<< Chapter < Page Chapter >> Page >
This module will define what a vector space is and provide useful examples to the reader.

Introduction

Vector space
A vector space S is a collection of "vectors" such that (1) if f 1 S α f 1 S for all scalars α (where α , α , or some other field) and (2) if f 1 S , f 2 S , then f 1 f 2 S
To define an vector space, we need
  • A set of things called "vectors" ( X )
  • A set of things called "scalars" that form a field ( A )
  • A vector addition operation ( )
  • A scalar multiplication operation ( * )
The operations need to have all the properties of givenbelow. Closure is usually the most important to show.

Vector spaces

If the scalars α are real, S is called a real vector space .

If the scalars α are complex, S is called a complex vector space .

If the "vectors" in S are functions of a continuous variable, we sometimes call S a linear function space

Properties

We define a set V to be a vector space if

  1. x y y x for each x and y in V
  2. x y z x y z for each x , y , and z in V
  3. There is a unique "zero vector" such that x 0 x for each x in V (0 is the field additive identity)
  4. For each x in V there is a unique vector x such that x x 0
  5. 1 x x (1 is the field multiplicative identity)
  6. ( c 1 c 2 ) x c 1 ( c 2 x ) for each x in V and c 1 and c 2 in
  7. c x y c x c y for each x and y in V and c in
  8. c 1 c 2 x c 1 x c 2 x for each x in V and c 1 and c 2 in

Examples

  • n real vector space
  • n complex vector space
  • L 1 f t t f t f t is a vector space
  • L f t f ( t )  is bounded f t is a vector space
  • 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]
  • 1 , 2 , are vector spaces
  • The collection of functions piecewise constant between the integers is a vector space

  • + 2 x 0 x 1 x 0 0 x 1 0 x 0 x 1 is not a vector space. 1 1 + 2 , but α α 0 α 1 1 + 2
  • D z z 1 z is not a vector space. z 1 1 D , z 2 D , but z 1 z 2 D , z 1 z 2 2 1

Vector spaces can be collections of functions, collections of sequences, as well as collections of traditionalvectors ( i.e. finite lists of numbers)

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask