<< Chapter < Page
  Signal theory   Page 1 / 1
Chapter >> Page >
Description of vector spaces, subspaces, bases and spans.

Vector spaces

Definition 1 A linear vector space ( X , R , + , · ) is given by a signal space X (called vectors), a set of scalars R , an addition operation + : X × X X , and a multiplication operation · : R × X X , such that:

  1. X forms a group under addition:
    1. x , y X ! x + y X , (closed under addition)
    2. 0 X such that 0 + X = X + 0 = X . (additive identity)
    3. x X y X such that x + y = 0 , (additive inverse)
    4. x , y , z X x + ( y + z ) = ( x + y ) + z . (associative law)
  2. Multiplication has the following properties: for any x , y X and a , b R :
    1. a · x X , (closure in X under multiplication)
    2. a · ( b · x ) = ( a · b ) · x , (compatibility)
    3. ( a + b ) · x = a · x + b · x , (distributive law over R )
    4. a · ( x + y ) = a · x + a · y . (distributive law over X )
  3. The set R has the following properties:
    1. There exists 1 R s.t. 1 · x = x x X , (multiplication identity)
    2. There exists 0 R s.t. 0 · x = 0 x X . (multiplicative null element)

Example 1 Here are some examples of vector spaces:

  • X = R n (space of all vectors of length n ) over R = R is a vector space.
  • X = C n ( C is complex numbers) over R = C is a vector space.
  • X = R n over R = C is not a vector space, because closure in X under multiplication is not met.
  • X = C [ T ] (continuous functions in T ) over R = R is a vector space.

Subspaces

Definition 2 A subset M X is a linear subspace of X if M itself is a linear vector space. Note that, in particular, this implies that any subspace M must obey 0 M .

Example 2 Here are some examples of subspaces:

  • In X = R 2 over R = R , any line that passes through the origin is a subspace of X :
    M = ( x , y ) R 2 such that y x = c .
  • In X = C [ T ] over R = R , the followings are subspaces of X :
    M 1 = { f ( x ) = a x 2 + b x + c : a , b , c R } M 2 = { f ( x ) : f ( x 0 ) = 0 } .
    In contrast, the set M 3 = { f ( x ) : f ( x 0 ) = a 0 } is not a subspace.

Proposition 1 If M and N are subspaces of X , then M N is also a subspace.

Proof: We assume that M and N hold properties of linear vector space, and show that so does M N :

  1. x , y M N x , y M x + y M x , y N x + y N x + y M N
  2. M linear vector space 0 M N linear vector space 0 N 0 M N
  3. x M N x M y M s.t. x + y = 0 x N y N s.t. x + y = 0 y M N

The other properties are shown in a similar fashion.

Definition 3 A vector x X , where ( X , R , + , · ) is a vector space, is a linear combination of a set { x 1 , x 2 , ... , x n } X if it can be written as x = i = 1 n a i · x i , a i R . The set of all linear combinations of a set of points { x 1 , x 2 , ... , x n } builds a linear subspace of X .

Example 3 Q = i = 0 2 a i x i is a linear subspace of ( C [ T ] , R , + , · ) containing the set of all quadratic functions, as it corresponds to all linear combinations of the set of functions { x 2 , x , 1 } .

Bases and spans

Definition 4 For the set S = { x 1 , x 2 , ... , x n } X , the span of S is written as

[ S ] = span ( S ) = x : x = i = 1 n a i x i , a i R .

Example 4 The space of quadratic functions Q is written as Q = [ S 1 ] , with S 1 = { x 2 , x , 1 } . The space can also be written as [ S 2 ] with S 2 = { 1 , x , x 2 - 2 } ) , i.e. [ S 1 ] = [ S 2 ] . To prove this, we need to show [ S 2 ] [ S 1 ] and [ S 1 ] [ S 2 ] . For the former case we have

x = a 1 + a 2 x + a 3 ( x 2 - 2 ) = ( a 1 - 2 a 3 ) + a 2 x + a 3 x 2 ,

which means that every element that can be spanned by S 2 , can also be spanned by S 1 , and hence [ S 2 ] [ S 1 ] . The latter case can be shown in a similar manner.

Definition 5 A set S is a linearly independent set if

i = 1 n a i x i = 0 a i = 0 , i { 1 , 2 , ... , n } .

Otherwise, the set S is linearly dependent .

