So let me, let me, uh, let me first recall what we discussed last time.     Briefly, I gave a brief overview of formal systems.     By the way, I have included two links on courses under modules to articles about formal systems. In case you would like to learn more, this will not be tested, It's just for your information,     I gave you an idea of what the formal system is.     There is a formal language, there is grammar in that language, there are sentences in that language.     Then there is a special collection of sentences, called axioms, which are taken as valid statements within this formal system without proof.     After that, we use rules of inference to produce new sentences that we consider as valid.     You could say, as mentioned last time, that there is a subtle difference between theorem and a true statement.     In fact, Gödel's First Incompleteness Theorem says that in any sufficiently sophisticated formal system, there will be true statements that cannot be proved by the tools, by the methods that we are discussing.     There's a subtle difference between true and provable, but in first approximation, you can think of those statements as the true statements in our formal system.     I prefer the term valid, valid in our formal system. Provable in our formal system.     Okay, This way we obtained by purely syntactic manipulation, by derivation, the way I explained that you have.     A sequence of sentences starting with an axiom and ending with a particular sentence.   Where to go from each sentence to the next. We use rules of inference.     I will give specific examples of that in a moment.     The beauty of it is that when you do that, you don't need to know which field or which set you're talking about. You're talking about abstract concepts which are introduced in your formal language.     And you are driving statements about these objects by using a very precise set of rules, Precise in their objective.     In other words, all mathematicians have agreed that these are valid methods.     Now, how should I say renegades? There are other schools.     For instance, I mentioned that for most mathematicians today, the formal system that describes set theory, which is at the foundation of all of modern mathematics, most mathematicians use the so called formal system, CFC,    but there are some who use a different version. I am right now, about the mainstream, so to speak, mathematics. That 99% However, most mathematies, even if they use different axioms, they still agree on rules of inference, so on.     Even if they don't agree, the disagreement is objective. There is no disagreement about what the disagreement is. You see what I mean? In other words, you can always get to the bottom. It's not about who's right, who's wrong, it's about consistency of the tools that are being used. In particular, the tools that we are using are consistent.     And these are the tools that are used by everybody else which enables us to check each other to verify each other's proofs, and eventually might enable computer computer programs to verify proofs.     To what extent they can actually come up with interesting them is open to debate, but certainly the idea is that computer programs eventually will be very, we will be able to use more precisely computer programs to verify proofs.     It's a technique, very powerful formalism.     Okay, Now, as I explained, the first formal system services describe set theory.     That's the one which is at the foundation of all, most mathematicians today subscribe to FC, the formal system. Fc in particular,     since it is a formal system, as I explained, it comes with its own formal language, right?     So it has an alphabet, it has some words in its vocabulary, and it has its own grammar.     In particular, the words which are used, the words set a member of a set, the map between sets from one set to another.     Those are the expressions which are valid, which makes sense in the formal language of set theory. In the formal language of this formal system.     Then we use for set, typically we use capital letter. For a member of a set, we use a small letter, this symbol to indicate this membership. Small maybe like this to emphasize that it's small. Small is a big. The set could be the set of real numbers. Small could be zero or one, or pi, et cetera.     The map I explain also. I recall the notion of a map or a function from one set to another. We denote by, we use these symbols as one to S two, where S one and S two are two sets is a rule which assigns to every member. When we say member as another term is element.     Obviously, if I wanted to program this in a way that would enable a computer program to verify proofs, I would have to be a little bit more precise. For instance, I would have to choose this term or this term when we speak about it with each other. We often cut corners a little bit and we're a little loose.     For instance, we use synonyms like element and member, or function, and map. I wouldn't want to say, okay, I'm only going to use the word element from now on, because at some point I'll say member and it will lead to confusion. I am allowing myself to go between these two terms.     This is just to indicate that ultimately must say still human activity. When it is a human activity, we are a little bit loose sometimes as long as we are aware of it and we know that we could go back and make things absolutely 100% air tight. We are okay.    Likewise, my upper function. These are the terms that are allowed in set theory.     Set theory is built to make statements about sets, about elements of sets, about functions between sets. There is one more thing which is important, which is a Cartesian product of two sets of two sets, which is denoted as one cross two here, which is a blackboard left over from last Tuesday.   Nicely, you see an example of a Cartesian product. The Cartesian product.  In a case when both sets, the set are of real numbers, also denoted by R, two can be geometrically represented at a set of points on a plane.  Once we choose a coordinate cross like this, you see technically members of the set A ordered pairs AB, where A and B elements of the original. Likewise, for general Cartesian product, this is quite sufficient for us to proceed.  What do I mean by proceeding?  