MATH 423: Vector Spaces
Definition:
Let F be a field.
A vector space over F
is a set V on which two operations + and · are defined,
called vector addition and scalar multiplication.
The operation + (vector addition) must satisfy the following conditions:

Closure:
If u and v are any vectors in V, then the sum
u + v
belongs to V.
 (1)
Commutative law:
For all vectors u and v in V,
u + v = v + u
 (2)
Associative law:
For all vectors u, v, w in V,
u + (v + w)
= (u + v) + w
 (3)
Additive identity:
The set V contains an additive identity element,
denoted by 0,
such that for any vector v in V,
0 + v = v
and
v + 0 = v.
 (4)
Additive inverses:
For each vector v in V, the equations
v + x = 0
and
x + v = 0
have a solution x in V,
called an additive inverse of v,
and denoted by  v.
The operation · (scalar multiplication)
is defined between elements of F and vectors,
and must satisfy the following conditions:

Closure:
If v in any vector in V,
and c is any element of F, then the product
c · v
belongs to V.
 (5)
Distributive law:
For all elements c in F and all vectors
u, v in V,
c · (u + v)
= c · u
+ c · v
 (6)
Distributive law:
For all elements c,d in F and all vectors v in V,
(c+d) · v
= c · v
+ d · v
 (7)
Associative law:
For all elements c,d in F and all vectors v in V,
c ·
(d · v)
= (cd) · v
 (8)
Unitary law:
For all vectors v in V,
1 · v = v
Subspaces
Definition:
Let V be a vector space over F, and let W be a subset of V.
If W is a vector space over F with respect to the operations in V,
then W is called a subspace of V.
Theorem:
Let V be a vector space,
with operations + and ·,
and let W be a subset of V.
Then W is a subspace of V if and only if the following conditions hold.
 Sub0
W is nonempty:
The zero vector belongs to W.
 Sub1
Closure under +:
If u and v
are any vectors in W, then
u + v
is in W.
 Sub2
Closure under ·:
If v is any vector in W, and c is any element of F, then
c · v
is in W.