All right, let's start the first lecture.  Well, hello and welcome to math 110. We used to call this class linear algebra. Now we call it abstract linear algebra, okay?  So it's like, you know, improved and upgraded, okay? But basically the same.  I want to start, I want to start by telling you about some administrative stuff, okay?  You will see that there will be lectures, obviously here on Tuesdays and Thursdays, 5-63510 to 630. There will also be section meetings on Fridays. Each of you will meet with a GSI on Friday. You know your schedule. There will be office hours, which are listed on this page of courses. And you are all welcome to attend all office hours, including those of the GSIs who are teaching other sections.  The textbook that we will use is called Linear Algebra Done Right. What could go wrong? Right. It's actually the first time I'm teaching from this textbook. It's by Sheldon Axler. Everybody loves it. I wanted to try it. I like it so far. But keep in mind that this book has several editions, and we are going to use the latest edition, which is the fourth, fourth edition. It is different from the previous editions. For instance, it has more material and it has more exercises from which we will draw homework assignments. It is essential for you to have the latest edition, which is the fourth edition, the book version. The PDF file is linked to on B courses. You will also be able to buy a paper version if you like. I think it should be available at the cal store but also at Amazon everywhere. You do not need a paper version, it's up to you. And the book version is available on courses for you to download, to print out whatever you want to do with it. Okay?  There will be two exams, so this is the most important part, okay? Because I'm frankly getting tired of e mails from students asking about various alternative arrangements for exams and quizzes. Let me say it once and for all. There will be no alternative arrangements for exams or quizzes. End of discussion. I have to be fair to everyone. We cannot, in a class of this size, organize alternative things. We simply do not have resources for this. Please make sure there is no conflict. You have no conflict with the mid term exam, with the final exam, and with the quizzes. Now, sometimes things arise, as far as quizzes are concerned, that you have to miss a quiz. For instance, you are ill or something, some kind of emergency. For this reason, we will drop 22 worst scores of every student, and that should cover it for everybody. So please do not bother the GSI's. Do not ask me about alternative arrangements. Even if you have a valid excuse for missing your quiz, there will be no alternative arrangement for quizzes. I hope it's clear. It's written here as well in red. Okay. Now, that's about the quizzes, no make up quizzes. About the exams, there will be one mid term exam, which will be right here Okay. During the time slot on Thursday, February 29. This is something I do every four years. Okay. And I guess it went over. Okay. I guess it's all about delivery, so Okay. I'll have to work on that. Some new material, you know, last time was four years ago, obviously. Anyway, the midterm exam will be on February 29, the 30th. Okay. On the 2095 to 630. Right here. Okay. And that week I will also give a review lecture. The final exam will be on Friday on the last day of the exam week. May 10, 1130-230 It's exam group 18. Absolutely. No makeup exams under no circumstances. Okay. Let's just understand this once and for all. There's one small thing that I want you to remember. Things can happen, student may get ill, it could be some emergency during the mid term exam. Have to provide sufficient documentation for that. If that's the case, then I will excuse you from the mid term exam. Your grade will be computed in a different way. This is explained on that page. I'm not going to get into this to save time. If you miss the final exam, you fail the course, or you get an incomplete. If up until that point you have completed all other assignments and have a passing grade C or better, that's the deal. Okay? Make sure you are available on those days and you don't miss those exams. Homework, I will assign homework after every lecture. This is posted on the courses side. You can already find there the reading assignment and the homework assignment for this lecture. Okay? On Thursday, I will post more material, more stuff, reading assignment, homework assignment. Every Thursday, I will also post what we will be doing the following week so that you can read ahead, which I think is a good idea. You don't have to understand everything, but just grasp a little bit red for five, 10 minutes even. I think when you come to class and you hear me talk about it, you'll be prepared. You'll be better prepared for it. So that's my advice. You don't have to do it, but that's a useful investment of your time. Homework will not be collected or graded for a very simple reason that solutions are available online, okay? In this day and age, we can't do it the old fashioned way. But homework, doing homework is extremely important. This is one of your main resources. Your main resources are these lectures, your section meetings, the book, and the homework. Last but not least, because homework will give you a sense of what to expect on the quizzes and the exams. And I actually have a habit of including homework problems on my exams. Just tell you upfront so it is in your interest to do the homework even though you the homework will not be graded.  To learn the material, I will post the solutions by Monday morning of the following week. So, for instance, this week we have lectures on Tuesday and Thursday. I have already assigned the homework for Tuesday. I will assign the homework after additional homework, after the lecture on Thursday, and I will post solutions either Sunday night or Monday morning.  But don't fall into the trap of just reading solutions and not doing the problems on your own. It's very easy to pretend that you understand the material this way. Try to do it by yourself first. If you can't ask fellow students, ask GSI's, ask me when all else fails, read the solution.  Don't try to look as they say in the back of the book right away. What else the grades written on? I will keep this. So this lectures are going to be recorded and I can see myself in there, so I feel like it's happening. They are going to be in the media gallery. I think that on courses, they made some small changes visa the US. But I will make an announcement this evening. When I am able to post it on courses, I will send an announcement to everybody so we will know where to find it. This way, videos of lectures will be posted on Tuesdays and Thursdays. I do not recommend to use this resource for skipping lectures for two reasons. First of all, I know myself, if I say, okay, I'll just watch it later, 99% of time, I don't. Okay. So let's be honest about it.  