Proceeding means that now we are building on top of this foundation a more sophisticated formal system, which will include this formal system. But there will be more. In other words, I will introduce new objects. Now I will introduce a new term, a new word in the formal language. I will introduce various additional axioms which describe the objects that are called by that name.  The first step here is what I just talked about last time, is we consider formal system of fields. We introduce new word in our formal system. We're building a bigger formal system, right? If you will. We can say like this. This is set theory. Specifically, let's say CFC, Formal System of fields is one which includes it. We will take all the axioms, because a field is a set plus something else plus additional structure, right?  So that means that to speak about a field, at least we need to know how to speak about sets. So we include the formal language, the axioms of FC. But then we, we adjoin other words and rules of grammar, perhaps servicing those words to the original formal language of FC, we add more axioms that gives us the formal system of fields.  This is new word, but in a textbook. In a mathematical textbook is not going to be written as computer code is going to be written in a more traditional language, we say English language.  The way this procedure that I'm talking about, of enlarging your formal system is handled by giving a definition. A definition adjoining a definition is a process, you could say, of enlarging your formal system.  Because the definition always has the form that such and such thing is a set with some operations or with some addition structure satisfying these actions. It precisely means that you are join a new word and you adjoin more axioms.  In this particular case, the definition we give is the following. We say a field is a set equipped with two functions FF. Remember, F crosseF is already introduced. It is a notion that applies to sets. To know what is F crosF, you do not need to know what is the field, you only need to know what is a set.  But I have just explained that we are assuming everything that can be done with sets. In particular, this makes sense for us already. Ff is just a set of pairs, AB or a and B, R F. The function is going to be like, one of them is called addition and is denoted by the one is called multiplication by dot or just empty space. You see brackets as a small footnote.  Yesterday, two days ago, I referred to these functions as binary operations. If you have a function from x crossF, it is customary to call that a binary operation on. However, I didn't want to introduce too many terms, I might as well just use the word function, which is already built into set theory. Okay? Two then there, I'm not going to repeat them, because I gave them last time. Those are precisely the properties of the field of real numbers you see.  Historically, this is how it usually happens. Mathematicians come up with some object which has some interesting structures. They study properties of the structures, then they look for other objects of similar nature.  For instance, we study real numbers. We realize that it is said which has two functions, which comes with two functions of this form, addition and multiplication, and those functions satisfy such and such properties.  After that, we start looking for other objects of the same kind.  We formalize it by this definition. Now taking those properties that we could verify in the case of real numbers as axioms, you see once we do that, certainly the set of real numbers will satisfy these axioms.  But then there will be other examples, for instance, the field of complex numbers. All right, and there are actually many other examples. Don't think that there are only two fields available. Real numbers, complex numbers. In fact, there also for instance, familiar one is the field of rational numbers.  Rational numbers are ratio of integers, right over here is not zero and both A and B are integers. That's why they're called rational, because they are ratios. That's a field. There are also finite fields and so on. Okay, that takes care of fields.  Finally, as far as we are concerned in this class, we go to the even bigger formal system, which is a formal system of vector spaces. In other words, vector spaces subsumed is a special case of a vector space, just like a field is also a special case of a set. That's how it usually works, is a hierarchy.  But in fact, things could also overlap and not just grow. I'll explain in a moment.  But first, since this is the main topic of this course, let me give the definition of a vector space. Again, giving a definition simply means enlarging our formal system of fields by adjoining new words and new axioms.  In this case, the new word is actually not a word, it's a term, vector space. Okay? And so we, again, I'm going to state it as a definition instead of saying he's a formal system and so on. So a little bit more, a little closer to normal natural language rather than formal language definition is that vector space.  So first of all, to give a definition, a vector space, there is no such thing as a vector space.  To specify a vector space, you have to first specify a field. In other words, it's dependent. A notion of vector space is dependent on the choice of a field.  The proper way to say X vector space is over here is a field. You see, you can't even give a definition of a vector space is without knowing what is a field.  This is why I started with fields on Tuesday. This is also indicated on this diagram.  In fact, the proper notion is not a vector space, but a vector space over a given field.  We have to fix a field, this is fixed.  The axioms are going to be otherwise very similar, but they will be pegged to a particular field.  For instance, could be field of real numbers, or field of complex numbers, or rational numbers, and so on. For each there is a separate theory, there are some links between them, but at the outset they just separate theories.  Okay, the definition a vector space over where again, is a, is a specific field, is a set. You see how. How it works. This is why I said set theory gives the foundation for mathematics because the way we talk about mathematics today is exactly like this.  