124 - מרחב וקטורי: משפט בסיסי

Summary

In this illuminating lecture by Dr. Aliza Malek from the Technion Institute, the intricacies of vector spaces are explored through foundational axioms in linear algebra. Dr. Malek revisits the definition of a vector space, emphasizing its ten core properties. This session focuses on enhancing those properties with five additional axioms, enriching our understanding of vector space theory. The lecture systematically proves each axiom, fostering a deeper comprehension of concepts like uniqueness of addition-indifferent elements, vectors' opposites, and conditions under which scalars and vectors equal zero. These axioms are vital for proving more advanced concepts in vector spaces, setting a robust groundwork for future exploration in linear algebra.

Highlights

• Chapter 4 delves into vector spaces, with Dr. Aliza Malek leading an engaging session on foundational axioms 🔍.
• The lecture revisits the definition of vector spaces and showcases standard versus non-standard examples for a diversified understanding 🌈.
• A key focus is on proving the uniqueness of addition-indifferent elements and vectors' opposites to ensure consistency within vector spaces 🧩.
• Dr. Malek demonstrates proofs of five new axioms that extend the ten defining properties of vector spaces, covering scalar-vector relationships and more 🎓.
• The session concludes with a solid foundation laid for tackling more advanced topics in vector spaces, emphasizing the educational value provided by these axioms 📚.

Key Takeaways

• Vector spaces are defined by ten core properties, forming the backbone of the course 🤓.
• Dr. Malek introduces five new axioms that build upon these properties ✔️.
• Each axiom is thoroughly proven, reinforcing the foundational understanding of vector spaces 👩‍🏫.
• Uniqueness plays a critical role, from addition-indifferent elements to vectors' opposites, ensuring consistency 🌟.
• The relationship between scalars, vectors, and zero is pivotal, with 'zero' acting as a unique transformative element in vector spaces ⚖️.

Overview

In this engaging session led by Dr. Aliza Malek, vector spaces are explored with precision and clarity. The lecture revisits the fundamental definition of vector spaces, highlighting ten core properties that form the bedrock of linear algebra. Dr. Malek introduces five additional axioms to enrich these foundational principles, setting the stage for a deeper exploration into the subject.

The heart of the lecture lies in demonstrating the proofs of these new axioms. Each axiom is meticulously proved, emphasizing critical elements such as the uniqueness of the addition-indifferent element, the uniqueness of opposites in vectors, and the pivotal role of zero in scalar-vector equations. These proofs not only solidify the understanding of vector spaces but also prepare students for advanced topics in the field.

Concluding the session, Dr. Malek ensures that students are well-equipped with both ten properties and five axioms now available at their disposal. This comprehensive grounding empowers students to explore more complex concepts within vector spaces, making this lecture a critical resource for anyone delving into linear algebra.

Chapters

• 00:00 - 30:00: Chapter 4 - Vector Spaces Chapter 4 covers the concept of vector spaces, exploring both standard and non-standard examples. The chapter begins with a review of the definition of a vector space, which involves a group V with addition and a field F for scalar multiplication. It sets the foundation for understanding the theory related to vector spaces within linear algebra.