And number two, even if you do, it's different when you are watching it on your laptop and you get interrupted and stuff like that, as opposed to when you're here in the classroom. Now, I'm not going to take attendance. You're adults. You decide for yourselves whether you want to attend lectures or not. It's just my recommendation to attend lectures in person rather than fall back on. Oh, I will watch 20 hours of videos, you know, before the exam. Maybe it will work for you, but unlikely. All right. Any questions? By the way, you guys in the back, feel free to come here. There are plenty of seats in the front. Yeah. Especially on this side. So unless you prefer to stand, it's fine. All right. No questions. Oh, God, been 10 minutes. 12 minutes. All right. So, linear algebra. What is this all about? Linear algebra is a very important subject for several reasons. La algebra study is what we call vector spaces. I don't like this. Let me see if my choke is bad. I don't like that one. Yeah, it's better. Is a study of vector spaces and maps between them, which are called linear transformations and maps between. Okay, so what is the vector space? So the simplest example of a vector space is a line. The history of the vector spaces was such that mathematicians decided to start with the simplest geometric objects that could possibly exist. What are the simplest mathematical objects? The simplest are points, right? A point is just a dot with no inner structure, and we call it a zero dimensional space. In this subject, there's not much to talk about. The first non trivial geometric figure is going to have dimension one. The simplest one is a line. A straight line, okay? Maybe loose a little crooked. But I'm trying to show. Imagine idealized line which is completely straight, okay? So something like this. If I just drew this line, there is not much structure. But I can add some structure to it which will make it a much more meaningful object. First of all, I can choose orientation. In which direction does it go from the left to the right or from the right to the left? There are two possibilities, but traditionally, we put the orientation from left to right. It's just a convention. Okay, that's good. Next we pick a reference point, and we call it zero. Once we do that, we get a notion of a vector. A vector in the first approximation is a directed interval which starts at this point, at the initial point, goes to another point. It has an orientation. It could go like this, or it could go like that. Each vector, therefore a direction and the magnitude or length. In the case of a line, there are only two directions it can go. We will soon consider the next example, which is a plane. And there will be a whole infinity of directions to choose from. It also has a length, now is not something that is given to us. We have to choose a unit length, we have to choose the standard length to speak. Let's say this is one. Then once I have this point and echo it, one, I will have a 0.2 which I just put this same length from one in this direction, I get this 0.2 then 0.3 and so on. And likewise, here I will have negative one, negative two. Suddenly, this line becomes alive. It is what we call the number line. Every real number has a home on this line as a point. For instance, number zero is here. Number one is here. Here is 12. Okay? Then somewhere here is 3.156 and so on is number pi. What can we do with numbers? With real numbers, of real numbers, we have a set of real numbers. The line, now points of the line get identified with real numbers. We can think of points as numbers. Now, the totality of all points is what we call the set of points of the line. That's the line from the genetic point of view, if you want to think of this line as a collection, as a totality of all of its points. But likewise, you can think more abstractly of the set of real numbers, the set of real numbers. As follows, the letter R, but with a double line. Sometimes you use bold face for it, but I will use this notation. The set of real num is very important if you just look at it. As I said, this is just a collection of things. It's not very interesting. But it has two operations. Namely, if you have two real numbers, x and y. I used here the notation for member of belongs to have two real numbers, for instance two and pi. Then you can attach to it it's sum, also this sum or addition, addition operation of addition, but you can also multiply them, okay? We'll write it either x Y or maybe x dot y. Usually, we will not put dots just to simplify things, but if you prefer to write dots fine as well, this is a multiplication, right? This is true for every pair. For every pair X and Y, there's a well defined pair which is called, obtained as a result of addition of X and Y and the product obtained as a multiplication of X and Y. That's not all. There exists a special element. This call there exists, it's a, it's a symbol for there exists a special element which is called zero. There exists a special element, one such that if you take the sum of anything with you will get x. If you take the product with one, you will get x. They are called neutral elements. Neutral for addition and multiplication respectively. I'm going slowly because this will be my blueprint for what will come next when I give a rigorous definition of a vector space, this is what mathematicians call a formal system. This is a framework in which all of mathematics unfolds. These are the data they satisfy. Axioms, it's called axioms. I will explain this more precisely in a moment. For now, think of them as the properties which these two operations satisfy. Okay? What are these properties? First of all, there is commutativity for both addition and multiplication. X plus y is y plus x. X times y is y times x commutativity. Then you have what's called associativity, which is the following. If you have three elements here we're talking about different order of applying the operation. Either x is the first argument and y is the second or the other way round, we're saying the result is the same. Likewise for multiplication, as social activity is about something else. We don't change the order in which xyz