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Dimension

Let V be a vector space with basis B . The dimension of V , denoted dim V , is the cardinality of B .

Every vector space has a basis.

Every basis for a vector space has the same cardinality.

dim V is well-defined .

If dim V , we say V is finite dimensional .

Examples

vector space field of scalars dimension
N
N
N

Every subspace is a vector space, and therefore has its own dimension.

Suppose S u 1 u k V is a linearly independent set. Then dim < S >

    Facts

  • If S is a subspace of V , then dim S dim V .
  • If dim S dim V , then S V .

Direct sums

Let V be a vector space, and let S V and T V be subspaces.

We say V is the direct sum of S and T , written V S T , if and only if for every v V , there exist unique s S and t T such that v s t .

If V S T , then T is called a complement of S .

V C { f : | f is continuous } S even funcitons in C T odd funcitons in C f t 1 2 f t f t 1 2 f t f t If f g h g h , g S and g S , h T and h T , then g g h h is odd and even, which implies g g and h h .

Facts

  • Every subspace has a complement
  • V S T if and only if
    • S T 0
    • < S , T > V
  • If V S T , and dim V , then dim V dim S dim T

Proofs

Invoke a basis.

Norms

Let V be a vector space over F . A norm is a mapping V F , denoted by , such that forall u V , v V , and F

  • u 0 if u 0
  • u u
  • u v u v

Examples

Euclidean norms:

x N : x i 1 N x i 2 1 2 x N : x i 1 N x i 2 1 2

Induced metric

Every norm induces a metric on V d u v u v which leads to a notion of "distance" between vectors.

Inner products

Let V be a vector space over F , F or . An inner product is a mapping V V F , denoted , such that

  • v v 0 , and v v 0 v 0
  • u v v u
  • a u b v w a u w b v w

Examples

N over: x y x y i 1 N x i y i

N over: x y x y i 1 N x i y i

If x x 1 x N , then x x 1 x N is called the "Hermitian," or "conjugatetranspose" of x .

Triangle inequality

If we define u u u , then u v u v Hence, every inner product induces a norm.

Cauchy-schwarz inequality

For all u V , v V , u v u v In inner product spaces, we have a notion of the angle between two vectors: u v u v u v 0 2

Orthogonality

u and v are orthogonal if u v 0 Notation: u v .

If in addition u v 1 , we say u and v are orthonormal .

In an orthogonal (orthonormal) set , each pair of vectors is orthogonal (orthonormal).

Orthogonal vectors in 2 .

Orthonormal bases

An Orthonormal basis is a basis v i such that v i v i i j 1 i j 0 i j

The standard basis for N or N

The normalized DFT basis u k 1 N 1 2 k N 2 k N N 1

Expansion coefficients

If the representation of v with respect to v i is v i a i v i then a i v i v

Gram-schmidt

Every inner product space has an orthonormal basis. Any (countable) basis can be made orthogonal by theGram-Schmidt orthogonalization process.

Orthogonal compliments

Let S V be a subspace. The orthogonal compliment S is S u u V u v 0 v v S S is easily seen to be a subspace.

If dim v , then V S S .

If dim v , then in order to have V S S we require V to be a Hilbert Space .

Linear transformations

Loosely speaking, a linear transformation is a mapping from one vector space to another that preserves vector space operations.

More precisely, let V , W be vector spaces over the same field F . A linear transformation is a mapping T : V W such that T a u b v a T u b T v for all a F , b F and u V , v V .

In this class we will be concerned with linear transformations between (real or complex) Euclidean spaces , or subspaces thereof.

Image

T w w W T v w for some v

Nullspace

Also known as the kernel: ker T v v V T v 0

Both the image and the nullspace are easily seen to be subspaces.

Rank

rank T dim T

Nullity

null T dim ker T

Rank plus nullity theorem

rank T null T dim V

Matrices

Every linear transformation T has a matrix representation . If T : 𝔼 N 𝔼 M , 𝔼 or , then T is represented by an M N matrix A a 1 1 a 1 N a M 1 a M N where a 1 i a M i T e i and e i 0 1 0 is the i th standard basis vector.

A linear transformation can be represented with respect to any bases of 𝔼 N and 𝔼 M , leading to a different A . We will always represent a linear transformation using the standard bases.

Column span

colspan A < A > A

Duality

If A : N M , then ker A A

If A : N M , then ker A A

Inverses

The linear transformation/matrix A is invertible if and only if there exists a matrix B such that A B B A I (identity).

Only square matrices can be invertible.

Let A : 𝔽 N 𝔽 N be linear, 𝔽 or . The following are equivalent:

  • A is invertible (nonsingular)
  • rank A N
  • null A 0
  • A 0
  • The columns of A form a basis.

If A A (or A in the complex case), we say A is orthogonal (or unitary ).

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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