Bob Bailey
August 2004
Definition vector space
We call a set V a vector space over a field F (e.g., the field of real numbers, complex numbers, rational numbers, Galois field, etc.) if
1. There is a sum “+” defined between the elements of V (we call these elements “vectors”) that satisfies:
Let ∈ V , then
(a) a + b = b + a (commutative)
(b) ( a + b ) + c = a + ( b + c ) (associative)
2. There is an element 0 of V that satisfies:
(a) a + 0 = a
(b) There exist an element of V , “– a ”, such that a + (– a ) = 0
3. There is a product “ ⋅ ” defined between the elements of F (which we call “scalars”) and the elements of V that satisfy:
λ , µ ∈ F and a , b ∈ V , then
(a) ( λ µ ) ⋅ a = λ ( µ ⋅ a )
(b) ( λ + µ ) ⋅ a = λ ⋅ a + µ ⋅ a
(c) λ ⋅ ( a + b ) = λ ⋅ a + λ ⋅ b
(d) 1 ⋅ a = a
The essential property of vector spaces is that we can form linear combinations of the elements of a vector space. This is guaranteed by the above axioms. Note that it is possible to have a vector space without defining concepts such as distance, length, or angle.
Examples
Vector spaces may be either finite dimensional (the usual meaning ‘vector’) or infinite dimensional
(as a set of infinite sequences or a set of functions). Examples of vector spaces:
1.
2.
A space of n -dimensional real vectors:
The set of matrices M ∈ mn v ∈ n = V
3.
V is the set of all real-valued functions defined on the closed interval [a,b], – ∞ < a < b < + ∞ .
If f , g ∈ V , define h = f + g as a function whose values are h ( t ) = f ( t ) + g ( t ), t ∈ [a, b].
If f ∈ V and λ ∈ , define h = h ( t ) =
λ ⋅ f (or simply λ f ) as the function whose values are
λ ⋅ f ( t ). Let “0” be the function that is identically zero on [a, b]. These definitions, then, establish V as a vector space.

4.
The set of all continuous finite-energy functions known as the L 2 (– ∞ , + ∞ ) space; that is,
−∞
∞ f t 2 dt < ∞ , where | ⋅ | represents the absolute value of a real value or the magnitude of a complex value. These functions are also known as square-integrable functions.
5.
The discrete version of 4: all finite-energy discrete sequences, where f ( t ) is replaced with x n
,
{ x n
| x
1
2 + x
2
2 + < ∞ }where .
This is referred to as the l 2 space. Both L 2 and l 2 are important in signal and image processing; in fact they form the basis of the mathematical domain of continuous (or analog) and digital (or discrete) signal/image processing, respectively.
6.
In the above examples, real values for vectors can be replaced by complex values.
See also the links on Metric, Normed Linear, and Inner Product Spaces for uses of vector spaces.
Links
Metric Spaces
Normed Linear Spaces
Inner Procuct Sspaces
Vector space - Wikipedia, the free encyclopedia