appear, but we look at different orders in which they could give us something meaningful. Operations of addition and multiplication are what's called binary operations. Binary because there are two inputs, X and Y. But suppose you have x, y, and z. How can you possibly produce something by using, say, operation of addition? Well, you can take the sum of x plus y that would be an element of your set, or in this case, Si set of your numbers. And then take the sum with z, right? In other words, you're trying to think of the operation of addition as a black box, which has two inputs and two outputs. The inputs are called x and y, and the output is called x plus y. Likewise for multiplication. But now if you have x, y, and z, what you can do is you can feed them into the box. So that you get X, X plus Y, and then feed the result. Actually, let me put it like this, X plus Y again, into the box. That's the left hand side. X plus Y plus, whereas the right hand side is going to be the other way where you first apply, feed Y and Z to the box, right? And then feed the result together with X into the box. That's the right hand side of associativity x plus y plus. There are two ways in which you can get something out of three numbers that corresponds to putting brackets in two different ways. As axiom of associativity, or property of associativity says that the result will be the same. And it is a very important state statement because after that we don't have to worry about how we put brackets, no matter, in other words the expression, it will be meaningful to simply write x plus y plus z. Because in general, this is not meaningful. Because you will ask, in what order do I do the operations? But because of a sociativity property, it doesn't matter. The result will be the same. Choose the one you like. This sum of three becomes a well defined element, right? So instead of an operation whose input is two elements, you've now got a well defined operation whose input is three elements. If you think about it, the same argument works for any number of inputs. This will also be well defined, will not depend on how you put brackets, Because by using this property, you can always rearrange brackets in any way you like. It is an important property in some sense. It's more important than commutativity because actually we will see examples of algebraic objects in this course called matrices. Of course, you've seen them already because I didn't mention it, but goes without saying that the prerequisite for this course is my 54, which is a calculus version of simplified version of linear algebra. You already know what matrices are. Matrices with respect to multiplication. They do not satisfy. They satisfy a sociativity, but not commutativity. It turns out that commutativity is not really that important. It is the most important property. This is the most important property. And likewise for multiplication, I'm not going to write it. I think it's clear, right? So, these are the first two axioms. Then you could say that the are also, these are also two axioms, actually. So, in other words, it's better to say Axiom, put it here, okay? Because these two are also properties that we require, which are satisfied in the case of real numbers. And there is one more, which is distribute how the multiplication and addition interact. If you have x t, this is, I think that's it. Let me check. You always have to be careful with Axios, miss any of them? Oh yes, I forgot something important to write. Of course. Then we also have inverses, right? Then there is also inverses for any x, for every, this simple means for every x have already used it. Another element, which is called minus x, such that x plus minus x is zero, is this element that we started with. That's the additive inverse. We call this additive inverse then, for any x, which is not x in r, but here it's without the element zero. Every non zero element has a multiplicative inverse. Right? There exists an element which is called x inverse, such that x times x inverse is the fictive unit or alive identity. What do you call it? Afltive identity one. Okay, now here's an important point which I have not yet mentioned. This course is interesting because it's supposed to be one of the very first courses that you take where the idea is to actually present mathematics in a rigorous way. Rigorous way means that we prove everything and not just take things on. Fate one of the Themes here of this course is to really understand what is a proof? How do mathematicians prove things and then learn to understand, and ultimately to create your own proofs? Okay, how does mathematics proceed? Mathematics proceeds in the context of what's called a formal system. What I have just described is an example. And this is why I wanted to go slowly and to describe it in detail. So bear with me. A formal system is an important point, is an important notion. This applies to all of mathematics. Today, maybe one day mathematicians will come up with a different way of formalizing what we do, but for now, this is how we do it. In every subject, in every part of mathematics. Mathematics develops in the context of a formal system. What is a formal system? A formal system starts with its own language. You have certain symbols and alphabet, and then you have what's called well formed words. This is very similar to natural languages. But perhaps an even better analogy would be a programming language. If you ever had a chance to work with a programming language that's a metaphor or a great example of a formal language. First of all, there is an alphabet. Everything is expressed in terms of strings of symbols from this alphabet, including the space, so that you can have spaces between words. Then some of the sequences are meaningful. They're called well formed words, and some of the sequences are not meaningful. Right? That's clear. After that, we have sentences. Sentences are obtained by putting well formed words together and separated by spaces. So far, so good. That's a language now from all sentences, formal language, I'll put a link to something. I'll find some resources if you want to read more about this, but it's not going to be essential for us. I'm giving an overview to set the stage for what we're going to do here. I think it's very important to understand how Mats proceeds and what exactly is a proof Course is advertised as one in which we will learn how to understand and write proofs. But what is a proof? Okay, I'm explaining that I'm giving a very brief overview. Proofs arise in formal systems in the