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The second problem posed in the introduction is basically the solution of simultaneous linear equations [link] , [link] , [link] which is fundamental to linear algebra [link] , [link] , [link] and very important in diverse areas of applications in mathematics, numericalanalysis, physical and social sciences, engineering, and business. Since a system of linear equations may be over or under determined in a varietyof ways, or may be consistent but ill conditioned, a comprehensive theory turns out to be more complicated than it first appears. Indeed, there isa considerable literature on the subject of generalized inverses or pseudo-inverses . The careful statement and formulation of the general problem seems to have started with Moore [link] and Penrose [link] , [link] and developed by many others. Because the generalized solution of simultaneous equationsis often defined in terms of minimization of an equation error, the techniques are useful in a wide variety of approximation andoptimization problems [link] , [link] as well as signal processing.

The ideas are presented here in terms of finite dimensions using matrices. Many of the ideas extend to infinite dimensions using Banachand Hilbert spaces [link] , [link] , [link] in functional analysis.

The problem

Given an M by N real matrix A and an M by 1 vector b , find the N by 1 vector x when

a 11 a 12 a 13 a 1 N a 21 a 22 a 23 a 31 a 32 a 33 a M 1 a M N x 1 x 2 x 3 x N = b 1 b 2 b 3 b M

or, using matrix notation,

A x = b

If b does not lie in the range space of A (the space spanned by the columns of A ), there is no exact solution to [link] , therefore, an approximation problem can be posed by minimizing an equation error defined by

ε = A x - b .

A generalized solution (or an optimal approximate solution) to [link] is usually considered to be an x that minimizes some norm of ε . If that problem does not have a unique solution, further conditions, such as also minimizing the norm of x , are imposed. The l 2 or root-mean-squared error or Euclidean norm is ε T * ε and minimization sometimes has an analytical solution. Minimization of other norms such as l (Chebyshev) or l 1 require iterative solutions. The general l p norm is defined as q where

q = | | x | | p = ( n | x ( n ) | p ) 1 / p

for 1 < p < and a “pseudonorm" (not convex) for 0 < p < 1 . These can sometimes be evaluated using IRLS (iterative reweighted least squares) algorithms [link] , [link] , [link] , [link] , [link] .

If there is a non-zero solution of the homogeneous equation

A x = 0 ,

then [link] has infinitely many generalized solutions in the sense that any particular solution of [link] plus an arbitrary scalar times any non-zero solution of [link] will have the same error in [link] and, therefore, is also a generalized solution. The number of families of solutions is the dimensionof the null space of A .

This is analogous to the classical solution of linear, constant coefficient differential equationswhere the total solution consists of a particular solution plus arbitrary constants times the solutions to the homogeneous equation. The constants are determined from the initial(or other) conditions of the solution to the differential equation.

Ten cases to consider

Examination of the basic problem shows there are ten cases [link] listed in Figure 1 to be considered.These depend on the shape of the M by N real matrix A , the rank r of A , and whether b is in the span of the columns of A .

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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
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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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Basic vector space methods in signal and systems theory. OpenStax CNX. Dec 19, 2012 Download for free at http://cnx.org/content/col10636/1.5
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