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Introduces tools and formulas to use when dealing with Linear Vector Spaces. Topics covered include: linear vector spaces, inner product spaces, norm, Schwarz inequality, and distance between two vectors

One of the more powerful tools in statistical communication theory is the abstract concept of a linear vector space . The key result that concerns us is the representation theorem : a deterministic time function can be uniquely represented by a sequence of numbers.The stochastic version of this theorem states that a process can be represented by a sequence of uncorrelated random variables.These results will allow us to exploit the theory of hypothesis testing to derive the optimum detection strategy.

Basics

A linear vector space S is a collection of elements called vectors having the following properties:

  • The vector-addition operation can be defined so that if x y z S :
    • x y S (the space is closed under addition)
    • x y y x (Commutivity)
    • x + y z x y + z (Associativity)
    • The zero vector exists and is always an element of S . The zero vector is defined by x 0 x .
    • For each x S , a unique vector x is also an element of S so that x x 0 , the zero vector.
  • Associated with the set of vectors is a set of scalars which constitute an algebraic field. A field is a set of elements which obey the well-known laws of associativity and commutivity forboth addition and multiplication. If a , b are scalars, the elements x , y of a linear vector space have the properties that:
    • a x (multiplication by scalar a ) is defined and a x S .
    • a b x a b x .
    • If "1" and "0" denotes the multiplicative and additive identity elements respectively of thefield of scalars; then 1 x x and 0 x 0
    • a x y a x a y and a b x a x b x .

There are many examples of linear vector spaces. A familiar example is the set of column vectors of length N . In this case, we define the sum of two vectors to be:

x 1 x 2 x N y 1 y 2 y N x 1 y 1 x 2 y 2 x N y N
and scalar multiplication to be a x 1 x 2 x N a x 1 a x 2 a x N . All of the properties listed above are satisfied.

A more interesting (and useful) example is the collection of square integrable functions . A square-integrable function x t satisfies:

t T i T f x t 2
One can verify that this collection constitutes a linear vector space. In fact, this space is so important that it hasa special name - L 2 T i T f (read this as el-two ); the arguments denote the range of integration.

Let S be a linear vector space. A subspace of S is a subset of S which is closed. In other words, if x y , then x y S and all elements of are elements of S , but some elements of S are not elements of . Furthermore, the linear combination a x b y for all scalars a , b . A subspace is sometimes referred to as a closed linear manifold .

Inner product spaces

A structure needs to be defined for linear vector spaces so that definitions for the length of a vector and for thedistance between any two vectors can be obtained. The notions of length and distance are closely related to the concept ofan inner product.

An inner product of two real vectors x y S , is denoted by x y and is a scalar assigned to the vectors x and y which satisfies the following properties:

  • x y y x
  • a x y a x y , a is a scalar
  • x y z x z y z , z a vector.
  • x x 0 unless x 0 . In this case, x x 0 .

As an example, an inner product for the space consisting of column matrices can be defined as x y x y i 1 N x i y i The reader should verify that this is indeed a valid inner product (i.e., it satisfies all of the properties givenabove). It should be noted that this definition of an inner product is not unique: there are other inner product definitions which also satisfy all of theseproperties. For example, another valid inner product is x y x K y where K is an N x N positive-definite matrix. Choices of the matrix K which are not positive definite do not yield valid inner products ( property 4 is not satisfied). The matrix K is termed the kernel of the inner product. When this matrix is something other than an identity matrix, the innerproduct is sometimes written as x , y K to denote explicitly the presence of the kernel in the inner product.

The norm of a vector x S is denoted by x and is defined by:

x x x 1 2

Because of the properties of an inner product, the norm of a vector is always greater than zero unless the vector isidentically zero. The norm of a vector is related to the notion of the length of a vector. For example, if the vector x is multiplied by the constant scalar a , the norm of the vector is also multiplied by a . a x a x a x 1 2 a x In other words, "longer" vectors ( a 1 ) have larger norms. A norm can also be defined when the inner product contains a kernel. In this case, the normis written K x for clarity.

An inner product space is a linear vector space in which an inner product can be defined for allelements of the space and a norm is given by . Note in particular that every element of an inner product space must satisfythe axioms of a valid inner product.

For the space S consisting of column matrices, the norm of a vector is given by (consistent with the first choice of an inner product) x i 1 N x i 2 1 2 This choice of a norm corresponds to the Cartesian definition of the length of a vector.

One of the fundamental properties of inner product spaces is the Schwarz inequality

x y x y
This is one of the most important inequalities we shall encounter. To demonstrate this inequality, consider the normsquared of x a y . x a y 2 x a y x a y x 2 2 a x y a 2 y 2 Let a x y y 2 . In this case: x a y 2 x 2 2 x y 2 y 2 x y 2 y 4 y 2 x 2 x y 2 y 2 As the left hand side of this result is non-negative, the right-hand side is lower-bounded by zero. The Schwarz inequality is thus obtained. Note that the equality occurs only when x a y , or equivalently when x c y , where c is any constant.

Two vectors are said to be orthogonal if the inner product of the vectors is zero: x y 0 .

Consistent with these results is the concept of the "angle" between two vectors. The cosine of this angle is defined by: x , y x y x y Because of the Schwarz inequality, x , y 1 . The angle between the orthogonal vectors is 2 and the angle between vectors satisfying the Schwarz inequality with equality x y is zero (the vectors are parallel to each other).

The distance between two vectors is taken to be the norm of the difference of the vectors. d x y x y

In our example of the normed space of column matrices, the distance between x and y would be x y i 1 N x i y i 2 1 2 which agrees with the Cartesian notion ofdistance. Because of the properties of the inner product, this distance measure (or metric ) has the following properties:

  • d x y d y x (Distance does not depend on how it is measured.)
  • d x y 0 x y (Zero distance means equality)
  • d x z d x y d y z (Triangle inequality)
We use this distance measure to define what we mean by convergence . When we say the sequence of vectors x n converges to x ( x n x ), we mean n n x n x 0

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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