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Vector Spaces

Module by: Clayton Scott

Summary: This module introduces real and complex vector spaces, with examples. Notions of linear independence, linear span, basis, dimension, subspaces, and direct sums are covered.

Vector spaces are the fundamental object of study in linear algebra, and are prerequisite for the study of many advanced topics in both pure and applied mathematics. Vector spaces generalize the notion of Euclidean space, in which vectors are viewed as arrow emanating from the origin. When the collection of scalars is RR, we call VV a real vector space. When the collection of scalars is CC, we call VV a complex vector space.

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

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

Figure 1
Figure 1 (vecsp_f1.png)
  • 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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