WHAT IS ANALYTIC GEOMETRY? Perhaps a more fundamental question is: What is geometry? It is interesting to realize that the answer to this second question has changed a great deal over the centuries. Euclid, and his Greek contemporaries, had a rather clear idea about what they meant by geometry, as did Newton, Descartes, Klein, Einstein .... The answers that each of these people would have given to the question ”What is Geometry?” would have been very different and depended on the language they used to describe their answer (and by language I don’t mean only Greek, English, French, German etc. but also the mathematical vocabulary they would have at their disposal.) In fact, since we really don’t have such a great mathematical vocabulary yet, I will not try to answer this question directly now. At the end of the term I will offer a “tentative” answer which I hope you will then find interesting. However, two questions I will try to answer today are: where will we do “geometry”? and what will be the objects we ‘play’ with in our geometry? To the first question I will give a precise answer. We will do our geometry in the sets Rn , where R, denotes the real numbers and n can be any positive whole number. So, let’s remember how Rn is defined: Definition: Rn := {(a1 , . . . , an ) | ai ∈ R}. Remarks: 1) If n = 1, then it is hard to see what is so different between R and R1 , except that 2 ∈ R but (2) ∈ R1 and 2 6= (2)! But, despite that difference we will often think of R1 as the real numbers and we will further think about them as being ‘strung out’ on a line. We have made an important ideological jump here. We are thinking of real numbers as being points on a line. Real numbers, quantities that measure something (how many people are there in this room, how many atoms are there in the universe, how tall is the person sitting alongside of me,.... all these very different ideas and ways to think about a number we are now thinking of as a point on a line. 2) If n = 2, then the elements of R2 are pairs of real numbers like (3, 5), or (−2.379, 1/3). Just as in the case of n = 1, we will have many different ways to think about these pairs of numbers. I will come back to talk about that later, but one simple way to think about 1

a pair of numbers is as describing a point on a plane. You are familiar with that from Linear Algebra but you were aware of that idea for a very long time. (Explain) 3) If n = 3 then the element of R3 are triples of real numbers like (3, 5, 7) and you have also seen, in Linear Algebra, how to identify such a triple with a point in space. 4) If n > 3, then it is no longer so easy to “see” the place in which we will be doing geometry, but we will find that one can actually ‘see’ a great deal by analogy and by other mental ‘tricks’. It may be that sometimes we will do geometry that we cannot see! It is a fact that one of the great geometers of the 20th century, Pontrayagin, was blind. For us, in this course, the objects we will be interested in will be certain special subsets of Rn . The key word in that sentence is special. It’s not hard to think that not all the possible subsets of Rn are equally interesting. There is no ‘democracy’ here about subsets. We will be interested, in the first place, in subsets of Rn that arise for some reason. Let me illustrate some different “reasons” for which interesting subsets can arise (we won’t necessarily study all of these either): Examples: 1) the set may be the graph of some nice function. E.g. if we consider the graph of the function f(x) = x2 + 1, then the graph is the set of points {(x, f(x)) | x ∈ R} (draw picture) 2) a graph can be the graph of a function f : R2 → R, e.g. f(x, y) = 3x + 2y + 1. In this case the graph is {(x, y, z) ∈ R3 | z = 3x + 2y + 1} 3) or, things could get a bit more complicated and we could have a function f : R2 → R2 like f(x, y) = (x + 2y + 2, x − y + 3). In this case, the graph lives in R4 and is {(x, y, z, w) ∈ R4 | z = x + 2y + 2, w = x − y + 3} 2