following way. You got formal language, you got sentences in that language. But some of the senses are special, and they are called theorems or propositions. These are the ones which we prove are special. They're called theorems. Propositions. Sometimes when we do mathematics, we also have a hierarchy of special sentences. Some of them we call lemmas, some of them we call propositions, some of them we call corollaries theorems, and so on. Here I lump them altogether into this name, theorems just to simplify things because we're not putting any value on how important it is A theorem is. Oftentimes people say theorems are true statements in the formal system. But in fact, I want to warn you as the first approximation. A good way to think about it, that theorems are statements in a given language. But in fact, this becomes more and more important as artificial intelligence gets developed. It's very important to understand that there is a mathematical theorem which is due to a mathematician who was oppressor. Here his name is Alfa Tarski. He started the School of Logic, which is world famous. If you look at the rankings, university math programs or rankings, we almost always get number one in logic. Alfred Tarski was one of the first famous logicians in Burke at Berkeley and started the tradition. Perhaps there were others, but he's very famous. Tarsky's Theorem says that you cannot encapsulate the notion of truth within a formal system. It's a very important statement. A system is something that has to do with syntax, it has to do with symbolic manipulation. You are putting things together and you are transforming difference one sentence into another by certain rules. This is something that the computer can do. Meaning or semantics is an entirely different thing. It's a certain value that you assigned to your statements. At the beginning of the 20th century when mathemaologic was developing, most mathematicians, mathematicians believed that in fact there is no difference between the two, That semantics or meaning could capture it by symbolic computation. They were proven wrong. The notion of a truth in a formal system would be assigning the value, true or false, to every sentence, satisfying some obvious rules. The question would be, can you design those rules algorithmically? Can you design them within the parameters of this formal system? Tarski theorem says it is not possible, at least for sufficiently complicated formal systems. This theorem is just as important in my opinion as another theorem which is much more popular in our culture, which is called Godes Incompleteness Theorem. Actually, there are two Godes incompleteness Theorems. This is why I'm reluctant to speak about theorems as true statements. Because first of all, for a given system, there are different notions of true statements. There are different ways to put meaning to sentences. In one way, some statements will be true and the other will be false. Well, it's actually obvious because it can away switch true and false. It's like taking the negative of a photograph. It's still a bona fide way of assigning the values of true and false. You see, so there is no unique notion of truth is something that has to do with what we call the model of a formal system when we're trying to model things. Model those sentences as in some concrete applications. But there are different applications and those models will give different notions of truth. Be careful when people say theorems are the true statements that they may well be with respect to a particular model, but the notion of truth is much more subtle. Okay, Now, what are theorems are not true statements. Theorems are things which can be derived from axioms. Theorems are produced in two steps. First, you have to start with some initial data, Initial theorems, which we call axioms. Axioms are not proved. You see, you cannot do much starting from nothing. You have to start with some sentences which you have to take on fate. These are called axioms. In other words, the statements which do not require proof. Every formal system has those because otherwise, it would be empty. If you have nothing to start with, there is no seed. What can you prove? To prove something, you have to start with something that has already been taken as a valid statement. Those are the axioms. That's the first step. The second step is what's called rules of inference. Rules of inference allow you to produce new valid statements from axioms in a controlled, rigorous fashion, okay? And the result of that process is the set of all remaining theorems, okay? In other words, here's how it works. First you have a list of axioms. Let's say axiom one, axiom two, axiom, and so on. Those are the statements which are declared to be theorems. Axioms are special cases of theorems. Every axiom is a theorem, is just a theorem which has no proof, which is taken as valid. After that, you start creating things by using rules of inference. For example, let's suppose that a is an axiom, a implies B is another axiom. Okay? One of the rules of inference is that if a is a valid statement and a implies B is a valid statement, then is a valid statement. It makes sense, right? That's, that's how logic operates. Sometimes the rules of logical inference then is the, there you see how it works. May not be on the list of axioms, long as a is on the list and a implies B is on the list. We got ourselves a new statement which we consider as valid. Can see. Let me, this is this clear by any questions? Yeah. Okay. Am I going to slow? Yes. All right. Let me go even slower. Sometimes important to go slow anyway. Now in general, this is going to look like this. Maybe. Let's give one more example. Let's suppose after that that implies is also an Ax or another theorem obtained. In a similar fashion. Then is a there on it's a recursive definition, there is a recursion going on. App is a list of sentences which makes sense in your formal language, which looks like this. Sentence one, sentence two, sentence three, and so on. Where each sentence is obtained from the previous is either an axiom, or is obtained a serum that has already been proved or is obtained from previous sentences by the rules of inference. And that's how you proceed and gradually prove more and more things. Finally, you arrive at some sentence S, n, then this is called the proofN. See at all previous steps, you either axioms or things that you have already proved before by a similar procedure or something that you have obtained. Well, I guess that's it. Yes. Either an axiom or a statement