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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
is it a question of log
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
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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