Definition 6 A finite set S of linearly independent vectors is a basis for the space X if [ S ] = X , i.e. if X is spanned by S .

Definition 7 The dimension of X is the number of elements of its basis | S | . A vector space for which a finite basis does not exist is called an infinite-dimensional space .

Theorem 1 Any two bases of a subspace have the same number of elements.

Proof: We prove by contradiction: assume that S 1 = { x 1 , ... , x n } and S 2 = { y 1 , ... , y m } , m > n , are two bases for a subspace X with different numbers of elements. We have that since y 1 X it can be written as a linear combination of S 1 :

y 1 = i = 1 n a i x i .

Order the elements of S 1 above so that a 1 is nonzero; since y 1 must be nonzero then at least one such a i must exist. Solving the above equation for x 1 yields

x 1 = 1 a 1 y 1 - i = 2 n a i x i .

Thus { y 1 , x 2 , x 3 , ... , x n } is a basis, in terms of which we can write any vector of the space X , including y 2 :

y 2 = b 1 y 1 + i = 2 n b i x i .

Since y 1 , y 2 are linearly independent, at least one of the values of b i , i > 2 , must be nonzero. Sort the remaining x i so that b 2 is nonzero. Solving for x 2 results in

x 2 = 1 b 2 y 2 - b 1 b 2 y 1 + i = 3 n b i b 2 x i .

Therefore, { y 1 , y 2 , x 3 , ... , x n } is a basis for X . Continuing in this way, we can eliminate each x i , showing that { y 1 , y 2 , ... , y n } is a basis for X . Thus, we have y n + 1 = i = 1 n c i y i , or equivalently:

c n + 1 y n + 1 + i = 1 n c i y i = 0 with c n + 1 = - 1 .

As a result, S 2 is linearly dependent and is not a basis. Therefore, all bases of X must have the same number of elements.

Basis representations

Having a basis in hand for a given subspace allows us to express the points in the subspace in more than one way. For each point x [ S ] in the span of a basis S = { S 1 , S 2 ... S n } , that is,

x = i = 1 n a i S i ,

there is a one-to-one map (i.e., an equivalence) between x [ S ] and a = { a 1 , ... , a n } R n , that is, both x and a uniquely identify the point in S . This is stated more formally as a theorem.

Theorem 2 If S is a linearly independent set, then

i = 1 n a i S i = i = 1 n b i S i

if and only if a i = b i for i = 1 , 2 ... n .

Proof: Theorem 1 states that the scalars { a 1 , ... , a n } are unique for x . We begin by assuming that indeed

i = 1 n a i S i = i = 1 n b i S i .

This implies

i = 1 n a i S i - i = 1 n b i S i = 0 , i = 1 n ( a i - b i ) S i = 0 .

Since the elements of S are linearly independent, each one of the scalars of the sum must be zero, that is, a i - b i = 0 and so a i = b i for each i = 1 , ... , n .

Example 5 (Digital Communications) A transmitter sends two waveforms:

S 1 ( t ) = 2 / T cos ( 2 π f c t ) t [ 0 , T ] if bit 1 is transmitted,
S 0 ( t ) = 2 / T sin ( 2 π f c t ) t [ 0 , T ] if bit 0 is transmitted.

The signal r ( t ) recorded by the receiver is continuous, that is, r ( t ) C [ T ] . Assuming that the propagation delay is known and corrected at the receiver, we will have they the received signal must be in the span of the possible transmitted signals, i.e., r ( t ) span ( S 1 ( t ) , S 0 ( t ) ) . One can check that S 1 ( t ) and S 2 ( t ) are linearly independent. Thus, one can use a unique choice of coefficients a 0 and a 1 that denote whether bit 0 or bit 1 is transmitted and contain the amount of attenuation caused by the transmission:

r ( t ) = a 1 S 1 ( t ) + a 0 S 0 ( t ) .

The uniqueness of this representation can only be obtained if the transmitted signals S 0 ( t ) and S 1 ( t ) are linearly independent. The waveforms above are used in in phase shift keying (PSK); other similar examples include frequency shift keying (FSK) and quadrature amplitude modulation (QAM).

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Signal theory. OpenStax CNX. Oct 18, 2013 Download for free at http://legacy.cnx.org/content/col11542/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signal theory' conversation and receive update notifications?

Ask