When we introduce a new notion, we say something in the vector space over ef. It means that it's a set with some addition structures, or something in the field. If it's a set with some addition structures, this is not the only way to set up foundations of mathematics. Just to be sure to be clear, there are other approaches, but this is the prevalent one today. Most mathematicians work in this framework. It's a set V equipped with two functions. Also, one is called addition, and it goes from V cross V to given two elements, U and V. This function gives us an element which we denote U plus V. But the other apparation is not the same as here, function from crossF. Now it's going to be from cross v to v rather than if it were from V, cross V to V, It would look very similar to fields, but this is where the crucial difference is. This is called addition, this is called scalar multiplication. You're multiplying vectors by number. Think of these elements of V as vectors. In fact, I give you example already last time where this pointed intervals from the origin, which is the point of the intersection in this coord cross to some other point. They can be added to each other by using the familiar parallelogram rule. But also every vector in this sense can be multiplied by scalar. If it's a positive scalar, you get a vector pointing in the same direction, but with magnitude multiplied by that number. If it's negative, then it goes in the opposite direction multiplication. It is an analog of the familiar scalar multiplication of vectors on the plane or in n dimensional vector space given some element lambda. I want to emphasize here, traditionally we use small letter U, V W and so on. To make it easier to read, by default, we'll use Greek letters like lambda muni, et cetera. For the lambda v goes to lambda V. Again, we don't put a dot normally if there is no ambiguity. But sometimes we do just to emphasize this is a product and not some typo.  Okay, now, The axioms, what are the axioms of a vector space?  By the way, all of this is in section one of the book. You don't have to copy what I'm writing because you can read it in the book. But I'm going to write it because it's important.  I'm going to reference this when I prove a couple of theorems based on the acts. Will take a few minutes, but I think it's worth it.  So the first one is the axiom of commutativity. I will not write the words for the name of the axiom, I'll just label them by numbers. I remember introduce this symbol also last time. This is a symbol from Set Theory. Actually, for any element we have UV is equal to V. This is what we call commutativity. I'm not writing the word commutativity down just to save time.  The second one, again introduce a shorthand. Okay? $u + (v + w)=(u + v) + w$ I will write U plus V plus equals U plus V plus.  If I just write a formula like this where it's clear that I'm not talking about specific element U, it's generic element. The proper way to write it would be for, for every dad. But then this will take forever to write, okay? So I'm not going to do it.  That's called a associativity Far, it's very similar to fields, right? But that's a associativity for addition.  There are two parts actually is like two A and two B. Which is that associativity also for the scalar multiplication is not always Greek letters, maybe more like Latin letters, but starting from ABC and on to distinguish from UVW. I'm following the same notation as the book. No additional confusion can arise.  Ab times V equals A times V Here, actually it's worth saying, where do these elements belong? It's for a and B, and F V V, right? Also associativity, but not quite because they're not in the same set.  Next, the additive identity,  There exists an element zero v such that V plus zero equals V.  Then I will actually say, for every, to emphasize that is not a statement about a particular, it's a statement about exist, existence now of a particular element.  The previous three axioms applied to arbitrary elements. This one speaks about specific elements. Okay? To emphasize that zero and V here have very different roles.  Zero is fixed and V is arbitrary. I added for every V and V. This is called, maybe I should say it identify, okay.  Number four additive inverse.  For every, there exists a v such that V plus zero.  That's the zero we have introduced in, at number three. The, that's called additive inverse.  But in the case of this is our model example, think about UVW, et cetera are going to be vectors on the plane, which graphically, which we represent by point at intervals starting at the origin and going somewhere else to another point. Then zero is a vector which goes from zero to itself. We can't even draw it. It's like a point like vector, its end point is the same as the initial point.  The additive inverse is just a vector obtained by reflecting with respect to the origin. Okay, so then you have multiplytify identity, which is that if you take one, now what is one is one.  What are you talking about? Well, remember, we have fixed a field which is a field. Now, look at the axioms of a field then.  In the axioms of a field, it is stipulated that there exists a special element, zero, which is the additive identity such that x plus zero is zero, right? Then there is an element which is called one, which is the mark Tifidnstan, which is one times x is x. Okay, Since we have already fixed in this definition is already given. We can use whatever structures exist in. One of those structures is an element one. There's also element zero, that's the one. This is the one from f as an element of it can multiply an arbitrary vector according to the operation of scalar multiplication. Here, I will actually use the dot to emphasize that fact. If you, the axiom says, if you multiply any vector by one, you get back. Yes. Okay?  Finally, there are dist, distributive laws which are similar in generalized, the ones which we had in the one which we had in the case of field, first of all, a times U plus V is a U. Plus V for every second. The second here you have additional vector multiplied by scalar. Here you do, of scalars and multiplied by a vector a and v in V. Yes, so I'm doing autopilots. Thank you. Yes, I meant a and B, first of all. Yeah, you see there's some glitch in the program. Okay. So I switched, was supposed to be here. What I meant here is this. Yeah. So yes, please check, I might off. That's right. See how easy it is to get lost. But luckily I have you guys it will not let me down. I hope for any A and B. V. F and V. V, yes. Do I have your approval? Okay. Thank you. Okay. Any questions? Yes. Yes, also. Wow, I think it's my unlucky blackboard. You see, I've been so good on all other blackboards. And this one. Come on, give yourself. Thank you. Any other questions? Not a correction, but I was wondering. System. Uh huh. Let's go back to this consistency completeness introduced the systems of the previous acid. Yeah. Yeah.  