Chapter 1 & 2 definitions

Question 1

Define a field

Answer 1

Let F be a set in which two operations + : F ⇥ F ! F (called
addition) and · : F ⇥ F ! F (called multiplication) are defined. That is, to each pair
(x, y) of elements of F, there correspond elements x+y and x ·y of F. Then F is said
to be a field if the following axioms are satisfied:

(1) (x + y) + z = x + (y + z) for all x, y, z 2 F (associativity of addition)
(2) x + y = y + x for all x, y 2 F (commutativity of multiplication)
(3) there is an element 0 2 F such that 0 + x = x for all x 2 F (additive identity)
(4) for every element x 2 F there is an element −x 2 F such that (−x) + x = 0
(additive inverse)
(5) (xy)z = x(yz) for all x, y, z 2 F (associativity of multiplication)
(6) xy = yx for all x, y 2 F (commutativity of multiplication)
(7) there is an element 1 2 F \ {0} such that 1x = x for all x 2 F (multiplicative
identitiy for F \ {0})
(8) for every element x 2 F \ {0} there is an element x−1 2 F such that x−1x = 1
(multiplicative inverse in F \ {0})
(9) x(y + z) = xy + xz for all x, y, z 2 F (distributivity)

Question 2

Define a vector space:

Answer 2

Definition 2.1 A vector space V over a field F is a set V with two operations: vector
addition + : V ×V → V ; and scalar multiplication · : F×V → V . These operations
are required to obey the following eight axioms:
(1) (a + b) + c = a + (b + c) for all a, b, c ∈ V (associativity of addition)
(2) a + b = b + a for all a, b ∈ V (commutativity of addition)
(3) there is a vector θ ∈ V such that θ + a = a for all a ∈ V (additive identity)
(4) for every vector a ∈ V there is a vector −a ∈ V such that (−a) + a = θ
(additive inverse)
(5) α(βa) = (αβ)a for all α, β ∈ F and a ∈ V (associativity of scalar multiplication)
(6) 1a = a for all a ∈ V (multiplication by multiplicative unit of F)
(7) (α + β)a = αa + βa for all and α, β ∈ F and a ∈ V (distributivity)
(8) α(a + b) = αa + αb for all α ∈ F and a, b ∈ V (distributivity).

Question 3

Define a subspace

Answer 3

A nonempty subset, S ⊆ V, of V is a subspace it satisfies the following conditions:
i. x+y ∈ S whenever x,y ∈ S,
ii. ax ∈ S whenever x ∈ S and a ∈ F,
iii. 0 ∈ S.

Question 4

Define span

Answer 4

For every subset S ⊆ V , the span of S, denoted <S>, is the set of all
vectors of V which can be expressed as a linear combination of vectors from S</S>