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

1-2017 More on the randomness of randomness.

2 months ago

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