So there is a, so there is a big elephant in the room in some sense, which is called second incompleteness theorem.  We haven't, I haven't talked about the notion of consistency. All right, maybe I mentioned that.  So the idea is that in a formal system, you should not be able to prove every statement.  One of the things that we can talk about FC and actually even in just propositional logic area which is underlies FC is the notion of negation,  that for every every sentence we can also make the opposite statement.  Obviously, both statements cannot be true, at least in traditional mathematics.  But like I said, there is always a footnote.  There is a so called intuitionist mathematics, which actually, uh, this is called the law of excluded middle.  You have either the statement is true or the opposite statement is true.  In the intuitionist mathematics, they allow the possibility that the both statements are true.  But it certainly everybody agrees that you don't want to say that x is true and the opposite of x is true. Okay? In other words, these are statements which contradict each other. In other words, we don't want to say that two is an even number, and also to say two is not an even number.  A formal system which does not produce a statement and its negation is called consistent.  A system which does that, you could say it's one in which axioms contradict each other.  If you have such a situation, the ability to prove a statement and a statement negation of, then if you use the rules of inference, you can actually prove every statement.  The formal system becomes useless.  Every statement is a theorem. It's not interesting.  It's very important to understand what the purpose of the formal system is. You want to prove some but not all statements, or in principle, you don't want all statements to be provable because that would mean that it's inconsistent that you're proving things which negate each other and then it's a complete chaos.  But how do we know that a given formal system is consistent?  And the answer is that, for most of them, we don't.  And that is the subject of the famous Gödel's second incompleteness theory.  Most people here heard about the first incompleteness theory, which is already shocking, which says that in a given formal system. Sufficiently sophisticated.  Which means that in the language of this formal system, you can talk about natural numbers and the addition and multiplication. It includes the axioms of natural numbers. In such a formal system, there will be some statement which cannot be proved but will be true.  In some model of the system, you see that is a very disconcerting. A lot of people take it as a disconcerting thing because it means that you cannot, in principle, reach every true statement by simple deduction from axioms, by using rules of inference, by purely syntactic procedure. It's already concerning.  But the second incompleteness in some sense is even more so in my opinion because it says that not only that, but the formal system cannot prove its own consistency.  Therefore, how do we know that FC is consistent? We don't, if it's not consistent, none of them is consistent. In fact, there are people who, who believe that that's the case.  The way mathematicians wiggle out of it is by saying that if you add some axioms to CFC, you consider a bigger system. Not by introducing fields and so on, but by introducing larger sets or things which are large cardinals.  Then from that big system, but most import you step out of FC. If you step out of FC, you can actually prove its consistency. You prove it's consist in a bigger formal system in which additional atoms are joined. But how do you know that that one is consistent? If it's not whatever you've proven it, you can prove anything in it so it has no value. You see the just kick the can down the road.  Now the question becomes about that bigger system. Let's say C prime. How do you know it is consistent? Well, somebody could come up with additional axioms and get FC double prime in which you will prove consist of FC prime, but then you will not know it's consistency and so on.  In fact, we have to take it on faith. It's actually, it's better than that in my opinion, because what gives us confidence that things are consistent is that as mathematical practice ish, depending on what exactly you're doing, either hundreds or maybe thousands of years of mathematical research which has not revealed any inconsistency To prove inconsistency is very easy to produce a proof of some statement and a statement which is its negation. That's easy in the sense that it's like giving counterexample.  If there are such, then the question is why we haven't found them yet? It gives the fact that we have, if we started doing mathematics yesterday or on Tuesday, they could say, okay, well, there is no confidence. But since we started doing it at least 2,200 years ago in this sense, because that's when Euclid wrote his book, Elements, which established that tradition.  And even before him, pagers and others did something very similar. 2,500 years ago, maybe. Okay, that gives you some confidence, but it's not ironclad.  Proving consistency is much harder. You see, it's showing that it's impossible to prove a statement and its negation.  Now we know that for most interesting formal systems, including CFC and anything that is built on top of FC, it is impossible without stepping out of it.  Simpler formal systems, for example, Euclidean geometry, it actually can be proved. In other words, it doesn't apply to every formal system.  It applies to formal systems which are, which include arithmetic of natural numbers, which includes infinity. In some sense, infinity is a culprit, in some sense, the possibility of operations that run indefinitely.  If you have studied theoretical computer science, or have heard about Turing Machines, you will know that there is an analogue of that in the theory of Turing Machines. There is a so called halting problem. A problem which cannot, in principle, be solved by an algorithm. And again, the culprit is infinity. Because you don't know which programs will halt, in other words, and which ones will go into a loop. Okay, well, I can talk about this all day, but let's go back.  