Vectorspace

2-5-2008

Vector Spaces

Vector spaces and linear transformations are the primary objects of study in linear algebra. A

vector space (which I’ll define below) consists of two sets: A set of objects called vectors and a field (the

scalars).

Definition. A vector space V over a field F is a set V equipped with an operation called (vector) addition,

which takes vectors u and v and produces another vector u + v .

There is also an operation called scalar multiplication, which takes an element a ∈ F and a vector

u ∈ V and produces a vector au ∈ V .

These operations satisfy the following axioms:

1. Vector addition is associative: If u, v, w ∈ V , then

(u + v ) + w = u + (v + w ).

2. Vector addition is commutative: If u, v ∈ V , then

u + v = v + u.

3. There is a zero vector 0 which satisfies

0 + u = u = u + 0 for all

u ∈ V.

4. For every vector u ∈ V , there is a vector −u ∈ V which satisfies

u + (−u) = 0 = (−u) + u.

5. If a, b ∈ F and x ∈ V , then

a(bx) = (ab)x.

6. If a, b ∈ F and x ∈ V , then

(a + b)x = ax + bx.

7. If a ∈ F and x, y ∈ V , then

a(x + y ) = ax + ay.

8. If x ∈ V , then

1 · x = x.

The elements of V are called vectors; the elements of F are called scalars. As usual, the use of words

like “multiplication” does not imply that the operations involved look like ordinary “multiplication”.

Example. If F is a field, then F n denotes the set

F n = {(a1 , . . . , an ) | a1 , . . . , an ∈ F }.

F n is called the vector space of n-dimensional vectors over F . The elements a1 , . . . , an are called the

vector’s components.

1

F n becomes a vector space over F with the following operations:

(a1 , . . . , an ) + (b1 , . . . , bn ) = (a1 + b1 , . . . , an + bn ).

p · (a1 , . . . , an ) = (pa1 , . . . , pan ),

where

p ∈ F.

It’s easy to check that the axioms hold. For example, I’ll check Axiom 6. Let p, q ∈ F , and let

(a1 , . . . , an ) ∈ F n . Then

(p + q )(a1 , . . . , an )

=

((p + q )a1 , . . . , (p + q )an...