which has been proved by a similar procedure. Then the last sentence in this list is called a new theorem. The the whole text is called the proof of that statement. I hope it's clear this is how Matmis operates. You start with axioms and then you derive this can be called the proof of SN. But say SN is derived from the previous statements as one, as two, S and minus one. It is a linear step by step procedure. That's why it can actually be programmed. It is very close to the notion of an algorithm which goes back to the great alenturing obviously. So let's go back to the set of real numbers Here I start with a particular model. In other words, I start with a certain object. It's an idealized object. It's not something that we can literally touch. For one thing, my line was not exactly straight line. It had a little bend. But we imagine that there is such a thing as a straight line. We call the points of this line as real numbers. Then we see that set of real numbers, the collection of real numbers satisfies certain properties. The way, the way usually my mine proceeds is what we discover an interesting example of something, some interesting object like the set of real numbers. Then we say, oh, that's interesting. Are there any other objects of a similar nature? Okay, That they will have the same qualities, same properties, same structures that can be found in this object that we had found. In this case, the properties lead us to the notion of a field. Field. Field is a mathematical notion which is described by a formal system. Okay, so what is the formal system? Maybe fields of which this set of real numbers will be just one example. In this formal system, you wanted to be pedantic. I would have to introduce the language and well formed words and sentences. Yeah. Okay. It's called, it's called a shortcut, I guess. Okay, I'm in. This will take forever. I mean, this is the thing. In other words, as a teacher, I have to find a balance between giving the ideas but not getting too bogged down on details. And giving you the opportunity maybe to go and dig deeper on your own if some things are not clear or you would like to get more details. This is where I'm going to skip. The definition of the formal language is what the words and so on. And I will cut straight to the axioms. I have already used, the word axioms. In fact, I don't have to even write anything new, but just a few comments. This is going to be one of the symbols in our alphabet. Remember I said formal language starts with the alphabet. Obviously, that's so for every language, natural or programming. So for instance, this is one of the symbols. This is another symbol. The member of right there exists is also a symbol and so on. This is a symbol which is allowed in our language, in our alphabet. Then there are various statements that plus and equality is two other symbols that we use, and so on. In reality, working mathematicians do not specify every step. In other words, as you become more proficient with this cut to the chase, you don't every time go through every step. But it's important to know that it is rigorous construction, that it is objective in the sense that everyone else will be able to verify it and ascertain it, and everybody will agree that this makes sense or it doesn't make sense. This is how Mathemas operates. They should not give you the impression that there is no subjective element in mathematics. First of all, there is an obvious objective element, that it's human mathematicians who have verifying proofs. Well, eventually, maybe computers will do it. If it's a human mathematician, there is a subjective element that people make mistakes. They are famous examples of mathematicians believing that they have proved something only to be proven wrong. But that's not a very serious way in which subjective enters mathematics. Because you could say that ultimately you can actually walk through every step and eliminate the possibility of loopholes, quite possibly, true. But there is another place where the subjective enters, That is, when we choose axioms, Nobody can tell us which axioms to choose, and there are different options. A famous example of that is Euclidean geometry. Euclidean geometry was the first example in history, as far as I know of a formal system. Euclid, who was a Greek mathematician who lived about 2,200 years ago, wrote his famous books called Elements, in which he tried to formalize the geometry on the plane. When I say a plane, think about this blackboard extended to infinity in all directions. Euclid came up with five axioms. First of all, there is a form of language. There are words in that language. What are the words in Euclidean geometry, at a line, a triangle, a circle, and so on. A sentence could be point is a point of it belongs to line l or something like that. Okay? Then there are five axioms. Then he was able to derive many interesting theorems from these five axioms by using the rules of inference that I mentioned earlier. For instance, you could derive Pitagar's theorem. You could formulate the notion of right triangle in this way. In the formal language, the one which has 90 degrees, you could explain what 90 degrees is by constructing bisectors and so on, by comparing angles to each other. And that's not hard. Then you have a notion of a right triangle. Then you have lengths of the sides. The two sides of the right angle are x and y. And the third side is the length of it is z. The hypotenus, X squared plus y is critical, Z squared. He literally derived it in his book of elements, actually, collection of books. Okay? But there was one thing which was nagging him, which is that out of these five axioms, four were very natural. For instance, if you have two points which are distinct on the plane, that there is a unique line which passes through them or which contains them. Well, that seems very obvious, right? Three others were like that. But the fifth axiom, the famous fifth postulate, was something that made him a little uncomfortable because it wasn't clear, why should you believe that? And that was the so called fifth postulate about the parallel lines. That if you have a line on the plane and you have a point outside of it, then there is a unique line passing through it and parallel to this one. Parallel means that they never intersect. There is such a line for every point outside of your original line, and it's unique. That was the fifth axiom. For the next 2000 years, generations of mathematicians tried to prove it from the first four, right? Because I just explained how proofs