Algebra 3

Question 1

Holomorphism or Linear Map

Answer 1

A function between vector spaces that preserves the operations of addition and multiplication

Question 2

Isomorphism

Answer 2

2 vector spaces V and W over the same field are said to be isomorphic if there is a bijection T:V→W which preserves addition and scalar multiplication

T(u+v) = T(u) + T(v)
T(aV) = aT(V) 

T is an isomorphism of vector spaces

Question 3

Inner Product

Answer 3

A generalisation of the dot product ⇒ multiplying vectors in a vector space with the result being scalar

4 properties:
1. <u> = <u> +  
2.  = a
3.  =  
4.  ≥0 and  = = iff. v = 0 
</u></u>

Question 4

Irreducible Polynomial

Answer 4

A polynomial f∈k[x] is irreducible if:

1. Deg(f) ≥ 1
2. If f=gh, then deg(g) = 0 or deg(h) = 0

Irreducible polynomials are analogous of prime numbers in ℤ