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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

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s. Reply
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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