Matrices and Computer Programming
Date: 11/05/96 at 23:01:21 From: Robert Subject: Matrices We are doing matrices in my Algebra II class and I was wondering why matrix multiplication and matrix Algebra are the way they are. I understand how these formulas are used in computer programming but it wasn't made for computers. I really need to know why matrices are the way they are and the history of them.
Date: 11/06/96 at 11:24:10 From: Doctor Tom Subject: Re: Matrices Hi Robert, It seems a little strange to me that an algebra II book would talk about matrices. It is true that you know enough to be able to multiply them (because it is just a simple pattern of additions and multiplications), but what's the point of learning to do an operation if you have no idea how to use it? I can pretty much guarantee that you won't need them for a few years. I took my first course in linear algebra (where you learn about the details of matrices) in my sophomore year (as a math major) in college. But I'll give it a crack. Let's just look at 3x3 matrices. You know, I suppose, that you can identify points in three-dimensional space with a collection of 3 numbers (x,y,z) that give the displacement of the point from the origin. Well, suppose you have a bunch of points in 3 dimensions, each described with its own 3 numbers, and you'd like to find out where they wind up if you rotate them about the x-axis by 45 degrees. It turns out that there is a matrix that represents a 45 degree rotation about the x-axis, and if you multiply the (x,y,z) by that matrix, the result will be rotated by 45 degrees. Things like rotation, scaling, shearing, and so on are called linear transformations, and every 3x3 matrix corresponds to a linear transformation in 3-space. (And every linear transformation in 3-space corresponds to a 3x3 matrix.) Now suppose you want to rotate first around the x-axis by 45 degrees, then about the y-axis by 22 degrees. Suppose the matrix M represents the 45 degree rotation, and a matrix N represents the 22 degree rotation (about a different axis). Then the matrix MN (matrix multiplication) corresponds to the combination of the operations. In other words, if you work out the 3x3 product of M and N, you just have to multiply your points (x,y,z) by that product to get the combination of rotations in one operation. I work in computer graphics, and as you can imagine, I do this sort of thing all the time. For information on the history of matrices, search the MacTutor History Archive for the word matrix: http://www-groups.dcs.st-and.ac.uk:80/~history/ Look for an article on "Matrices and Determinants." -Doctors Tom and Rachel, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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