124 - מרחב וקטורי: משפט בסיסי Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Space - Basic Axiom Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. We have already seen seen what a vector space is, and we have seen standard and non-standard examples. The standard ones will be used later on, the non-standard ones were for seeing that we can also define things that are a little weird. And now it's time for the theory, so let's get started. First of all, as usual, we recall the definition. I have the group V with addition, I have the field F that provides us with scalars to multiply by a scalar,
• 00:30 - 01:00 and we have a vector space if all ten properties hold. What we will do now is we will define, formulate, and prove a basic axiom. an axiom that will enrich these properties a little more for us, so that when we have a vector space, in addition to all these ten properties, we will have five more additional properties which will stem out of the axiom we will now prove. So here is the axiom: Let V be a vector space over the field F.
• 01:00 - 01:30 If you ask me, I recommend you prepare a stamp. From this moment on, every axiom and every definition will begin the same: Let V be a vector space over F. We will get situated inside a vector space, and will begin to define concepts, formulate axioms, and prove them. So here is the first axiom with five clauses: The first clause says, the addition-indifferent in V is unique.
• 01:30 - 02:00 Remember, the property says: "exists within V, addition-indifferent". It doesn't say how many there are. Here we prove that if we have already proved that we have a vector space, the addition-indifferent must be unique, there is only one as such. The opposite of v in V is unique as well. Do you have a vector? It must have an opposite one. We didn't say how many, the axiom will state: "There is always only one". Note that if you ever want to check
• 02:00 - 02:30 whether you have a group that is a vector space, and you discovered two terms that are addition-indifferent, you can immediately say that it will not be a vector space. But, when proving a vector space, there will be no need to prove uniqueness. If there will be ten properties, using the ten properties I will prove that is addition-indifferent is unique, and that the opposite is unique, let's continue. For every v in V there exists 0v=0. Many zeros, it's confusing indeed.
• 02:30 - 03:00 So let me help you read this. I don't always help to read, sometimes I just ask, what is this zero, and what is that zero? Since this is the first time, I will help you read this. How do I help? I'm just anotate, here: The zero that is written here is 0F, which means that it is the scalars addition-indifferent in F, times any vector v, no matter which one, we get 0V,
• 03:00 - 03:30 The vectors addition-indifferent in the group V. A zero scalar multiplied by some vector v, gives the zero vector, that's what it says here. For every v in V there exists: -1 times v equals -v. What does it say here? That if you take the opposite to the multiplication-indifferent, and multiply the vector v by it, you will get the opposite of v.
• 03:30 - 04:00 The opposite of the indifferent, multiplied by v, gives the opposite of v. And the last clause says that if alpha times v equals zero, then alpha = 0, or v = 0, but the opposite is also true, because it's an "if and only if" statement. Alpha equals zero, or v equals zero, means
• 04:00 - 04:30 that alpha times v equals zero. A scalar times a vector will give the zero vector, if and only if, at least one of them, either the scalar or the vector, is zero. We have no other way to get the product of a scalar with a vector, and get the zero vector. We will prove all five sections. So here is the proof: A, the addition-indifferent is unique, let's prove it. Let's assume by negation that there are two addition-indifferent terms in V,
• 04:30 - 05:00 we will call them e1, and e2. If e1 is addition-indifferent, it means that e1 plus e2, is e2. if e1 indifferent, we get the other one. Wait, but e2 is also indifferent, so e1 plus e2 will actually give me e1. e1 plus e2 on one side, e1 plus e2 on one side, are equal, and therefore the right side must also be equal.
• 05:00 - 05:30 In other words, if the left side is equal, so is the right side, therefore, e1 must be equal to e2. It looks familiar? Indeed, this proof is very familiar, we saw it when we proved the uniqueness of the addition-indifferent in the field, it's exactly the same proof, only there e1 and e2 were scalars in the field, This time e1 and e2 are vectors in V. Which are actually the same vector, because e1=e2,
• 05:30 - 06:00 and it is the addition-indifferent which we denote by zero. The opposite of v, that is, -v, which exists according to five, is unique. -v, we already have. If I would like to think for a moment that I have another one, I will come to the conclusion that it is the same one. So let's assume again by negation that v has another opposite, let's call it v1, for instance.
• 06:00 - 06:30 So I know that v+v1=0. What is this zero then? That's right, it's the zero of v. We add vectors, we get a vector. Let's add the opposite of v, that because of property five we know we have, on both sides of the equation. What do we get? Here is -v that I added to v+v1, and here is -v that I added to zero. Now we will use the properties of a vector space.
• 06:30 - 07:00 For example, we know that these brackets can be moved to -v plus v, and it's worth my while, because I know what -v plus v equals to. I also know that -v plus zero will give me -v, because each vector to which the addition-indifferent is added, gives the original vector. And so, we will get... First we will move the brackets. -v plus v is a vector, plus the opposite, will give me the zero,
• 07:00 - 07:30 and here, I will have a zero as well. And so, we get 0 plus v1, 0 plus v1, on one side, and on the other side, -v plus 0, that is the indifferent, is -v. But again we have zero plus v1. Zero plus v1, is v1. And so we get that v1 is actually -v. Zero plus v1, is v1, equals to -v.
• 07:30 - 08:00 I had an opposite -v, I thought I had another opposite which I called v1, and I came to the conclusion that they are both the same, so we only have one opposite term. Section C, for every vector there exists: 0*v=0. I remind you that zero scalar times vector v, gives the zero of v, the zero vector. Sometimes it's a bit confusing, but if we know that this is v,
• 08:00 - 08:30 that zero must be a scalar, because we don't multiply vectors. And the result must be within V, so we must get a vector. So zero belongs to F, that is, this zero, within the context I understand that we are talking about the zero of the field. Meaning, it's the scalar. It is addition-indifferent, we already know that. That's why we work with a field we have already learned a long time ago,
