 mod p.
( Actual proof is worked out in the video )

What are the finite fields beyond Z/pZ ?
- the finite fields are of order p^n where p is a prime and n>=1, so there are finite fields of order 2, 3, 4, 5, 7, 8, 9, 11, etc. ( note 6 = 2*3, 10=2*5 not of type p^n )

Definition of a vector space ( V )
- Additive abelian group
- With a map f: VxF -> V which is called the scalar multiplication
- ( All rules are written down on board. )

Examples of vector spaces
- V={0}
- V=F
- V=F2
- V=Fn
- V=F[X], vector space of all polynomials p(x) with coefficients in F

Vector subspace
- A subgroup 'stable' under scalar multiplication

Vector space homomorphisms
- Linear transformations ( as we knew it ) are explained as group homomorphisms stable under scalar multiplication
- So for T: V-> W we can define the Kernel of T as a subspace of V and the image of T as a subspace of W. We can also define the quotient space V/W analog to the quotient group

Book of beauty: Visual Symmetry

Visual Symmetry by Magdolna and Istvan Margittai is a beautiful book with hundreds of pictures about symmetry. Aftrer reading this book you will look at the world through different eyes, forever. This book is also an excellent conpanion to M336 imo.

Watched lecture 7 of Abstract Algebra E-222

Gross formulated the following question: Can we put a group structure on the set of cosets {aH} for a subgroup H in G? He subsequently based the entire lecture on answering this ( simple ) question. The answer is ( of course ) yes if H is normal in G.

At the end, briefly for students with an interest in Algebraic Topology, Gross mentioned sequences like 1 -> H -> G -> G' -> 1.  With examples 1 -> Z3 -> Z6 -> Z2 -> 1 and 1 -> A3 -> S3 -> {1,-1} -> 1 which show that by knowing  Z3 ~ A3 and Z2 ~ {1,-1} does not say anything about the resultgroup.

Basicly a long abstract theoretical discussion about factorgroups, with basicly zero examples. How will group theory develop in people exposed to such lectures? I am not sure if I want to think about that.

The prototypical mathematician ( is NOT ).

A picture of the prototypical Linux type of person. As one gains experience in Linux one tends to start looking like the Ultimate Nerd. I wish I could understand why. Thank G_d, there are all sorts of mathematicians, maybe it's because mathematics in itself is so rich in subjects that there is no such thing as the prototypical mathematician. - I read somewhere that it is rather not done to say " I am a mathematician", the proper thing is " I studied math ", or something like that. One becomes a mathematician not before others ( in the field ) are calling you one. - I agree there is a difference between simply having done some math courses ( even if they add up to a B.Sc. or M.Sc ) and practicing mathematics at the research level.

Galois Theory

Galois Theory is a topic which is, at least in the algebra books I have, covered in the last chapter as the most beautiful result of algebra. I know that Galois introduced group theory and proved that it was impossible to solve an equation of type f(x)=0, where f(x) has a term of x in the 5th degree or higher, by means of  a formula. ( Solving the quintic by radicals is how it is described. ) What bothers me is that I still can't follow the proof, or worse: I simply don't get it.

I found a hint though. The Galois Group of x^2-1=0 is C2 and of x^4-2=0 the Galois Group is the Dihedral Group of order 8 ( symmetry group of the square ). Will play a bit with these examples, I hope it will break some ice.

Update: the field we work in is Q.

Watched lecture 6 of Abstract Algebra E-222

( A lecture by Peter again ).
Arithmetic congruent mod n.
Addition
Multiplication
How a congruent b mod n is in fact an equivalence relation.
And thus induces a partition of the integers.
Cosets are nZ, 1+nZ, 2+nZ, ... (n-1)+nZ
Addition can be defined on these cosets and then they have a group structure.
The map Z -> nZ is then a homomorphism with 0 as kernel.

