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Vector spaces are the principal object of study in linear algebra. A vector space is always defined with respectto a field of scalars.


A field is a set F equipped with two operations, addition and mulitplication, and containing two special members 0 and 1( 0 1 ), such that for all a b c F

    • a b F
    • a b b a
    • ( a + b ) + c a + ( b + c )
    • a 0 a
    • there exists a such that a a 0
    • a b F
    • a b b a
    • a b c a b c
    • a 1 a
    • there exists a such that a a 1
  • a b c a b a c
More concisely
  • F is an abelian group under addition
  • F is an abelian group under multiplication
  • multiplication distributes over addition



Vector spaces

Let F be a field, and V a set. We say V is a vector space over F if there exist two operations, defined for all a F , u V and v V :

  • vector addition: ( u , v ) u v V
  • scalar multiplication: ( a , v ) a v V
and if there exists an element denoted 0 V , such that the following hold for all a F , b F , and u V , v V , and w V
    • u + ( v + w ) ( u + v ) + w
    • u v v u
    • u 0 u
    • there exists u such that u u 0
    • a u v a u a v
    • a b u a u b u
    • a b u a b u
    • 1 u u
More concisely,
  • V is an abelian group under plus
  • Natural properties of scalar multiplication


  • N is a vector space over
  • N is a vector space over
  • N is a vector space over
  • N is not a vector space over
The elements of V are called vectors .

Euclidean space

Throughout this course we will think of a signal as a vector x x 1 x 2 x N x 1 x 2 x N The samples x i could be samples from a finite duration, continuous time signal, for example.

A signal will belong to one of two vector spaces:

Real euclidean space

x N (over)

Complex euclidean space

x N (over)


Let V be a vector space over F .

A subset S V is called a subspace of V if S is a vector space over F in its own right.

V 2 , F , S any line though the origin .

S is any line through the origin.

Are there other subspaces?

S V is a subspace if and only if for all a F and b F and for all s S and t S , a s b t S

Linear independence

Let u 1 , , u k V .

We say that these vectors are linearly dependent if there exist scalars a 1 , , a k F such that

i 1 k a i u i 0
and at least one a i 0 .

If only holds for the case a 1 a k 0 , we say that the vectors are linearly independent .

1 1 -1 2 2 -2 3 0 1 -5 7 -2 0 so these vectors are linearly dependent in 3 .

Spanning sets

Consider the subset S v 1 v 2 v k . Define the span of S < S > span S i 1 k a i v i a i F

Fact: < S > is a subspace of V .

V 3 , F , S v 1 v 2 , v 1 1 0 0 , v 2 0 1 0 < S > xy-plane .

< S > is the xy-plane.


If S is infinite, the notions of linear independence and span are easily generalized:

We say S is linearly independent if, for every finite collection u 1 , , u k S , ( k arbitrary) we have i 1 k a i u i 0 i a i 0 The span of S is < S > i 1 k a i u i a i F u i S k

In both definitions, we only consider finite sums.


A set B V is called a basis for V over F if and only if

  • B is linearly independent
  • < B > V
Bases are of fundamental importance in signal processing. They allow us to decompose a signal into building blocks (basisvectors) that are often more easily understood.

V = (real or complex) Euclidean space, N or N . B e 1 e N standard basis e i 0 1 0 where the 1 is in the i th position.

V N over. B u 1 u N which is the DFT basis. u k 1 2 k N 2 k N N 1 where -1 .

Key fact

If B is a basis for V , then every v V can be written uniquely (up to order of terms) in the form v i 1 N a i v i where a i F and v i B .

Other facts

  • If S is a linearly independent set, then S can be extended to a basis.
  • If < S > V , then S contains a basis.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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