123 - מרחב וקטורי: דוגמאות לא סטנדרטיות

Summary

In this lecture, Dr. Aliza Malek from the Technion introduces non-standard examples of vector spaces, demonstrating how different operations, when meeting all ten vector space axioms, can still define a vector space even if the resulting spaces are not practically usable. Using real numbers greater than zero (R+) and ordered pairs in R², she illustrates how combining these with unusual addition and scalar multiplication operations satisfy the vector space criteria. These examples, while complex and non-intuitive, underline the flexibility and abstract nature of vector space definitions in linear algebra.

Highlights

• Dr. Malek explores non-standard vector spaces using R+ and odd operations. 🤓
• Vector space axioms can be fulfilled with unconventional addition and scalar multiplication. ✨
• Examples might be complex, but they highlight the abstract flexibility in algebra. 📚

Key Takeaways

• Vector spaces can be non-standard, defined with unusual operations yet still satisfying all axioms. 🚀
• Real numbers greater than zero and ordered pairs in R² can form vector spaces with custom operations. 🌐
• Understanding the theory can be challenging, but it's crucial for advanced studies in linear algebra. 🔍

Overview

Welcome to an intriguing exploration of non-standard vector spaces in linear algebra, guided by Dr. Aliza Malek from the Technion. Imagine spaces defined not by familiar operations, but by abstract ones that still conform to the rigorous requirements of vector space axioms.

Dive into examples where R+ sets the stage for a vector space using unique methods of addition and multiplication. Despite these operations being non-intuitive, Dr. Malek shows how they adhere to each axiom, transforming theoretical concepts into creative expressions of mathematics.

Through complex proofs and thoughtful explanations, this lecture demystifies the abstract nature of vector spaces, preparing students for deeper future studies. These non-standard examples, although not practical, demonstrate the breadth and depth of possibilities in the mathematical universe.

Chapters

• 00:00 - 00:30: Introduction to Non-Standard Vector Spaces In this chapter, Dr. Aliza Malek introduces non-standard examples of vector spaces. The focus is on understanding what constitutes a vector space by examining how groups and fields can form these spaces under unusual operations. Despite these spaces meeting all ten axioms required for vector spaces, they are not practical or used in typical scenarios, highlighting the flexibility of defining vector spaces.
• 00:30 - 01:00: Ten Axioms of Vector Spaces The chapter titled 'Ten Axioms of Vector Spaces' covers the essential axioms required for defining vector spaces. It starts with a brief review of what constitutes a vector space, emphasizing the importance of fulfilling all ten axioms. The discussion involves a group with addition and a field, denoted as F, which provides scalars. By selecting three vectors from the vector space V and two scalars from the field F, the chapter outlines the ten axioms of vector spaces. It is expected that readers are already familiar with these axioms, as the chapter proceeds to explore some non-standard examples of vector spaces.
• 01:00 - 03:00: Defining the Set R Plus and Non-Standard Operations This chapter begins with the definition of the set R plus, which consists of all real numbers greater than zero, i.e., all positive real numbers. It clarifies that zero is neither positive nor negative.
• 04:30 - 05:00: Closure and Axioms 2-3 Proof for R Plus The chapter titled 'Closure and Axioms 2-3 Proof for R Plus' discusses the concept of fields, specifically R (real numbers) and C (complex numbers), in mathematical operations. It emphasizes the importance of not manipulating the field, as opposed to groups where more freedom is allowed. The chapter points out that usual operations won't work because R functions as a vector space over itself, requiring a different approach to defining operations.
• 05:00 - 10:00: Axioms Proofs for Scalar Multiplication in R Plus The chapter discusses the axioms and proofs related to scalar multiplication within a set V that is a subset of the real numbers R. It emphasizes the need to redefine operations within V to maintain mathematical consistency when only part of R is considered. The operations involve terms x and y, which are positive real numbers, and a scalar alpha, which can be any real number. This approach ensures that V maintains its structure as a vector space over R.
• 10:00 - 15:00: Introduction to Non-Standard Example with R2 The chapter introduces the concept of a non-standard operation denoted as 'x circled plus y', which differs from conventional addition. This non-standard operation is defined in a unique manner compared to usual arithmetic operations. It emphasizes the distinction by using a circle to signify that this is not normal number addition but a new type of 'odd addition'. Furthermore, the chapter explains the standard multiplication without any special notation, signifying regular multiplication as known traditionally. An example provided is 'three circled plus five equals three times five, which is 15', highlighting the application of this unique operation.
• 15:00 - 16:00: Defining Non-Standard Operations in R2 The chapter discusses defining non-standard operations within the set of positive real numbers (R+). In this world, the operation of addition is redefined such that three plus five equals fifteen. The transcript also touches on the constraints of this new system, highlighting that negative numbers, such as minus three, do not belong to R+ and hence cannot be added to positive numbers under this system. It briefly introduces the concept of scalar multiplication, mentioning that a scalar within the real numbers will be multiplied by a real positive number.
• 16:00 - 20:00: Closure and Axioms 2-4 Proof for R2 The chapter 'Closure and Axioms 2-4 Proof for R2' covers mathematical operations involving exponents, specifically focusing on powers and their implications. It explains the concept of a number x raised to an alpha power, including examples such as 'two circled times five' resulting in 'five squared,' which equals 25. It further delves into negative exponents, providing an example with minus two times five, leading to 'five to the power of minus two,' which equals one twenty fifth. The chapter notes that alpha can also be zero, and questions the result of zero times x, emphasizing flexibility with real numbers.
• 20:00 - 26:00: Scalar Multiplication and Axioms Proof for R2 This chapter discusses the concept of scalar multiplication and the axioms proof for R2. It begins with explaining the rule that x to the power of zero equals one, given that x is a non-zero element within R+. The speaker ensures that everything is clearly defined and operates smoothly as a vector space, although not all ten axioms are detailed, leaving some for personal exploration.
• 26:00 - 27:00: Conclusion of Non-Standard Examples The chapter titled 'Conclusion of Non-Standard Examples' focuses on the mathematical concept of closure in addition within a set described as V, where V is the set of positive real numbers (R plus). It discusses that for any two terms within V, their sum (x + y) is treated as their product (x times y), maintaining the property of closure as the result remains within V. The text highlights the movement from this non-standard addition to exploring the regular multiplication of real numbers, indicating a transition in mathematical exploration and reasoning from axioms related to addition to those involving multiplication.

