Importance of Linear Functions

Date: 02/06/2002 at 16:38:51
From: Abreeka Moore
Subject: Importance of Linear Equations

Hello, 

I was wondering what the importance of linear functions in the "real 
world" is, and how they are used.

Thank you.

Date: 02/07/2002 at 09:46:12
From: Doctor Ian
Subject: Re: Importance of Linear Equations

Hi Abreeka,

Most 'real world' functions are approximations, as the world usually 
contains too many complications to be modeled exactly by mathematical 
functions. 

The really nice thing about linear functions is that they are easy to 
work with. They are easy to solve, easy to plot, and easy to 
understand. So when you're looking for a function to approximate the 
behavior of something in the real world, you usually try to use a 
linear function first; and only if that proves to be too simple a 
model do you look for other kinds of functions to use. 

Often, in order to use a linear function to model something, you need 
to find the 'best' line that models your data. For example, if your 
data look like

  |             *
  |         * *
  |   *  *  *
  |    * * *
  |  *   
  |
  +-----------------

each pair of points that you could choose would give you a different 
line; but there is _no_ line that contains all the points. What you 
can do in a case like this is find the line that is 'closest' (in some 
sense) to all the points:

  |                  L
  |             *
  |         * *
  |   *  * L*              The 'best' line is the one that passes
  |    * * *               through the L's. 
  |  *   
  |L
  +-----------------

This line is normally found using a process called 'least squares' 
analysis, or 'linear regression'.  (The basic idea is this:  Choose a 
line; find the distance of each data point from the line; add those 
distances up to get a score for the line. Try other lines, until you 
find the one that gives you the lowest score. That's the 'best line'.)

Once you've found a linear function that you think models your data, 
you can use the function to make predictions. For example, if the data 
above showed the amount of wheat harvested as a function of inches of 
rain in past years, then knowing how much rain has fallen _this_ year 
can let you predict the amount of wheat that will be harvested, even 
before it's finished growing. 

Note that when you've reached this point, one little function

  wheat = m*rain + b

replaces a whole collection of data, so it's more efficient in terms 
of space. 

Of course, there is a danger in doing this: It might turn out that the 
function doesn't consider all the relevant factors. (For example, 
interest rates and subsidies have as much to do with the production of 
wheat as rain does.) In which case, any predictions that it makes 
might be wrong. But this is always the trade-off - ease of use versus 
precision - that you make when you try to use math to describe the 
real world.  

As someone once said, you want to make your mathematical models as 
simple as possible... but no simpler. A linear function is about the 
simplest kind of model that it's possible to make.  

Does this help?  Write back if you'd like to talk more about this, or 
anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/07/2002 at 23:21:53
From: Abreeka Moore
Subject: Importance of Linear Equations

Thank you! That was a huge help to me. I look forward to sending 
questions in the future.