Question

Hello friends.. I have some confusion on Linear Equation and Linear system.... will anyone help me to clear my confusion.. ? I want some discussion with me.. please..

Answer 1

A linear equation is an equation of the form \(y = a_1x_1 + a_2x_2 + a_3x_3 + ... \)

A system of linear equations or a linear system is more than one of these equations where a solution must satisfy all of them.

Answer 2

yes.... the system has to satisfy the homogenity and and additivity , but an equation can be with out satisfying these two right. was that mismatch came due to historical definitions or anything else...? And why exactly we changed our definition of linearity to the systems? and if someone asks is Y= mX + c is linear equation or not? what i have to say.. somewhere I saw some answers regarding regions and function mappings also.. please explain me in that direction also.. thanks for answering my question.

Answer 3

Well yes I should say that \(T(x) = ax + b\) is not a linear transformation. It is known as a affine transformation because it includes a translation \(+b\). I wouldn't be too worried about the particularities of the terms. Usually when you are reading material you will be able to tell from the context what is being discussed.

Answer 4

It is a "linear equation" but I don't really use that term very much.

Answer 5

http://en.wikipedia.org/wiki/Linear_equation

Answer 6

I think "linear equation" is used often in high school curriculum.

Answer 7

you are right .. but while i am going through my technical subjects.... most of the practical approaches are based on the evaluation of these equations and systems only.... what the bad thing is... one part of the subject is based on systems... and other was based on equations.... if u take.... all our Differential equations are linear equations only.. and all our Matrices are those only... and more over if +b term is the origin of non-linearity.. we will use transformation for that term also in matrices.. let us take in augmented matrix.. ok i didn't cross checked me when i am thinking on this.. but of course we are equally giving importance to that High school term also...

Answer 8

To be very clear a linear system MUST be of the form \(Ax = b\). A matrix is really just a linear transformation expressed with respect to basis for the domain and target space. So there is no translation involved.

Answer 9

Does that clear things up?