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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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
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
Abhi
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?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
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salma
Commplementary angles
Idrissa Reply
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Sherica
im all ears I need to learn
Sherica
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Tamia
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Uday
hi
salma
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
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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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