One more thing I want to say about this picture is that I have shown you how to enlarge, go from CFC to fields and so on.  But it should not give you the impression that that's the only way in which mamas can develop.  Because we can add axioms, but we can also remove axioms.  For example, in the case of fields, there is a additive inverse. I explained it last time. And also multiplicative inverse, say every real number, every non zero real number has a multiplicative inverse such that if you multiply them, you get one, right? That's one of the axioms.  One already exists. One exists as an element such that if you multiply any number by one, you get that number back, Right?  One is there, but existence of multiplicative inverse is one of the axioms and you can remove it. Okay? If you remove it, you're going to get something which is smaller. It's, it still contains ZFC, but it's like this. That's extra things that you get from the existence of multiplicative inverse.  You get a theory of what's called a theory of rings. Those of you who have taken Th 113 will remember this notion.  An example of a ring is the ring of integers. All integers, meaning a positive and all negative whole numbers, and zero. This set has two operations, addition and multiplication. They are commutative and associative, and there is a zero, and there is a one additive inverse for every number. You take its negation, but there is no multiplicative inverse for every non zero number only for one and negative one.  There is no multiplicative inverse for two, or three, or four, and so on.  Now, there is a way to enlarge it to get a field, and that's the field of rational numbers.  But the set of integers is not a field. It is what's called a ring. The Crisponianformal system is smaller than right.  But then you can also go in the opposite direction. You can adjoin something to the axiom of ring. You can remove and you can join.  This way, you're going to get a, it's more like a tree of systems. It's like a Russian doll. It's not just a hierarchy, just to give you a sense, okay? Now these are the axioms. That's not all of all. This would be the end of the class, okay? So we have given the definition of vector space, and that's the interesting part, begins. We have to prove theorems based on the axioms. It is assumed that this is consistent, so we're not going to be able to every possible there. But we will be able to prove interesting there. Okay? Any questions? There we go. The first, The maybe a couple of examples first. Okay? Before we get to theorems, it's also important to know that there are examples because you could also build a formal system for which there are no examples, right? It's essential. This is also one more thing I want to say.  This is where mathematical practice also has a very important place. Because you say, okay, well look at all these weird formulas. How do we know that these are good axioms? If we tweak a little bit, then we get something that's inferior to, it's not as interesting.  There a sense of formal systems are not created equal. Some formal systems are interesting. They have applications. They appear in many places in mathematics and elsewhere. And some are dead end. Not really that interesting how to decide it. You cannot decide it before you try. People have tried just out of the blue. How could we modify it? For instance, what if I wrote instead of ABV B or B V? If you just look impartially, purely syntactically doesn't look worse. But you get a very different formal system and you're not getting very interesting examples.  It's important to understand that, that this is a very powerful formalism. But in some sense, it only becomes useful when you actually have good examples in mind when it is applicable to something that people work with.  Vector spaces have incredibly important applications actually, for instance, large language models, neural networks, you know, machine learning, and so on all the rage today. Such a GPT? You have a question. Yes. Why? Why is what? Yeah. Okay. Okay. You're right.  Then I would have to also relax competitivity, I suppose. Look, I haven't really given it a lot of thought. I was just kind of giving a generic. Example, okay? Maybe there's a better way to. Yeah, Probably better to look at that social activity in the field and try to sweak there. This maybe is not the best example, okay, but you got my point, okay. There is applications galore.  For instance, all of these models, like for instance, large language, let's just be more specific, large language models. The first step is, in a given language, you take what's called tokens, which are more or less syllables or something parts of words. You represent them by a vector space of very large dimension. Okay, like 4,096 That's the first step. After that, you start doing something with those vectors. In other words, the idea is to represent things which could be parts of natural language, could be something else, represent them as vectors in the structures that vector space has turned out to be very powerful for analyzing data.  Let's just say that's not all in physics, quantum physics, quantum in quantum mechanics.  Quantum systems are represented by vectors in the vector space, but over complex numbers that vector space may actually be infinite dimensional. Again, instead of representing things as objects in three dimensional space or four dimensional space time, which is what traditional classical physics used to do in quantum physics, we represent things by vectors in the vector space. How interesting In the year 2024, our most powerful theories in physics and in data processing, in computer science, in artificial intelligence, if you will, are based on representing things as vectors in vector space. That I think is a very powerful argument why we should study vector spaces. If you tweak an axiom, maybe not the way I was trying to do it, but in a more clever way. You might still get some interesting things, but not nearly as interesting.  So this is where the practice of mathematicians and those who apply mathematics in other areas and other areas also.  