22M:13

Spring 09

J. Simon

Program to solve systems of linear equations.

Example:

2x + 3y + 4z = 5

x - y + 5z = 2

-3x +4y +z = 3

First just write the coefficients. If we keep rows and columns carefully, we don't have to keep on writing the x's, y'x, etc.

with(linalg):

Warning, the protected names norm and trace have been redefined and unprotected

M:=matrix([[2,3,4,5], [1, -2, 5, 2], [-3, 4, 1, 3]]);

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

rref(M);

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

That's it. For this system of equations, we get

z = 13/20, y = 7/10, and x = 3/10.

%%%%%%%%%%%

What if the system is inconstent?

M:=matrix([[1,1,1,1], [2,2,2,1], [3,4,5,6]]);

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

Notice the second equation looks like double the first, except we did not double the "output" number. So the first and second equations are inconsistent, so the whole system is inconsistent.

When we tell the computer to simplify the system, we get

rref(M);

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

Here the third equation says "0 = 1", which is impossible. This is how we learn that the original system was inconsistent.

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What if the system has infinitely many solutions?

M:=matrix([[1,2,3,4], [3,2,1,1], [5,6, 7, 9]]);

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(Note: I scrambled the system a little, so it is not obvious when we start whether this has one solution, no solutions, or infinitely many.)

Now do the reduction process...

rref(M);

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The last equation now says, "0 = 0", which is neither offensive nor informative.

All we have left are the first two equations. They tell us how to express x and y in terms of z.

So the set of solutions has one "degree of freedom" - for any value of z, there are numbers x,y such that (x,y,z) is a solution of the system. The system has as many solutions as there are numbers z; that is, the system has infinitely many solutions.

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There is another command, "gausselim", that will make the system look more like what you would get if you were doing the arithmetic by hand.

Here is the same system of equations as in the first example above.

M:=matrix([[2,3,4,5], [1, -2, 5, 2], [-3, 4, 1, 3]]);

gausselim(M);

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

Instead of being able to read off right away that z=13/20, the third equation now says,

(100/7)z = 65/7. We can solve this:

100 z = 65 ==> z=65/100 = 13/20.

Then plug that in to the second equation:

(-7/2)y + 3z = (-1/2)

(-7/2)y + 3(13/20) = (-1/2)

y = ((-1/2) - 3(13/20))/(-7/2), which is messy arithmetic. It should all reduce to y=7/10.

JustToCheck:=((-1/2)-3*(13/20))/(-7/2);

NiM+SSxKdXN0VG9DaGVja0c2IiMiIigiIzU=

Isn't math wonderful!!

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end of handout

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