4) We could have sets of points in Rn which arise as data points for an experiment or a process? E.g. suppose we have a chemical process which is being tested for outcomes. A number of experiments are done which each take 24 hours. There are 100 of these experiments. Each experiment consists of beginning with x grams of carbon, y grams of sea water, z grams of magnesium carbonate, w grams of a secret substance and a fixed temperature t. At the end of 24 hours we have u grams of magnesium carbonate, v grams of carbon dioxide and j grams of junk. We can display the outcomes of these 100 experiments as 100 points in R8 where each experiment would correspond to: (x, y, z, w, t, u, v, j) These data points are simply points on the graph of a function F : R5 → R3 , where F (x, y, z, w, t) = (u, v, j). 5) The points of our set X could be the path of a moving object. E.G. the flight path of an airplane can be viewed as a subset of R3 6) The points of our set X could be the solutions to some equations or system of equations; for example, {(x, y) ∈ R2 | x2 + y 2 = 1} or {(x, y, z) ∈ R3 | x − 2y + z = 5 and x + y − 5z = 7} In this course the subsets of Rn which we will consider will be very simple. Simpler than many of the examples we have just looked at. But, we will have to “pay” for that simplicity in that we will want to know a great deal about these subsets. **************************** The elements of Rn are usually called vectors, but the word really doesn’t have to mean anything other than —- an element of Rn . The important thing will be how we want to think about these elements. Let’s first concentrate on R2 : recall R2 = {(a, b) | a, b ∈ R}. 3

First way to think about R2 : We draw two perpendicular axes (assi) with equal measure on each of them. Then we can think of the vector (for example) (1, 3) ∈ R2 as the point with coordinates 1 and 3, i.e one unit along the positive x-axis and 3 units along the positive y-axis. If we use this interpretation we have a bijection: { vectors of R2 } ↔ { points of the plane } I want to emphasize that in this interpretation the vectors correspond to points. A Second way to think about R2 : In this interpretation we will think about vectors as directed line segments. The process we will use is the following: take two points P = (a, b) and Q = (c, d) (using the first way to think about R2 ) and we form the line segment P Q (which will mean the line segment which starts at the point P and finishes at the point Q). I want to associate a vector to this line segment. I.e. I want to associate an element of R2 to this line segment. The way I will do that is the following: to the line segment P Q I will associate the vector (c − a, d − b) ∈ R2 . E.g. if P = (2, 3) and Q = (3, 5) then we associate (1, 2) to the line segment P Q. In this way we can associate a vector of R2 to any directed line segment. But, this is not exactly like what we did in the First Way of Thinking about R2 , because the directed line segment from P 0 = (4, 7) to Q0 = (5, 9) is ALSO associated to the vector (1, 2) ∈ R2 . I.e. we do not have a bijection from { directed line segments in the plane} AND {vectors of R2 }

How can we remedy this situation. There is a graceful way out of the situation and that is the following: we form an equivalence relation on the set of directed line segments. 4

We do this in the following way: if P is the point (a, b) and Q is the point (c, d), P 0 the point (a0 , b0 ) and Q0 the point (c0 , d0 ) then Definition: The directed line segment P Q is equivalent to the directed line segment P 0 Q0 if and only if c − a = c0 − a0

AND

d − b = d0 − b0 .

If you don’t remember what an equivalence relation is, you should look it up. In any case an {exercise} is to prove that the relation I defined on directed line segments is, indeed, an equivalence relation. The nice thing about this equivalence relation is that now we do have a bijection between { vectors in R2 }

AND

{ equivalence classes of directed line segments}

You should think about how to set up the bijection between the set of vectors in R2 and the equivalence classes of directed line segments. Let me give you a hint about how to do that: in every equivalence class there is a directed line segment that starts at the point (0, 0). It is an easy exercise to see that we can do the same thing I did above in Rn for any n. I.e. we can think of vectors in Rn as points or we can think of them as equivalence classes of directed line segments. Of course, we can picture this in R2 and in R3 but we have some difficulty in drawing pictures in Rn for n > 3. But, that shouldn’t stop us from trying to make the analogies. Algebraic Operations on Rn In your linear algebra course you have already seen two algebraic operations you can define on the vectors of Rn , namely scalar multiplication and vector addition. Recall that these were defined as follows: scalar multiplication on Rn : if u = (a1 , . . . , an ) ∈ Rn and λ ∈ R then, by definition, λu := (λa1 , . . . , λan ). 5

vector addition on Rn : if u1 = (a1 , . . . , an ) and u2 = (b1 , . . . , bn ) ∈ Rn then u1 + u2 = (a1 + b1 , . . . , an + bn ). Notice that both of these operations are very formal and are defined on the vectors of n

R for every n. It remains to see what these operations “mean” when we take the different interpretations of what a vector is. (Do some examples in R2 , see what it means for the interpretation as “points” and then for the interpretation as “directed line segments”.)

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