MATH 240: Vector Spaces
Definition:
A vector space
is a set V on which two operations + and · are defined,
called addition and scalar multiplication.
The operation + (vector addition) must satisfy the following conditions:
-
Closure:
For all vectors u and v in V, 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 all vectors 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)
must satisfy the following conditions:
-
Closure:
For all real numbers c and all vectors v in V, the product
c · v
belongs to V.
- (5)
Distributive law:
For all real numbers c and all vectors
u, v in V,
c · (u + v)
= c · u
+ c · v
- (6)
Distributive law:
For all real numbers c, d and all vectors v in V,
(c+d) · v
= c · v
+ d · v
- (7)
Associative law:
For all real numbers c,d 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, and let W be a subset of V.
If W is a vector space 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 real number, then
c · v
is in W.