April 25th

#class
Introduction to Vector Spaces in Abstract Algebra. Needed for April 28th and Extension Fields. Need understanding of Introductory Ring Theory prior to reading.

Warm up:

$F={\mathbb{Z}}_2$ <- Note

1. Is $x^2+x+1\in F[x]$ irreducible
2. Is $F[x]/<x^2+x+1>$ a Field?
3. How many elements are in $F[x]/<x^2+x+1>$
   Ans on phone, check all pictures

Class

After warm up
Notion: Classic $GF(p^n)$ = Finite Element w/ $p^n$ elements
GF(4) is a finite field with 4 elements

Modern (preferred):
${\mathbb{F}}_4$
Note, after warm up you can see why this is important.

Abstract Linear Algebra 101:

Vector Spaces"Linear Properties"

1. Vector Spaces
2. Matrices
   1. Representations of linear maps
3. Basis

Def:

Vector Space over F=field is

1. Abelian group ("addition of vectors")
   usually VxV -> V s.t. (v,w) -> v+w
2. Operation of multiplication by elements of F
   (i.e. "multiplication by scalars")
   VFxV -> V st (a,v)-> a*v

Main:

Now we need to show

1. $\alpha (\beta v)=(\alpha \beta )v$
2. $(\alpha +\beta )v=\alpha v+\beta v$
3. $\alpha (u+v)=\alpha u+\alpha v$
4. $1V=V$

1 and 2 are important for final (like group action). Group action is def on final

Def:

"Maps between Vector Spaces" =maps preserving VxV-> V and the action of F
note: some sort of a map that preserve the given structure.

Mathematically:
A linear map $\psi :v\to w$ where $v,w$ are F-vector spaces, is a map S.T:

1. Abelian groups homomorphisms
2. Preservers the multiplication by scalar:$$\psi (av)=a\ast \psi (v)$$

Final: lets introduce some weird new structure that has random maps that etc, those are just all maps that preserve homomorphisms of all structures.

Practically:
$V=<a_1,....,a_n{>}_F$ and $W=<b_1,.....,b_f{>}_F$
then you are thinking of $\psi :v\to w$ as a matrix.
picture on phone