123 - מרחב וקטורי: דוגמאות לא סטנדרטיות Transcription

• 00:00 - 00:30 Vector Space - Non-Standard Spaces Linear algebra course Dr. Aliza Malek Hello, thanks for coming back. In this lesson we will see non-standard examples of vector spaces. What does it mean? We will take a group, we will take a field, we will think of some very strange operations, and we will see that all ten axioms are met and therefore a vector space exists. They will be no useable spaces, so they will not be used later. It seems that such things exist and we can define whatever we want.
• 00:30 - 01:00 All that matters is that the ten axioms are fulfilled. So let's get going. First of all let's remember what a vector space is: We have a group with addition, we have the F field, which provides scalars. We take three vectors within V, we take two scalars in the F field, and here are the ten axioms that exist in vector space. I assume that at this point you already remember it by heart. Now we will go over the non-standard examples,
• 01:00 - 01:30 here is the first one... We will define group V as follows: R plus. What is R plus? A real R, plus means positive numbers, so R plus means: All real x's where x is greater than zero, are positive numbers. Remember that zero is not positive. It is not negative either. Positive is greater than zero, negative is less than zero,
• 01:30 - 02:00 and zero is equal to zero, so it is not on here. We will have the field to be R. Please note that the field cannot be toyed with. It may either be R, or C, or any field we have already seen to be a field in the past. With a group we are allowed to do whatever we want. So this is what we chose. We will now need to define operations. It is clear to us that the usual operations will not work here, because we know that R is a vector space over R.
• 02:00 - 02:30 So if I took only a part of R, I will not be able to leave the operations as they are, we will have to change them. And here are the strange operations that we will now define. V will be a vector space over R with the following operations: We will take two terms within V, remember that x and y will always be positive real numbers, and alpha within R. Alpha is any real number.
• 02:30 - 03:00 What is x circled plus y? How do I join in the V group? Now we will define something very strange, that's why I wrote a plus and marked it with a circle around it, so that we remember that this is not a normal addition of numbers. It is the odd addition that we will define here. And what will we define? x times y. Note that this is multiplication without a circle, what does that mean? Normal multiplication of numbers. For example, three circled plus five equals three times five, which is 15.
• 03:00 - 03:30 In this world three plus five is 15. What would happen if I wanted to add minus three to 5? You cannot. Minus three does not live within R plus. Now let's define scalar multiplication. Remember, a scalar within R, which we will multiply by a real positive number. Alpha times x will be...
• 03:30 - 04:00 x to the alpha power. For example, two circled times five is five squared, 25. Minus two times five will be five to the power of minus two, one twenty fifth. This time, alpha can also be a zero, why? Because we take a real number as we wish. So how much will zero times x be?
• 04:00 - 04:30 It is x to the zero power. How much is x to the zero power? That is right, it's one. Remember, x to the zero power is one, provided x is not zero. x is indeed not a zero, because it is within R plus. Everything is defined very nicely. And not only is it defined nicely, it also works, it's really a vector space, Let's see all ten axioms. Oh well, I won't show all ten, some you will complete by yourself.
• 04:30 - 05:00 So let's start from a closure to addition. I take two terms within V, I know they are positive. So x plus y is x times y. What happens when we multiply two positive numbers? Of course, we get a positive number that is also within V, remember? V is R plus. Therefore there is a closure to the addition. In axioms two and three, we will perform addition, that is, we will move to the world of the normal multiplication of real numbers. So we will use the fact that our addition is regular multiplication,
• 05:00 - 05:30 and therefore in normal multiplication there is a grouping and exchange. Let's see how this helps us prove axioms two and three. Let's start with axiom two. x circled plus, y plus z. Remember, each circled plus is replaced by regular multiplication. Therefore what is written here is actually x times, open brackets, x times z.