Another place where this theory is very important is of course, engineering and economics and so on. And if it has such applications, then we say, okay, it's worthwhile to study this formal system. So it's very important to understand that they're not all critical. It's not a shooting game. So I come up with my formal systems, you come up with your formal systems. They're all on equal footing. No, they're not. Show me what you can do with yours and I'll show you what I can do with mine. And people will look at it and they will say, okay, this one is more interesting.  The ability to discern what's interesting is, quite frankly, is what, for now at least really separates us from computer programs. For computer program you feed, these are all strings of symbols. How will they ever be able to discern that something is more interesting than something else? Maybe, but not yet. Okay, Now examples. Let me give you a couple of examples before I go to proving theorems. So first of all, before I give serious examples, first ask what is the simplest example, okay? So it's always a good idea what is the simplest vector space. But before we ask what is the simplest vector space, what is the simplest field, before we ask the simplest field, what is the simplest set? Okay? A simple set is what's called the empty set. It has no elements. Okay? What is the simplest field? There is no empty field. One of the axioms, one of the axioms of a field is that it has a zero element. And another axiom is it has element one. And this should actually be different. The simplest field has two elements. The addition and multiplication here is exactly the bull additional multiplication you use in computer programming. For example, right, one plus one is 01 times one is one. That's the simplest field. What about simplest vector space? Now, we don't have one because one lives in the, in the vector space. There is no axiom that there is an element one in the vector space. The only element that we stipulate exists is an element we call zero. The simplest vector space has one vector addition is simple. Zero plus zero is zero, which actually follows from the axiom 30 multiplied by any number is also zero. It is a bona Fide field. It will be included in next example. But it's important to orient yourself what are the simplest things. Now, there is a slight national issue here because Remember, we have fixed the field we work, all our theory is relative to that field. Considering vector spaces over, let's say vector space over r as opposed to vector spaces over C comes with two elements, 0.1 But V also has an element which we have denoted by zero. This is actually not a good way of using nation. Well, I always have to balance like you don't want to overload nations, so on. But be aware that the zero in V, this one is not the same as the zero. And sometimes to emphasize this point, I will underline it. If the 54, the first course of linear algebra, we use the vector notation, that's another way to distinguish the two. But here we try to generalize that theory. We are not putting vector is not to suggest that we are considering just the old fashioned vectors from 54. The notation which I like underlings call zero level obvious. They're not necessarily useless because we will see that vector space consists of one element. The useful important to have, they appear in various proofs and so on. But a simple, non trivial example is example. Rn is a number, let's first say from 1234 and so on. Rns elements, colons of real numbers, of real numbers which we do not x 1x2n, okay? So each X I is a real number. Addition and scalar multiplications is a familiar one. You simply, I'm not even going to write it, obviously. You know, if you have two such colons to add them up, you add the corresponding elements. Can you guys see, by the way, from that corner, is it okay? Or should move the podium? Move it. Okay. Let me move it. Feel like I was thinking of kind of putting it so that it's seen on the screen. But actually maybe it's better that it's not seen. All right. Better now. Right? Okay. Yeah. Should there be additive? Yes, it's one itself. Right? So additive. I'm sorry. It's again one itself. One plus one is zero. It's called addition. It's arithmetic. So, you can always do you know when we do hours, then, Module 12, if you start work at nine and you work for 5 hours, you don't say you finish at 14, you finish at two, so you take the remainder of the division. Here, it's additional module two. Suppose your day only has 2 hours, then you wake up, you work for 1 hour and go to sleep. Again, we're lucky to have 12, I suppose 2024, actually, to be honest, but in Europe they use 24. But we use 12. Anyway, you're all familiar with the addition of colon vectors, right? Scalar multiplication, also simple multiply each entry, each component by that number lambda. Then zero is the colon vector which has all components zero additive inverse as you put minus sine, right? And so on. That's the familiar example. In fact, we can also extend it to n equals zero by saying that R zero is the smallest field, right? Makes sense. R is a line and two is a plane. And what is 00 is just a point, just zero. Now, one of the reasons why this theory has legs is that in addition to purely algebraic interpretation of fields like so, there are also geometric interpretations that I already explained last time. This is a case when n22 can be realized as a set of vectors on the plane, where addition of two vectors acquires a more geometric interpretation. It's a parallelogram rule. And scalar multiplication means you extend the vector length by a certain number and then change orientation? If it's negative, yes. Right. That's right. Here I was, sometimes there is a clash of nation from different theories. I was writing what Cartesian product of two sets. Traditionally for Cartesian product two sets, we write them in horizontally. In linear algebra, we write them as colons. The reason is that when we talk about linear transformations, they will be represent Bimtris'll be multiplying a matrix. Vector will be on the left by matrix which is like a square matrix. If we were using we would have to rule, would have to multiply on the right. So it's a little weird that we multiply on the right. So that's one of the reasons. Now of course, in principle, you could always switch from one nation to another. It's a very common situation that in different branch of mathematics there could be conventions are conventions. This is a national