vector spaces recap

Question 1

binary operation

Answer 1

a rule under which each ordered pair of elements of V is associated with a unique element

Question 2

group

Answer 2

a set v with a binary operation * where

(1) * is an internal operation (V is closed under *)
(2) * is associative
(3) * has an identity
(4) all elements of V have an inverse element

Question 3

abelian group

Answer 3

a group where * is commutative

Question 4

field

Answer 4

a set V with binary operations +, * containing 0 and 1 where

(1) 0!=1
(2) (a+b)+c = a + (b+c) (associativity)
(3) a+b=b+a (commutativity)
(4) a+0=a (identity)
(5) there exists and x s.t. a+x=0 (inverse)
(6) (ab)c= a(bc) (associativity)
(7) ab = ba (commutativity)
(8) a1=a (identity)
(9) there exists an x s.t. ax=1 (inverse)
(10) a(b+c)=(ab)+(a*c) (distributivity)

Question 5

vector space

Answer 5

A set V with binary operation # and scalar multiplication . . with elements from a field F,+,* is a vector space is

(1) V, # is an abelian group (with identity 0)
(2) V is closed under .
(3) scalar multiplication w.r.t # in V is distributive
(4) Scalar multiplcaion w.r.t. + in V is distributive
(5) mixed associativty for . and * i.e. a . (b . c) = (a * b) . c a in V b,c in F
(6) The identity for F{0} is also the identity for scalar multiplication