• 08:30 - 09:00 and we know that zero is addition-indifferent. Therefore, we can write the following: The scalar zero times the vector v, instead of the scalar zero, we will simply write zero plus zero, times v. Now, we will use property seven, which states, that we are allowed to open the brackets. This is zero times v, plus zero times v. Let's see what we got... I take the beginning and the end, and compare.
• 09:00 - 09:30 Zero times v equals 0*v+0*v. What is zero times v? We don't know of course, this is exactly what we are trying to prove. But we know it's within V. And if it is within V, because of the closure for multiplication by a scalar, we know it has an opposite, which is the opposite of 0v? Very simple, -0v, here it is:
• 09:30 - 10:00 And we will add the -0v to both sides of the equation, and let's see what we get. -0v plus 0v, I added to one side, and -0v on the other side, plus what I had, 0v plus 0v. Of course now we will use our ability to move the brackets from 0v plus 0v, to -0v plus 0v. Let's do this, move the brackets.
• 10:00 - 10:30 Now we will use the known properties again. A term plus the opposite gives me zero. A term plus the opposite gives me zero, and so, we get zero equals zero, plus 0v. Once again I have zero plus something. Zero plus something is something, and so we got that zero is equal to 0v.
• 10:30 - 11:00 We took a scalar, multiplied it by a vector, and got the zero vector. Here, again I helped you to anotate, do not forget to do it. A scalar multiplied by a vector gives the zero vector. Section D, for every V there exists: The opposite of the multiplication-indifferent times the vector is the opposite of the vector.
• 11:00 - 11:30 How ill we do it? We will calculate v plus -1 times v. Let's see what is the result, let's do the math. v plus -1 times v, equals: Here is v, and here is v, we would very much like to take v outside of the brackets, but I will have nothing left over here: In order to have something left, I have to use property nine, which states that instead of v, we can simply write 1 times v.
• 11:30 - 12:00 How lucky we are to have property nine. Now, v appears both in the first and second added terms, therefore, according to property seven, we can take it out of the brackets, and get: The first scalar plus the second scalar, which is 1 plus -1, times the vector v. Wait, a term plus the opposite is the zero scalar in the field, and so we get zero times v.
• 12:00 - 12:30 And we just proved that the zero scalar multiplied by some vector v, gives the zero vector. So what happened here? I got that when I take v, and I add -1 times v to it, I get the addition-indifferent term. This means that -1 times v should be the opposite of v.
• 12:30 - 13:00 Wait, but which is the opposite of v? The opposite of v is denoted to be -1 times v, and how many opposites did we prove there are? In section B we saw that there is only one such, and therefore, whichever term here plays the role of the opposite, must be the opposite itself. In other words, -1 times v, which behaved like the opposite, must be the opposite, because there is only one like it. Therefore, -1 times v has no choice but to be the opposite of v, which is -v.
• 13:00 - 13:30 Section E, alpha times v equals zero, if and only if, alpha equals zero, or v equals zero. We have two directions to prove here. Let's start from the other direction. What does other direction mean? Well, since I start from the left side, and reach the right side. Let's say that alpha equals 0, or v equals 0,
• 13:30 - 14:00 and we would like to prove that alpha times v equals zero. If alpha equals zero, what do we get? According to section C, I will write zero here, and what will I get? Alpha times v equals zero times v, equals zero. I already know it's zero, why? We proved it in section C.
• 14:00 - 14:30 What happens if v equals zero? Now, alpha v should be replaced by alpha times zero. That we haven't proven yet. But, we will reproduce the exact same proof. What we did in section C with the scalar, we will do the same thing with the vector instead of the scalar. What does that mean? Take a look: What did we do in section C? We took the scalar 0, multiplied it by v,
• 14:30 - 15:00 and we split 0 into 0 plus 0. We multiplied by v, from there we flowed with it. What did we get in the end? 0F times v, equals 0v. What will we do now? We will take alpha times zero, we will divide this zero into zero plus zero, only this time, it will be 0v plus 0v. Here we wrote the scalar as 0 scalar plus 0 scalar,
• 15:00 - 15:30 here we will write the zero vector as zero vector plus zero vector. We will open brackets, take the beginning, take the end, add the opposite, recreate exactly the same proof. And what will come out in the end? That any alpha scalar times the zero vector, gives us the zero vector. A recreation of the proof. Again, I suggest that you solve it by yourself.
• 15:30 - 16:00 Now we return to section E, to prove the first direction. The other direction is in the proof, but in terms of the axiom it is the first direction. It is given that alpha times v equals zero, I want to prove that either alpha is zero, or v is zero. So let's assume... Pay attention to the zeros, 0 of V, 0 of F, because it's the scalar, and v is a zero of V.
• 16:00 - 16:30 The product alpha times v is the zero vector, alpha equals zero means the scalar is zero, or v equals zero means the vector is zero. So let's assume that alpha equals zero, and we are done, why? Because it was given that either alpha is zero, or v is zero, so alpha is zero, and we are done. Wait, if alpha is not zero? Not a problem, alpha is a scalar in the field, it is not zero, therefore it must have an opposite in the field, alpha -1.
• 16:30 - 17:00 Let's multiply both sides of our equation by alpha and -1. I remind you, this is the equation because this is what is given, and we multiply by alpha -1. Alpha -1 times alpha v, alpha -1 times 0v. Let's see what comes out. According to property eight, I can move the brackets from the scalar times a vector, to the scalar times a scalar.
• 17:00 - 17:30 So I will get alpha -1 times alpha. First we multiply the scalars, and then we multiply by v. But alpha -1 times alpha is 1, the 1 of the field. So I have 1v equal to alpha -1 times 0v. But what is 1 times v equal to? Remember property nine? Property nine states that 1 times v is v. And so we got that v is equal to to alpha -1 times 0v.
• 17:30 - 18:00 Wait, but we saw that it doesn't matter which scalar we take, a scalar times the zero vector is always the zero vector, and so for every beta scalar we get, that v is equal to alpha -1 times 0, and the result must be zero. In other words, what did we get? That v must be zero. And that's it, we are done proving the basic axiom.
• 18:00 - 18:30 Now, whenever we say: "Let V be a vector space over F", we will have all ten properties at our disposal, as well as all five axioms we proved here, and will be use them to prove much more complicated and complex things in the vector spaces. That's it for now, thank you very much.