LINEAR ALGEBRA (what is that?)

1. LINEAR ALGEBRA(what is that?) What is linear algebra? Functions and equations that arise in the "real world" often involve many tens or hundreds or thousands of variables, and one can only deal with such things by being much more organized than one typically is when treating equations and functions of a single variable. Linear algebra is essentially a "language for accounting" that's been developed just for this purpose. (Copied from webpage.)

2. Another useful quote is (same source) One can plausibly argue that linear algebra is the fundamental tool in modern applications of math-ematics, used to determine airline schedules, rank webpages in search engines, compress electronic data, model the flow of oil underground, and so on and on and on. Finally, my (deceased) PhD thesis director called Linear Algebra “double entry bookkeeping.” I’d like to add my own: “First order simplification of many problems.”

3. Let us start from scratch. The word “algebra” comes to us from … the Islamic world!, more precisely a treatise by the Islamic scholar al-Khowarizmi(9-th century, our word algorithm comes from his last name) titled Hisabal-jabr(whence algebra) dealt with procedures of rearranging and combining terms when solving equations. (I am indebted to Prof. Hahn’s marvelous book “Mathematics in Architecture” for this information.) That is exactly what we will be doing in this class, learn how to combine and rearrange terms when …

4. solving equations! (That takes care of the word algebra.) What about the word “Linear” ? Its meaning will justify my comment: “First order simplification of many problems.” Most problems boil down to a collection of unknowns (the first set of entries, may be enormous) that are supposed to satisfy a collection (the second set of entries, whence double entry bookkeeping, may be enormous) of equations. There is not much we can do about how many unknowns we have to find, but …

5. Hopefully we can make the equations a little bit easier. You have already done this. When ? Do you remember your Calculus I class? You learned something called “linear approximation and differentials” and there you learned that those morons of the Flat Earth Society were somewhat justified in their belief, because “any smooth curve can be approximated by a straight line” In algebraic terms, any equation with difficult terms like can be replaced with

6. the much simpler equation of a straight line, that looks like (you should be able to compute and for any given !) This procedure is called, for obvious reasons, linearization and • simplifies things. • is responsible for the term linear algebra. The upshot of this discussion is that henceforth

7. No matter how many unknowns we have, the only equations we will consider share the sim-plicity of that is “symbols representing unknowns will appear with exponent 1 (whence first order …) and no terms like Suppose you have seven unknowns and five linear equations. The problem becomes:

8. Find the solution(s) (if any) of This is called a system of linear equations, we will study these animals deeply and at length. Let’s start with a few remarks:

9. We need a better notation, right now we can only write 26 equations. (why?) • We need ways to simplify the equations, keeping the same solution set. Item 1 is addressed as follows: we write for the q-th coefficient in the p-th equation, and move the constant terms on the other side of the = sign. So a linear system of five equations in seven unknowns can be written long-hand as follows:

10. (note the change in the representation of the constant terms, we write instead of , etc.) It’s obvious that we need a short-hand. Any suggestions?

11. Yeah, the summation symbol (sigma)! We re-write the whole thing as: (ready ?) (a little shorter !) This may be somewhat new for some of you, so I will do a couple of examples on the board, then return to our presentation. The rectangular arrangement of the coefficients

12. is called by mathematicians a and by computer scientists an . Notice that a linear system is completely determined by the matrix of the coefficients and the constant terms. The rectangular arrangement

13. is called the augmented matrix of the linear system. Let me do a couple of examples on the board, then continue with the presentation.

14. The previous examples worked out on the board showed how A linear system determines an augmented matrix and An augmented matrix determines a linear system. Now we ask ourselves: How do we find all the solutions, if any? This question could be asked for all kinds of systems of equations, not just linear. For example has exactly three solutions.

15. For linear systems the situation is much easier (remember, linearity simplifies things.) We will learn that a linear system may • have no solutions at all. • have exactly one solution • have infinitely many solutions We will be able to • Tell when case 1 obtains. • Tell when case 2 obtains, and compute that solution. • Tell when case 3 obtains and precisely describe the infinitely many solutions.

16. Let me begin by remarking that the names we give to the unknowns are totally irrelevant. They do not appear in the augmented matrix. You could use , but also or any collection of 753 symbol that work for you. What is important is the augmented matrix, therefore we will concentrate on it. I trust you know what are called the rows of a matrix, as well as the columns. (quick reminder, columns are vertical)

17. The basic operations that we will apply to a linear system to determine whether 1, 2 or 3 obtain are very simple, but (as most simple things are) very powerful when used judiciusly. I will describe all of them (just 3, don’t panic !), and for each one of them I will ask you to convince yourselves (I’ll help you if you can’t) that the solution set stays the same. I will describe them in terms of equations first, then translate them in terms of the rows of the augmented matrix.

18. Consider a system containing any two equations (could be very complicated!) Clearly • Exchanging the two equations gives same solution set.

19. Multiplying either equation (say, No. 1) by a non-zero number gives Same solution set. The third operation is: • Replace one equation (say No. 1) with the sum of itself and a non-zero multiple of the other: same solution set. (Requires some thought.)

20. In terms of the augmented matrix the three operations (formally know as elementary row reduction operations) become • Exchange any two rows • Multiply a row by a non-zero number • Replace a row with the sum of itself and a non-zero multiple of another row. (You have done operation 3 in High School, but you called “substitution”.) I will do one example on the board, but you are now ready for the homework !