Can someone explain linear\nonlinear to me?

sAid

Hey

I've looked them up in wikipedia,google etc, but all the sites I have tried tend to give definiations rather than explanations. I know they might be trivial but I never really got my head around it. Like how to idenify a nonlinear equation from a linear one, and there characteristics etc.

So if anyone could explain them to me I would be extremely grateful

Cheers

Delphi91

Linear means "line".

Therefore a linear relatonship between two variables means that if you plot one against the other, the result is a straight line!


Remember from coordinate geometry that the equation of a line is in the form y=mx+c, then if you can arrange an equation in two variables into this form, it is a linear relationship.

For example, y = 5x is a straight line and therefore a linear relationship exists between x and y.

As a general rule, as long as there are no powers greater than 1 in the equation, it represents a linear relationship.

y = 2x : Linear
t = 4s + 6 : Linear
w = 4z^2 : Non-linear

Any help??

cunnins4

Just to go further on Delphi91's explanation, yes, a linear equation is any equation where no variable has a power greater than 1. That's how i've always recognised them.

They represent lines or vectors and can be used for many applications: you can place a number of n linear equations or vectors containing n unknowns and you can solve the equations using gaussian elimination or simultaneous equations. This works well for a problem where you have a large number of unknowns and by representing the unknowns in linear equations in a matrix you can solve them far easier this way.

Linear equations or vectors can be used to define planes, vectors projected onto planes, orthogonal projections, parrallelipedes (sp), the list goes on and on.

If you want to learn more about them and their applications, take a look at "elementary linear algebra: applications version" by anton (8th edition) it's very good, it's what i used in college for linear algebra.

hope this helps little.

Son Goku

I'll give you the definition of nonlinear from wikipedia and then explain it.

Wikipedia wrote:

Nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors.

I would define it as a situation where the superpositional principle cannot be applied.

The most basic example of linearity and non-linearity is the one Delphi91 gave. An example of linearity on its own is cunnins4's of elementary linear algebra.
(There is no such thing as nonlinear algebra.)

However when a mathematician usually says non-linear they aren't talking about these areas. For example polynomial equations (of power greater than 2) are far too simple to really deserve the term non-linear even though they technically are.

The real area they apply in is functional and real analysis. However this would take too much time to go into so I'll take an example from physics.

Electrostatics is linear, gravity is not.
What this means is that if I have two charged spheres, I can get their combined electric field, by adding (basically) their individual fields.
However their combined gravitational field can't be gotten by adding their individual ones. In fact the combined field is nothing like the original fields, either in strength or basic shape.

Another example from pure mathematics is equations where every term depends on every other one.

non-irish

cunnins4 wrote:

a linear equation is any equation where no variable has a power greater than 1. That's how i've always recognised them.

Not quite right, and equation can have all the variables with power 1, and not being linear, example y=x*z*t

cunnins4

non-irish wrote:

Not quite right, and equation can have all the variables with power 1, and not being linear, example y=x*z*t

would ya mind expanding on that? I realise that's correct, but i'd just like a bit of further clarification on it. I thought i had this down, not so sure anymore!

non-irish

cunnins4 wrote:

would ya mind expanding on that? I realise that's correct, but i'd just like a bit of further clarification on it. I thought i had this down, not so sure anymore!

Ok, the power of a equation is calculated as the maximum power of each of its terms. So in my example, it's 3, because 3=1+1+1...if you had y=z*t*h*x+x^2*z^2*t^2, it would be 6, becasue the first term power is 4(4=1+1+1+1) and the second term is 6(6=2+2+2). Hope that helps!
Maybe the poster should clarify on the context.