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


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


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


  • 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)

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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