work. You want to get to that fifth axiom by sequence like this, where you do not use the fifth, but only the first four and whatever you can derive from them. And they all failed miserably, but they kept trying until finally, in the 19th century, three great mathematicians around the same time realized that it's impossible. The fifth Axim was really independent from the first form, and therefore it will actually be changed by its negation. We could say that every line that passes through this point intersects this one. Now I cannot draw it on the plane, because the plane will no longer be a model for this new formal system. But a good model can be found. If you consider the geometry of a sphere instead of a plane, consider a sphere. Think of a globe. It has points, just like there are points on the plane. The analogs of lines are now the meridians and all other largest possible circles. In other words, all the meridians go through North and South Pole. But you also have a equator which has the same radius, you can always like tilt to the equator. You got many other circles of maximum radius on the sphere. Those are the lines so to speak, or what replaces the notion of a line in sphercle geometry. Guess what? Every two meridians intersect, every two large circles intersect actually at two different points. So that's how they realized that. Actually you do have the freedom to choose axioms differently. If you do so, you get totally different mathematics theorems of one formal system, even if you just replace one axiom by another one. The theorems of one formal system will not be the theorems in the new formal system and vice versa. That's where the subjective element is in the choice of the axioms. Anyway, this is, you could say, philosophical in some sense. But I think it's important if we really want to take seriously the idea that in this class we are going to give proofs and, and create proofs and understand proofs on. I think it's important at least to understand in a basic fashion what this is all about and what are the ingredients. But let's go back to the formal system of the theory of fields. As I said, we have to define the formal language and so on. But let's keep that step. Let's go straight to the axioms. These are the axioms, the sentences which are meaningful in the formal language of the theory of fields. We have two operations. First, we have some additional structures which are introduced. This is a definition definitions. When we declare that certain, certain properties are going to be called this, then we can use that going to be a word in our formal language which we will use to label objects which have those properties To simplify things, we don't have to each time list those properties. A field is a set, let's say, with two operations which are called plus and do binary operations in the sense I explained satisfying the following actions. Those properties are taken for granted that they are satisfied for this field. You see for now, note the shift. It's subtle but very important. Earlier half an hour ago, I talked about a specific set, the set of real numbers. I talked about specific appirations of additional multiplication. I talked about their properties. I did a slight slight of hand by calling the maxims. At that time there were not ideally properties. That's what a model is. We're talking about a specific model of a field. A specific field. In this case, we actually need to verify whether it's true or not, that x plus zero is equal to x. But it's easy to do just from the definition of those operations. But now we are taking an abstract point of view. We are not thinking about a specific set with specific operations. We are trying to set up a formal language in which we can speak about objects just like the set of real numbers, which will have the same properties and the same structures that the set of real numbers enjoys. Okay, that's how it's done. It's done in a formal language, it's done in a certain particular frame, controlled rigorous framework. And then what appeared to us before as properties of a particular field now become the universal properties of all fields. You see now we are taking them as axioms because in the general set up there will be, in the general set up of a formal system, there is no way to derive these statements. In fact, they are precisely the statements from which we will be deriving things. I hope it's clear, but please ask me if it's not the things upside down. Now, instead of starting with an object and some structures on it and verifying that they have certain properties, we start with the properties themselves, taking them as axioms and searching for other objects of a similar kind which satisfy those axioms. Those will be the other fields, so we get a whole universe of fields of which are the set of real numbers will be just one member. Okay? So that's how things are done. Now I have swept under the rug something important. As you see this kind of a Russian doll, there are more layers, but I promise you that there is only one essential layer and not more. Okay, and what I have swept under the rug is this notion. So in fact, you could have asked me. Okay, so you said mathematics develops within formal systems. But what about the foundations of mathematics? Here we're talking about specific specialized field, which is called zero fields, no pun intended, or specialized part of mathematics. Likewise, I talked earlier about Euclidean geometry, which is also specialized chapter or sub field of mathematics. Okay, But what is t max itself based on? How can we talk about anything without some kind of foundation? Today, that foundation is called said theory. The real foundation is what's called a sad theory and it is a formal system itself. This is also a formal cyst. It comes with its own language, formal language, and its own axioms. In fact, there are different versions, just like there are different versions for geometry, for two dimensional geometry, Euclidean non Euclidean and so on. But most mathematicians today use the agree on the same one, which is called, Of these two are owner of initials of two mathematicians, Zermelo and Frankel, no connection to me. As far as I know, C is for choice. Axiom of choice. There is a particular formal system whose basic structure is just like the axioms of fields, or the axioms of Euclidean geometry. But much more sophisticated, which enables mathematicians to talk about sets. Every other formal system is built on top of this one. You see that the true foundation, there's nothing underneath