conventions. It's nothing to do with a substance, it's just a convenience of how you denote things. Therefore, you have to come up with some conventions. For instance, we have a convention that we write things from left to right. And I'm sure you're aware that in some other languages, they write from right to left. Who is right, Who is wrong? It's more like who's left and who's right. You see, in other words, it's unequal footing. We've made a choice. Likewise, orientation, we typically use orientation like this. That first x and then y is a horizontal. You may remember from 53 that there are two different orientations. If you put x and y or y, x, you cannot transform one orientation into another because the arrows would have to cross. You cannot continuously transform two equivalent orientations. It always happens that you have to make a choice of convention. Sometimes the conventions clash. This is an example where in set theory we're used to writing like this, in linears we're used to writing like this. Then we have to translate from horizontal language to vertical language. But I'm glad you mentioned that. So this kind of thing is important to sort out. Okay, Now to make sure that I don't rush the proof, let me stop at this example, and then you will read in the book some other examples. But I want to really get to the point of proving a theorem from axioms, okay? I want to keep, I want to keep this, okay? I will keep the axioms here on those two blackboards. And the first that we will prove by using rules of inference and the axioms of space is what we will call the uniqueness of the additive identity. That's the name of this there uniqueness identity. In other words, this element that is stipulated here in Axiom Three is unique. It cannot be that in a given vector space there are two different elements. Let's say 0.0 prime, which both sides by the Axiom. Okay? It's important to know that because what if there were two? Then if there is unique, it means that a vector space, once you've defined all the structures, you don't have to specify which zero element you're talking about because anybody will be able to find it and find the same one. You see this unique is important. It's not obvious. It is not stated that there exists a unique element with the properties. This is not stated as an axiom. Now, you can say, why do we both, why don't we use it as an axiom? Here we come to the opposite question to the question we've discussed so far. On the one hand, we don't want to prove too many statements. For instance, we don't want to prove every statement right, because then the theory is useless. Every statement together with this negation. The other desire is to have as few axioms as possible. The idea is to prove as much as possible from as little as possible. You don't want to overload. For instance, we can take any theorem that we prove from these axioms and we can adjoin it as an axiom. If the original formal system was consistent, the new one will also be consistent, but it will be redundant. We want really the minimal possible set of axioms. That's why we are probing. Is it true that every vector space, in any vector space, the zero element proper term is identify, identity is unique. If, for instance true, we couldn't prove it from these axioms, then maybe we would adjoin it as a separate axiom that this element is unique to narrow down the possible choices of vector spaces. But this is exactly the place where we can, we can prove from the axioms we have already been given that this is. So that question then becomes moot, whether we should adjoin it to axioms or not. Of course not, We can prove it from these axioms, okay? But how to state it? What is unique? What does uniqueness? Here is a precise statement. Let a vector space over a given field, okay? If zero in V and zero prime in v both satisfy Axiom three, this one, then zero is equal to zero prime. That's what uniqueness means more precisely, right? You see now I explained last time that a proof is a sequence of sentences as one as n, where at each step you either use an axiom, the sentence is an axiom, or it is obtained by rules of inference from the previous statements that the last one is a sentence or the statement that you want to prove, then we say N is a theory. Now, I'm going to actually give you those sentences. In this particular case, I want to do it slowly. Normally we'll be skip to speak, we'll be doing some of that in our head. But I think it's worthwhile at least in one situation, to actually to trace every step. Okay, here's how we do it. Sentence one for sentence one, we will actually, let me write it like this, and here I will have commentary in yellow, we are going to use Axiom Three, which is allowed, because I said that statement can be either an axiom or is obtained by rules of inference from previous sentences. This is the first sentence. So the only way we can start is with an aim. There is nothing before in this proof, right? This statement is the Axiom. Axiom says V plus zero for V, right? But for sentence two, I will use a rule of inference, which comes from what's called first order logic, which is the following. That if you have a statement of the form for every something, something is a particular example of that, then you can substitute the substitution. I'm going to substitute for V zero prime. You see in the statement of the theorem, I'm saying that I am stipulating two elements, 0.0 prime. I'm operating with those elements. I wrote my first sentence using zero, but then zero as a neutral element, as identified identity. But now I'm going to use zero prime. For V, I'm simply substituting zero prime. I get zero prime, right? Next three is going to be at one. I should be using yellow to emphasize that is an explanation of how things progress here is going to be a one, A one is commutativity. Remember that means that z prime zero. I use it in a particular case. Now, strictly speaking, if I wrote one, maybe let's be more pedantic, that's actually write it in detail. V equals v plus u. Then four will be obtained by substituting u equals zero prime, right? Equals zero, then I get zero prime plus zero is zero plus zero prime, right? Five x five. I'm going to use Axiom three. But for zero prime, this was zero. This is going to be for zero prime, right? Then it says V plus zero prime equals V for all. It looks the same as this one. But now I'm