Hi my name is Stephanie and I am doing a homework project in math class and I was just wondering if you could please help me with a few questions if you do not know the answers and you think one of your friends might know can you please e-mail this e-mail to them so I can find out. If you wanted to know about the project I will tell you. Our class is defending math and I chose solving equations. And we will be speaking about how you use this form of math and how you can use it in your real life. So here are some of my questions! Thank you for helping me!

Hi Stephanie,

In school, we would start with something like:

and try to find x and y so that both equations are satisfied.

In modern times we deal with huge systems with hundreds of thousands of equations and variables. But let's start at the beginning.

The earliest record of such a problem that I know of is in an ancient Chinese book "Chiu-chang Suan-shu" written around 200 BC. The problem posed there involved calculating prices of grain. The Chinese recognized that the coefficients (numbers) of the system were what was important, and that the variables (letters) were merely place holders. At that time, numbers were represented by coloured bamboo rods, and they were placed on a square counting board. The rods were then manipulated by certain rules of thumb to obtain a solution.

If we look back at our initial example, we can simplify how we present it by putting the coefficients of x in the first column, the coefficients of y in the second column, and the numbers on the right hand side in the third column.

Notice that we've gone from needing 14 symbols to represent our system of equations, down to just 6. With larger systems, the gains are even more significant.

We refer to this array as a matrix. The word matrix was derived from the word "mother", and was first used by James J. Sylvester in the 1800's. He referred to certian special numbers (now called eigenvalues) associated with a matrix as "children coming forth from their mother's womb". By this time, a method called Guassian elimination, named after the German Mathematician Carl Friedrich Guass, had become the accepted way in Europe to solve systems of linear equations.

Here are the basic rules of Guassian elimination:

Note: A row is the collection of numbers you get by reading a line from left to right. For example "3 2 6" is the first row or our array. A column what you get when you read down a line from top to bottom. "2" 3 is the first column in our array.

Now, using rules (i),(ii), and (iii), over and over, the goal is to create a matrix with the following properties:

Let's try it with our example. We begin with

Try to make the first entry in the first row a 1. To do this, multiply the first row by 1/3 to get

Now we want the first nonzero entry in the second row to be in the second column. To do this add -2 times the first row to the second row

Now make the entry in the second row and second column a 1 by multiplying the second row by -3/13. We get

Now add -2/3 times the second row to the first row:

We've reached our goal matrix.

Remembering where our array came from, we write it out the long way again:

Now the solution is easy to see. Let x = 0 and y = 3. Check that this really works in our original equations.

In modern times, the use of computers is a motivating factor behind much of the research in matrix analysis. The method of Guassian elimination works realitively poorly when trying to solve large systems of equations on a computer. It requires too many steps, and errors from the computer rounding off decimals tends to lead to inacurate solutions. Modern methods tend to attack the problem quite differently. Interative methods and the use of matrix factorizations are two common techniques which tend to be much faster and more accurate. Many of the large systems generated in industry have a significant portion of their entries equal to zero, leading researchers to look for ways to cleverly store a matrix in a computer so that only the nonzero entries are recorded and used in the solution process. The charactersitics of the matrix dictate which of the many modern methods should be used, and this is a hot area of ongoing research. The methods are being improved even as you read this!

Solving linear systems comes up in almost every field that uses mathematics. Most big corporations such as Boeing, GM, Xerox, Phyllips, Bell Labs, AT&T, and transpotation and petrolium industries employ research mathematicians in the area of linear systems analysis. Other disicplines including statistics, engineering, biology, chemistry, physics, economics and buisiness rely on linear models in their analysis.

The company Mathworks is the leader in commercially packaged computer software for solving systems of equations. You will find lots of interesting information on their website:

http://www.mathworks.com/index.shtml

There are two text books you may want to try to get to look at. Carl Meyer's book "Matrix Analysis and Applied Linear Algebra" has lots of historical information, both in the book and on the CD that comes with the book. David Lay's book "Linear Algebra and its Applications" contains numerous sections with interesting applications. Both books present several modern methods for solving systems of equations. They're organized in such a way that you can read the history blurbs and the introductions to the sections on applicatons without needing to know all the mathematics in the book.