as a side node. As I said, there are different versions of FC, there are different versions of the set of set theory. For instance, there is a version in which infinite sets are not allowed and so on. So that's where our subjective contribution is. Some mathematicians choose FC. Some mathematicians you choose something else. So there is some actually ambiguity here as well. But once you have made the choice, you say, okay, ZFC here, we will make this choice. Okay? So it's like clicking on terms of service without reading the contracts, like you sign up with your Internet provider. If you have no objections, it means that you have accepted it, okay. This class, we have accepted FC even if you didn't know what it was until now. So that's the theory of sets. Now the interesting thing is the notion of a set has actually never been defined. You see how interesting the creator of the theory of sets, or shall we say discoverer, was a German mathematician, Gar Cantor, who lived in the 19th century. He gave a poetic definition. He said, a set is the many that allows itself to be thought of as one. A set that allows itself to be thought of as one. It's a nice intuitive definition like the set of students in this classroom. Or a set of old chokes on this blackboard, and so on. So it's like a collection of things which have some similar properties, but it's not a rigorous definition. Mathematically, this place, we agree that we have a similar intuitive understanding, at least in one notion. Mathematicians have to resort to trust. Trust, Your understanding of set is the same as mine. You see, We can have a discussion about it. We can also study many different examples, like the set of real numbers or sets of points on the vector space, which I'm coming to right now. But ultimately, it's something that we have to accept trust, okay? So when I say a field is a set with two operations and so on, it means I'm using the formal language of CFC, I'm using formal language of set theory in that language. There is also notion of a map between sets. There are sets like set of real numbers, but there are also maps or functions. It works like this. If you have and at two sets, then a function from S to t or a map is a rule which assigns to every member of S, one and only one member of T. Now once you have that in your formal language, you can speak about bin reparations. A binary reparation is a binary reparation is a map from cross to you see the first step is what's called Cartesian product. If you have two sets, you can form a Cartesian product. This appears an element from the first, an element of the second. In this case, both of them are the same. A function from rof is the same as the rule that attaches to a pair of elements of f, x and y. Some element of operation of addition is like that is a function from crossF. Likewise, operation multiplication is a function from crossF. This way I give you a formal definition of bind reparation in the framework of formal system of the theory of sets which FC, but that particular part is shared by all possible formal systems, all meaningful formal systems about set theory. That is more or less how things proceed. Okay, I don't want to get too much bog down in details now. What are the vector spaces? Now, in this whole universe of formal systems, a vector space is an attempt to generalize the notion of a field to higher dimensional objects. That's one way to put it. Here is what I mean. You have your theory of fields, which is great. But then we find a very nice example of field. You see, once you have the formal system, formal system enables you to speak about objects which have certain structures, sides, finds properties. In this case, those objects are called fields and foremost as a set. It starts as life as a set. Everything in mathematics today is defined as a set. As a side remark, there are other approaches now to foundations of mathematics which involve what's called category theory and so on, homotopic theory and so on. I'm not saying this is the only way, but today every mass book you open will always start with a definition. The objects of this theory are sets endowed with such and such operation, scifying such and such axioms that will be So if you study group theory, it will start with the definition of a group which will go like this. A group as a set with an operation bind, reparation, et cetera, et cetera, such and such axioms. Or you have a theory of differentiable manufolds. Differentiable manufolds a set. Such and such structure satisfying a Hilbert space Siyuch. Things you see underneath it all is set theory, which as I just explained, comes from a particular formal system. What we are considering here is a next level. If you study computing, you know that there is ultimately, everything is converted into 0.1 And then there is a lowest level programming language, Assembler, which is alter one but still has some symbols. And then you have more complicated languages like Java and so on. Formal system of field like C plus plus and set theories like Assemblers, the lowest level of programming language. But see how beautiful it is. This idea, once you develop your formal, the formal system of fields, you have already stipulated what your objects are. You have stipulated what is the seed? What are the statements which are taken as valid without proof? These are the axioms. After that, you have this machine of producing more theorems by rules of inference. I will show you how it works. Maybe I won't have time today, but beginning of next lecture. Okay, So in other words, these are not abstract words. That's exactly what we'll be doing, but that's what proof is. What is the value of that? The value of that is that as soon as you find another set which has these structures satisfying these properties, you know that every statement about fields is true for it. The bottom. You don't have to prove it each time for this field or for that field, as long as you know that the aims are satisfied for this new set. That's what I mean by examples of fields. See how much more powerful this is than just looking at the set of real numbers and studying the operations of addition and multiplication, and finding out what the properties are. It's much more powerful because it's universal. Every field now satisfies all of the strems that the field of real number satisfies because you've isolated the seed, the initial valid statements, the axioms than anything else that shares the same set of properties and other satisfies, the axioms will also satisfy all the valid statements