# 0.2 General solutions of simultaneous equations

 Page 1 / 4

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 $\mathbf{A}$ and an $M$ by 1 vector $\mathbf{b}$ , find the $N$ by 1 vector $\mathbf{x}$ when

$\left[\begin{array}{ccccc}{a}_{11}& {a}_{12}& {a}_{13}& \cdots & {a}_{1N}\\ {a}_{21}& {a}_{22}& {a}_{23}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& & \\ ⋮& & & & ⋮\\ {a}_{M1}& & & \cdots & {a}_{MN}\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\\ {x}_{3}\\ ⋮\\ {x}_{N}\end{array}\right]=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\\ ⋮\\ {b}_{M}\end{array}\right]$

or, using matrix notation,

$\mathbf{A}\mathbf{x}=\mathbf{b}$

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

$\epsilon =\mathbf{A}\mathbf{x}-\mathbf{b}.$

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

$q={||x||}_{p}=\left(\sum _{n}{|x\left(n\right)|}^{p}{\right)}^{1/p}$

for $1 and a “pseudonorm" (not convex) for $0 . 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

$\mathbf{A}\mathbf{x}=\mathbf{0},$

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 $\mathbf{A}$ .

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 $\mathbf{A}$ , the rank $r$ of $\mathbf{A}$ , and whether $\mathbf{b}$ is in the span of the columns of $\mathbf{A}$ .

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!