using the second one, supposedly different element. At the end, I'm going to show that zero prime is equal to zero, right? But if I'm saying that both 0.0 prime satisfy Axiom Three, then I have both this statement and this statement. Now, of course, you see where it's going. I'm going to substitute zero prime for V. That's my six substitute v equals zero, then I get zero prime plus 0000, you see Now, on the other hand, I can take this and this together in general, if you have A is equal to B and A is equal to two are the same, therefore the two should be the same, right? That's set, if you have three elements, B and C, and you know A to B, and you also know that as equal to C, B has to be equal to C. So that's not an axiom of fields, it is an axiom of set theory. That's why I said that we're building a formal system of field theory as a first floor on top of the foundation of formal system of set theory. So we can use all the axioms of set theory and bringing them together we get sentence 67, right seven, because it was six. I'm combining 2.4 maybe may use yellow. The fact that on the left hand side of these two formulas, we have the same expression. It implies one of the rules of inference of set theory that the right hand sides are also equal to each other. Which means that zero plus zero prime is equal to zero prime. Now, I'm going to use this in conjunction with this, in the same way. On the left hand side, we have 00 prime here and here. Therefore, the right hand sides are the same, which is that prime is equal to zero, right? Finally, eight is obtained by combining a 7.6 to give us, let me use a, a choke maybe to emphasize what we are doing here. The left hand sides are the same, 6.7 which means that the right hand sides should be the same. They are, We get zero prime. We have derived the statement zero prime equals zero, right, By using the rules of inference questions. Now to be more precise, you see my statement. I'm not proving zero equals zero prime. I'm proving that under the assumption that both of them satisfy Axiom Three. In fact, the way I proved that I use a slightly different procedure. I said, in general proof you have to start with an axiom, and then you either do rules of inference or you use more axioms. But in this case, we are proving a statement of the form If you use the which is in the F, which is a 0.0 prime are elements satisfying axiom three. I'm using it as one, as if it were an axiom. I am in the world in which that is a valid statement. I'm saying that if that is valid, then coupled with all the axioms and the rules of inference, zero, z prime is also valid. This is what we mean by statement. This is, so then this is. So, in other words, in this calculation, in this procedure, I have used the possibility of applying Axiom Three to zero, and I have used the possibility of applying Axiom three to zero prime. Why? Because that was my assumption in this statement. So that is a fair game. Okay, let's do one more. Axiom Four stipulates that for every element V, there exists some element such that V plus zero. Now we already know that zero is unique. We don't have to say one particular zero, but it's something that is inherently defined element of a given vector space. But it's not clear whether this inverse element is unique. We don't say there exists a unique such that V plus zero. That's our next task, is to show that this element, for a given there is only one element, that V plus W is equal to zero, okay? So again, the form will be similar, but I will go a little faster here. Literally implement every step. Now I'm going to go a little faster. For example, I will not emphasize the step of substituting. If there is something that's true for every V, I will freely substitute things without mentioning it. You see theorem two, Let V be a vector space over F, and V in is an element of V. If in V and W prime both satisfy Axiom Four, they are forgiven for our V. All right? So that both W prime are the additive inverses of V, then W is equal to W prime. Okay? So very similar here, the zero element is universal, it services everybody here, the inverse is for a specific element. Okay? So, first we use statement one. We'll use again, Axiom one. Sorry, Axiom Three. But for let's just say for V, there is a subtle difference in Axiom Three. We say for arbitrary V. I'm using now the slightly different sense, that is, a specific vector have to be pedantic. I would have to write this axiom like here and then substitute for our element here. Of course, you see also clash of notation because V now means ambiguous, it means either arbitrary element or our element. I'm following the book here and actually I think what saves the day is that I'm going to use it for not for V, then it's okay. Yes. But be aware of the subtle things. At some point we get so proficient with all of it that we're not going to go through all the steps, it becomes your second nature. But at the beginning, I wanted to do it in a precise way. Let me repeat. There is at three, which is for arbitrary V. But we're using it for a three. Yes, but we're using it for our element. Who is, is one of these two guys that we are assuming are the negatives, the additive inverses of. Right now I'm using this fact that W plus zero is equal to W, right? On the other hand, I'm using the fact that I'm substituting instead of I'm, instead of zero, I'm substituting first. I should use white to be consistent. Next, I'm substituting here V plus W prime because the fact that is inverse means that V is zero. But the fact that prime is V prime is 00 is unique. We have already proved that. There is no ambiguity. What I do here now is that V plus W prime is equal to plus zero, right? Of course. I hope you see where it's going next. I'm going to switch this by using competitivity. So I'm going to get W prime plus V. I switch them. Okay. And then, I'm sorry. I mean, being too pedantic sometimes is difficult. So what I'm trying to do is, so this is right. So this is this. Then I substitute this. Then I want to do a chain of operations which will lead me to double prime. How to do it? At the end, I will use associativity, but I can't do it right now. In this way actually, Yes, I do. Wait, am sorry. Okay, good. So let's use associativity here and put brackets differently. So it's plus V plus W prime is equal to this. Let me just be a little bit more quick. Okay, now this is zero.