What is the Purpose of a Matrix?

Date: 11/04/2003 at 11:12:16
From: Jamie
Subject: What exactly is the purpose of a matrix

Could you explain to me the exact purpose of a matrix?

I can do matrix math, but I'm confused about what they do and what 
they are for.

Date: 11/04/2003 at 12:08:41
From: Doctor Ian
Subject: Re: What exactly is the purpose of a matrix

Hi Jamie,

A matrix is just a compact notation, which allows you to specify
several linear equations at once without having to write them all out.
 
For example, instead of writing 

  3x + 4y + 5z =  7

  2x - 3y + 6z =  6

  2x + 5y - 9z = 11

I can write the same thing more compactly using matrices:

 | 3  4  5 | | x |   |  7 |
 | 2 -3  6 | | y | = |  6 |
 | 2  5 -9 | | z |   | 11 |

For a system this small, the savings in effort isn't so great.  But
the more equations and unknowns you get, the easier it becomes to deal
with matrices, instead of with the equations directly. 

Of course, as with all notations, once it's in place, people figure
out how to do new and interesting things with it, which wouldn't have
occurred to anyone before the notation was created.  That's certainly
the case with linear algebra.  But the basic idea is just capturing
the parallelism in a set of similar-looking equations.  

It's not unlike what we do when we collapse a set of sentences like

  Pat, who is 47 years old, makes $38,000 per year as a manager.

  Chris, who is 29 years old, makes $110,000 per year a stock analyst.

  Sandy, who is 11 years old, makes $58 per year selling cookies. 

into a table, 

   Name   Age   Earnings    Occupation
   ----   ---   --------    ----------
   Pat     47     38,000    Manager
   Chris   29    110,000    Stock analyst
   Sandy   11         58    Cookie salesman

The compactness reduces the amount of writing (or typing) that we have
to do, but it also makes similarities, differences, and patterns 
easier to spot by eliminating redundant information (i.e., information
that shows up in each equation/sentence without really adding anything
new).  

Does this help?

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