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Vector Spaces

By Michael Haag, Steven J. Cox, Justin Romberg

Introduction

Definition 1: Vector space
A vector space S S is a collection of "vectors" such that (1) if f 1 Sα f 1 S f 1 S α f 1 S for all scalars αα (where αR α , αC α , or some other field) 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 vector space, we need
  • A set of things called "vectors" (X X)
  • A set of things called "scalars" that form a field (A A)
  • A vector addition operation ( )
  • A scalar multiplication operation (* *)
The operations 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 (0 is the field additive identity)
  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 (1 is the field multiplicative identity)
  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
  7. c(x+y)=cx+cy c x y c x c y for each x x and y y in V V and c c in
  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

Examples

  • Rn=real vector space n real vector space
  • Cn=complex vector space n complex vector space
  • L 1 R= ft ft |ft|dt< L 1 f t t f t f t is a vector space
  • L R= ft ft f ( t )  is bounded L f t f ( t )  is bounded f t is a vector space
  • L 2 R= ft 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 0 T =finite energy functions on interval [0,T] L 2 0 T finite energy functions on interval [0,T]
  • 1 Z 1 , 2 Z 2 , Z are vector spaces
  • The collection of functions piecewise constant between the integers is a vector space

Figure 1
  • + 2= x 0 x 1 x 0 x 1 ( x 0 >0)( x 1 >0) + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 11 + 2 1 1 + 2 , but α,α<0:α11 + 2 α α 0 α 1 1 + 2
  • D= z ,|z|1:zC 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)