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Vector Spaces

Module by: Michael Haag, Steven Cox, Justin Romberg

Summary: This module will define what a vector space is and provide useful examples to the reader.

Introduction
Definition 1: Vector space
A linear vector space S S is a collection of "vectors" such that (1) if f 1 ∈S⇒α f 1 ∈S f 1 S α f 1 S for all scalars αα (where α∈ℝ α or α∈ℂ α ) and (2) if f 1 ∈S f 1 S , f 2 ∈S f 2 S , then f 1 + f 2 ∈S f 1 f 2 S
To define an abstract linear vector space, we need
  • A set of things called "vectors" (X X)
  • A set of things called "scalars" (A A)
  • A vector addition operator (+ )
  • A scalar multiplication operator (* *)
The operators need to have all the properties of given below. Closure is usually the most important to show.

Vector Spaces

If the scalars αα are real, SS is called a real vector space.
If the scalars αα are complex, SS is called a complex vector space.
If the "vectors" in SS are functions of a continuous variable, we sometimes call SS a linear function space

Properties

We define a set V V to be a vector space if
  1. x+y=y+x x y y x for each x x and y y in V V
  2. x+y+z=x+y+z x y z x y z for each x x, y y, and z z in V V
  3. There is a unique "zero vector" such that x+0=x x 0 x for each x x in V V
  4. For each x x in V V there is a unique vector -x x such that x+-x=0 x x 0 .
  5. 1x=x 1 x x
  6. ( c 1 ⁢ c 2 ) x= c 1 ( c 2 ⁢ x ) ( c 1 ⁢ c 2 ) x c 1 ( c 2 ⁢ x ) for each x x in V V and c 1 c 1 and c 2 c 2 in ℂ ℂ.
  7. cx+y=cx+cy c x y c x c y for each x x and y y in V V and c c in ℂ ℂ.
  8. c 1 + c 2 x= c 1 x+ c 2 x c 1 c 2 x c 1 x c 2 x for each x x in V V and c 1 c 1 and c 2 c 2 in ℂ ℂ.

Examples

  • ℝn=real vector space n real vector space
  • ℂn=complex vector space n complex vector space
  • L 1 ℝ={ft|∫-∞∞|ft|dt<∞} L 1 f t t f t f t is a vector space
  • L ∞ ℝ={ft| f ⁡ ( t )  is bounded } L ∞ f t f ⁡ ( t )  is bounded f t is a vector space
  • L 2 ℝ={ft|∫-∞∞|ft|2dt<∞}=finite energy signals L 2 f t t f t 2 f t finite energy signals is a vector space
  • L 2 0T =finite energy functions on interval [0,T] L 2 0 T finite energy functions on interval [0,T]
  • ℓ 1 ℤ ℓ 1 , ℓ 2 ℤ ℓ 2 , ℓ ∞ ℤ ℓ ∞ are vector spaces
  • The collection of functions piecewise constant between the integers is a vector space
vecsp_f1.png
Figure 1
  • ℝ + 2={ x 0 x 1 | x 0 >0∧ x 1 >0} ℝ + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 11∈ ℝ + 2 1 1 ℝ + 2 , but ∀α,α<0:α11∉ ℝ + 2 α α 0 α 1 1 ℝ + 2
  • D=∀z,|z|≤1:z∈ℂ D z z 1 z is not a vector space. z 1 =1∈D z 1 1 D , z 2 =ⅈ∈D z 2 D , but z 1 + z 2 ∉D z 1 z 2 D , | z 1 + z 2 |=2>1 z 1 z 2 2 1
note: Vector spaces can be collections of functions, collections of sequences, as well as collections of traditional vectors (i.e. finite lists of numbers)

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