18.085: My Summer at MIT - Week 6

Week 6 - The Matrix

by Amy Beth Prager

We deal with many complex concepts here at MIT :) My hope in this week's blog is to break one down for you. How many of you know about matrices and matrix multiplication? No, not the movie The Matrix, but the mathematical object called the matrix. A matrix is a sequence of numbers arranged in rectangular form. For example, [1 2][3 4] is a matrix. The elements
of the matrix have their own names, as does the matrix itself. I like to call the matrix Amy...no only kidding, let's call it A. Each individual element of A, in this case, 1,2,3,4, are given names as well. A(1,1) is 1 because it is the first row and the first column. Similarly, for all the other elements. Some people like to start with 0 instead of 1. There is actually a lot of debate on the topic. Most computers like to start with 0, and most people like to start with 1. So this is proof that humans and computes differ. Starting with “1” is just so much more emotionally satisfying than starting with “0”. Of course, Spock and Lt. Commander Data like to start with 0.

Computers use matrices in virtually every application they perform. For example, the word processor used to write this
essay doubtlessly stores the words in this essay as a matrix, essay[0]=We, essay[1]=deal, etc. So if you are reading this essay, thank a matrix! Shake one's (or zero's) hand!

The most common use of matrices is matrix multiplication. There is a procedure for multiplying matrices that makes little
sense unless you really think about it. The procedure is to go across the row of the first matrix and down the column of the second matrix. For example, [1 2] [3 4] =1*3+2*4=11.

Why is matrix multiplication defined this way you may ask? If the first matrix is how many of something, and the second matrix is that something, than this scheme makes perfect sense, doesn't it? If you have 1 dime and 2 quarters, then [1 2][10 25]=60
cents.

Like most mathematical concepts, matrices and the theory associated with them were invented because of necessity. Most of physics and engineering could not be accomplished without the idea of a matrix. From structures to networks to statistics to everything in between, matrices are what make our world understandable and predictable.

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About the Blogger

Amy Beth Prager is a mathematics educator interested in outreach efforts. She is currently expanding her knowledge of mathematics, computer science and engineering.She is spending the summer in the  mathematics department of MIT pursuing coursework in applied numerical methods; taking a class known as 18.085, Computational Science and Engineering. The course teaches methods of applied mathematics useful in engineering and science.