2Vectors

IA Vectors and Matrices

2.1 Definition and basic properties

Definition

(Vector)

.

A vector space over

R

or

C

is a collection of vectors

v ∈ V

,

together with two operations: addition of two vectors and multiplication of a

vector with a scalar (i.e. a number from R or C, respectively).

Vector addition has to satisfy the following axioms:

(i) a + b = b + a (commutativity)

(ii) (a + b) + c = a + (b + c) (associativity)

(iii) There is a vector 0 such that a + 0 = a. (identity)

(iv) For all vectors a, there is a vector (−a) such that a + (−a) = 0 (inverse)

Scalar multiplication has to satisfy the following axioms:

(i) λ(a + b) = λa + λb.

(ii) (λ + µ)a = λa + µa.

(iii) λ(µa) = (λµ)a.

(iv) 1a = a.

Often, vectors have a length and direction. The length is denoted by

|v|

. In

this case, we can think of a vector as an “arrow” in space. Note that

λa

is either

parallel (λ ≥ 0) to or anti-parallel (λ ≤ 0) to a.

Definition

(Unit vector)

.

A unit vector is a vector with length 1. We write a

unit vector as

ˆ

v.

Example. R

n

is a vector space with component-wise addition and scalar mul-

tiplication. Note that the vector space

R

is a line, but not all lines are vector

spaces. For example,

x

+

y

= 1 is not a vector space since it does not contain

0

.