Talk:Linear equations

I have a few suggestions for improving this. Unfortunately, I don't have time to do it myself -- too swamped with the MV Calc work. I'll try to get to it eventually, but if anyone else wants to work on it, here's what I suggest:

Keep in mind that nowadays people do all the matrix operations by computer, so the emphasis should move away from the matrix operations themselves, and more toward getting an understanding of how it relates to the problem itself. The problem is one of N simultaneous linear equations in N variables. The matrix is just a stripped-down representation of the coefficients in the equations.

So I would suggest coming up with an example (N=3 is about right) of 3 equations in 3 unknowns. Work out coefficients that make the solution not get into anything bizarre. Then show how one solves these equations the way we learned in high school or whatever, by multiplying some equations by various numbers and subtracting, until we are left with just 1.2x=3.5, 2.5y=4.7, and 3.1z=2.1, or whatever.

Then show that a matrix, that is a 3x3 array of numbers with a 4th column on the right, could have been used to hold all the information, and that all the multiplications and subtractions could be thought of as various row operations on the matrix, until we have

1.2 0 0 3.5

0 2.5 0 4.7

0 0 3.1 2.1

from which we read out the answer. Explain that those are called matrix row operations, that the whole process is called Gaussian elimination, and that we are putting the matrix into "echelon form" as we go. (Eventually we get it into "diagonal form".)

One other thing: At each step, the element that we choose to have its row multiplied by something and subtracted from the other rows to make those other rows zero in the column is called the "pivot". It's not just the first nonzero element. For reasons of minimizing roundoff error, we choose the pivot as the largest element. We then bring it into the position we want (an action called "pivoting", yes, pivot is a verb in this case, sorry, I don't make the language.) Relate that to things we do with the equations themselves -- when we swap matrix rows to bring the pivot to the top of the active part, we are really just swapping some equations, which we can do, right? And when we swap columns to bring the pivot to the left, we are just changing the order of the variables. Addition is commutative, right? And when we multiply that row by a constant and subtract from the other rows, we are just multiplying a true equation by a constant -- that leaves it still true -- and subtracting a true equation from another true equation, which still leaves us with a true equation, right? Choose the example so that it can exhibit pivoting.

I don't have time to write this up in formal and pedagogically proper language just now. I may get to it eventually.

SamHB 00:13, 25 February 2010 (EST)