vis or also known as strems of that formal language. The first example is, that's the one we just discussed, which is represented by a real line in which you have point 0.1 right? But here's another example. It's called a field of complex numbers. A field of complex numbers consists of pairs or expressions of the form x. Let's call it a B I, where I is a special element, who square is negative one. Then it also has operations of additional multiplication which satisfy all of these axioms. Because of that, the set of expressions like so where a and B are real numbers, it becomes a field. Therefore, every statement that you obtain, every theorem in the theory of fields, applies to it and is going to be valid for the field of complex numbers. Same way as it is valid for the field of real numbers. Complex numbers will be important in this course, our vector spaces that we will consider the underlying field, Every space will have underlying field. In fact, a vector space can be defined in a general context for an arbitrary field. But in this course, we will look at just the cases of real numbers and complex numbers. Complex numbers are important, and the definition of complex numbers, definition of the operations of additional multiplications, so on, is given in the very first section of our textbook by axlerIink. Actually raise your hand if you have had a chance to interact with such complex numbers. Okay, good. So that's what I thought. I guess it's not something new, so I decided to skip that. So this will be your reading assignment. Just a few exercises from the back of that section just to refresh your memory. I don't want to spend time on this because it seems that it's something that is already known. Okay, So why do we need complex numbers? I'll get to that at the very end. But these are the two example of fields. Now, what is the vector space? A vector space is an attempt to create a similar structure, but not for R, but for example, for R two. What do I mean by r two? R two is rrosR is a Cartesian product of r with itself. I will write it like this. There are pairs of real numbers. If I put curly brackets, think about it as part of my formal language. If I put curly brackets like this, it means I'm describing something. I'm describing the collection of all possible symbols on the left side of this divider, where the values of those symbols are described on the right side of the divider. This means that R two is collection of all pairs of real numbers, but ordered pairs, we know which one is the first, which one is the second. First one is called a second one called B. We don't identify AB and BA. The great French philosopher and mathematician, Rene Decart, realized that pairs of numbers, just like real numbers, could be thought of as points on real line points, elements of R two can be thought of as points on a plane. For that, you need to choose a coordinate system. A coordinate system looks like this. So it's a coordinate cross, right, Two lines, then you choose one. This is zero and then there is one on this side. I'm reluctant to use x and y because I used X and Y over there. I want to use different nation. If you have a pair A and B, you simply plot a on the horizontal axis, on the vertical axis. After that you do this, that's the point on the plane which corresponds to the spare A and B. This is familiar, obviously, right now the question is, can we introduce some operations on this which would make it into a field? For that, we would need operations of addition and multiplication, right? The operation of addition is very easy to define a set of pairs of real numbers. Real numbers AB. So you have two elements. Let's say this one is B and this one is C, D. Then we will define addition by taking the sum of the first and the sum of the second, right? That's the familiar rule. A plus C plus D. Okay? That's nice, and it has a nice geometric interpretation, which is called the paralogram rule. So suppose you have this, this is where vectors show up. That it is useful to think of elements of our two not as points, but as point at intervals like all of them, as the initial point, this point at the intersection of the two axis, which will call the center of coordinates, the end point will be our point. If you have another one, let's say like this, then we can follow the paralogram rule. Draw a line parallel to this one through the end point of this vector. Draw the line parallel to this one through the end point of this vector, we get a parallelogram. Then we connect the origin at the central coordinates two to this point. This is, if we call this x, we call this y. Now traditionally in a study of R two, we actually put an arrow above it, but let's keep that for now. Then this will be precisely the sum, right? Paragram rule. This is going to be x plus y. In other words, you can calculate the coordinates of it algebraically by simply taking the sum of the first coordinates of x and y and the second coordinates, x and y. But you can also present this, represent this geometrically by means of the parallelogram rule, that's separation of addition, it satisfies all the axioms. See the half of the axioms roughly are about addition, half of the x is about multiplication. And there is one which is called distribution distributivity, which involves both. If you look at all the axioms of addition, they will all be satisfied for two with respect to this binary operation, right? What about multiplication? This is where, this is where we hit a wall. If we try to multiply the components a times B times D, the axioms will not be satisfied is two field. It turns out that there is no way actually, to define a structure by multiplying, but in fact, there is a way. By interpreting a pair AB as a complex number, there is a way. But for instance, if you go to R three, then there is no way, there is no trick like complex numbers anymore. For R four, there is called quaternions and that's about it. Instead, we will relax slightly our structure. We will consider slightly less powerful structure, which is called scalar multiplication. A scalar multiplication will be a box where one of the inputs is a vector, that is an element of our two, say the other one is a scalar, and a scalar is an element of your field. Instead of being able to multiply two vectors will multiply a vector by a number. All right, I'm out of time. I'm almost at the point of giving the axioms of vector spaces. I'll pick up on that on Thursday. One more thing I want to say before you guys leave is in this course we have a head TA Alex Burka.