( Around min 35 or so I lost interest... I fast forwarded watching minutes here and there, just to make sure there was not introduced anything I did not know already. I hope I am not losing interest in the series all together. We'll see. )

Watched lecture 5 of Abstract Algebra E-222

Defines the equivalence relation on a set as a partition in disjoint subsets whose union is the set.
Properties of an equivalence relation:
- reflexive: a~a
- symmetric: a~b <=> b~a
- transitive: a~b and b~c => a~c.

A homomorphism f: G->H with kernel K which is a normal subgroup of G implies an equivalence relation on G where K is one of the equivalence classes. The other equivalence classes have the form aK = { ak; k in K, for some a in G}. aK is also called a left coset of K. ( Gross writes complete proof of this proposition on board. )
A bit of mathematical history about Lagrange ( born in Italy! ) who writes a letter to Euler at age 17 containing some very sophisticated mathematics. Euler immediately recognizes the genius of Lagrange and arranges further education for Lagrange who until that time learned his math through self-study.
(The famous) Theorem of Lagrange.
If G is a finite group and H is a subgroup of G then the order of H divides the order ( size ) of G.
More propositions are discussed.
- Groups of order p are simple.
- Groups of order p^2 are abelian.
- An is simple for n>=5.
- Any finite, non-abelian group has even order.

( Next lecture Peter. )

Watched lecture 4 of Abstract Algebra E-222

Definition of homomorphism.
Proof that e is mapped to e by any homomorphism.
Proof that inverses are mapped to inverses by any homomorphism.
Definition of Image.
Definition of Kernel.
Properties of the Kernel.
- subgroup;
- normal subgroup.
Any normal subgroup is the kernel of a homomorphism.
Example homomorphism.
f: GL(n,R) -> R_x
f(A) = det(A)
f has as kernel the matrices with det=1, also called SL(n,R). ( Special linear group )
Example homomorphism.
f: Sn->GL(n,R)
f(p)=Ap ( permutation matrix associated with p )
f( (1,2,3) ) = {{0,0,1}, {1,0,0}, {0,1,0} }
Definition center of G.
Example homomorphism G-> Aut(G) i.e. Klein4 -> S3

Watched lecture 3 of Abstract Algebra E-222

( Off-topic: Since I am ' in between jobs ', which happens if you are a freelance IT professional and there in an economic crisis, I should be studying new Oracle features or something like that. Instead I watched another Algebra lecture, well the day is still young. )

Watched lecture 3 of Abstract Algebra E-222. ( A lecture by Peter, Gross's assistant, if he isn't a professor yet, he will be soon, I suppose. )

Review of lectures 1 and 2.
- Groups. And examples of groups GL(n,R), Sn, Z+.
- Subgroups. Cyclic subgroups.
- ( Hom(Rn, Rn) has the structure of a vectorspace. )
- All subgroups of Zn are of the form bZ. ( Emphasis on importance of proof of this proposition.)
- ( Studying the subgroup structure of a group is in general very difficult. )
- Example of a cyclic subgroup of GL(2,R). The group generated by {{1 1}, {0,1}} is {{1 n}, {0,1}} n in Z.

Example of an isomorphism.
G1 = {i, -1, -i, 1}
G2 = {(1,2,3,4}, (1,3),(2,4), (1,4,3,2), ()}
G1 and G2 are isomorphic by i -> (1,2,3,4)
( Permutations are here in cyclic notation which are not introduced in the course yet. )

Example of an isomorphism.
G1 = {R,+}
G2 = {R\{0},*}
G1 and G2 are isomorphic by f: G1->G2; x |-> e^x
Proof:
f(x+y)=e^(x+y)=e^x * e^y = f(x)*f(y).

Klein4 group.
V={() , (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)} as a subgroup of S4.
V={ {{1 0}, {0,1}}, {{-1 0}, {0,1}}, {{1 0}, {0,-1}}, {{-1 0}, {0,-1}} as a subgroup of GL(2,R).

Definitions.
-Automorphism.
-Homomorphism.
-Image ( of a homomorphism)

Next lecture Gross on images of homomorphisms ( and more ).

( Thank you, Peter. )