Definition (Vector Space)
A (real) vector space V is a non-empty set equipped with an addition
operation and a scalar multiplication operation such that for all α, β R ∈
and all u, v, w V: ∈
1. u + v V (closure under addition). ∈
2. u + v = v + u (the commutative law for addition).
3. u + (v + w) = (u + v) + w (the associative law for addition).
4. there is a single member 0 of V, called the zero vector, such that for all
v V, v + 0 = v. ∈
5. for every v V there is an element w V (usually written as −v), ∈ ∈
called the negative of v, such that v + w = 0.
6. αv V (closure under scalar multiplication). ∈
7. α(u + v) = αu + αv (distributive law for vectors).
8. (α + β)v = αv + βv (distributive law for scalars).
9. α(βv) = (αβ)v (associative law for scalar multiplication).
10. 1v = v.
Exercise A
Determine if the following are vector spaces.
(1) Let V be the set of all real numbers x such that x
≥
0. Define an
operation of addition by x Å y = xy + 1 for all x, y
∈
V. Define an
operation of “scalar multiplication” by
α
ʘ x =
α2
x for all
α∈
R and
x
∈
V.
(2) Let V be R2, the set of all ordered pairs (x,y) of real numbers.
Define an operation of addition by (u,v) Å (x,y) = (u + x + 1, v + y
+ 1) for all (u,v) and (x,y) in V. Define an operation of scalar
multiplication by
α
ʘ(x,y) = (
α
x,
α
y) for all
α
∈
R and (x,y)
∈
V.
(3) Let V be R2, the set of all ordered pairs (x,y) of real numbers.
Define an operation of addition by (u,v) Å (x,y) = (u +x, 0) for all
(u,v) and (x,y) in V. Define an operation of scalar multiplication by
α
ʘ(x,y) = (
α
x,
α
y) for all
α
∈
R and (x,y)
∈
V.
(4) Let V be the set of all n x n matricesof real entries. Define an
operation of addition by AÅ B =
1
2
(AB + BA) for all A, B
∈
V.
Define an operation of scalar multiplication by
α
ʘA = 0 for all
α
∈
R and A
∈
V.
