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
.