Math 206, Fall 2004

Introduction and Expectations

When Sir Arthur Cayley , who graces the front page of your syllabus, in 1841 published the first English contribution to the theory of determinants, he was unknowingly (perhaps) starting on his way to becoming the modern father of matrix algebra. But Linear Algebra has humble beginnings much earlier in Chinese and Babylonian texts from 1000-100 B.C.. There we find the need to solve simple "linear systems of equations" disguised as problems of the following sort:

There are two fields whose total area is 1800 square yards. One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field?

It is likely that most students who have had a course in algebra could solve the problem above. But for large systems with many variables, ad hoc techniques must yield to a systematic understanding of linear systems of equations. Thus was born the need for matrix algebra.

The beauty is that out of this most practical of purposes has arisen an elegant subject, full of abstraction and worthy of mathematical inquiry for its own sake. Over the quarter I hope we can observe the evolution of the subject from the pedestrian to the sublime, as well catch fleeting glimpses of what lies ahead in courses like math 306, 406, and 481. Such scenic viewpoints are not reached without effort, however, and the journey will at times be arduous.

Should you be intimidated by the material? No. But each of us should maintain a healthy respect for the centuries of work that have brought us to this moment. And each of us should realize that this course will take a fair amount of dedication and concentration. To those hearty souls who accept the challenge with a smile, I say welcome. Now on with it.