• 05:30 - 06:00 Wait, in normal multiplication I can play with the brackets. Now I am in the world of normal multiplication, which I know. I can move the brackets and the result will not change, let's move them. Here, now I translate back to circled plus. Each multiplication is a circled plus. What comes out in the end? x circled plus y, circled plus z. That's it, the grouping principle is fulfilled. Once the brackets are on y and z, and once the brackets are on x and y, thus we have the grouping principle.
• 06:00 - 06:30 Let's see the exchange principle. What is x plus y? Let's move to the familiar world of normal multiplication, it's x times y. Now I have an exchange principle, I will use it. It's y times x. But normal multiplication we convert to a circled plus. Therefore we will get y circled plus x. Next, we will now move to R. I need to show that within V I have an indifferent to addition term. Of course it won't be the zero we are used to, zero is not in our group at all.
• 06:30 - 07:00 What will it be? Let's think for a moment. I want a term for which I will do x circled plus y, and get x. x times something should give me x, what is it? Of course, it's the positive number: one. Luckily that one is positive, otherwise it would not be in our group at all. Let's see that it indeed works, x plus one equals x times one, which is actually x.
• 07:00 - 07:30 We took x, added the one, nothing happened. Therefore... Zero of our group V is the regular number: one. Zero V marks the zero of our group. We have an indifferent to addition zero within V, and we have an indifferent to addition zero in the field. It's not the same zero. That's why sometimes, when I want to help, I dot the zero of V.
• 07:30 - 08:00 Next, we will continue with axiom five. Now I want an opposite. What is an opposite? I want to take a term within V that is real positive, add something to it, and get a zero. When I add, do I multiply? Is getting a zero means getting a one? Who performs this task? Correct, one divided by x will do this for us.
• 08:00 - 08:30 Is it allowed to divide by x? Of course, because we are within R plus. No matter what x you take, it will not be zero. So every x term you take, it will always be possible to calculate one divided by x. Therefore we did not make any mistake here, thanks for reminding us. So this is what x plus y equals to. In our world, x plus y is x times y. Now I'm going to take this y, and I'm going to place the candidate to be the opposite.
• 08:30 - 09:00 It's x times one divided by x. What is x times one divided by x in the normal world? Because it's normal multiplication. It is equal to one. Wait, x plus y, meaning x plus one, divided by x, is one,. And what is one? That is our zero. Therefore y is the opposite of x. One divided by x is the opposite of x, because when we added them we got the indifferent to addition term.
• 09:00 - 09:30 It's a bit confusing. Indeed, non-standard examples are a bit confusing. And here is a scalar multiplication closure. I take scalar alpha, and I calculate alpha times x, getting x to the power of alpha. This is always greater than zero. What determines the sign of the result when raising to a power? It is determined solely by the base of the power and not by the exponent. Since the base is positive, it does not matter to what power you raise:
• 09:30 - 10:00 Positive, negative, or a zero, we will always get a positive number. That's it, thus we have closure for scalar multiplication, meaning the result is within V. Moving on. Now we need to examine seven through ten. Again, as before, you have to remember: Addition is in fact multiplication. Scalar multiplication is in fact a power. That's why we will use the axioms of:
• 10:00 - 10:30 The division, the exchange, and the grouping of real numbers in multiplication. Let's look at two example axioms, seven and nine. I'll leave the rest to you to complete on your own. Here is number seven. Alpha plus beta, times circled plus x. Pay attention, here it says regular alpha and beta, because alpha is a scalar, and beta is a scalar. This is an operation performed in the R field.
• 10:30 - 11:00 That's why it is a regular addition, two plus three. Now we want to multiply by x. Remember that when we multiply, we exponentiate. Therefore, alpha plus beta times x means, x to the power of alpha plus beta. Now we will start using power axioms. What is x to the alpha plus beta power? It's basically x to the alpha power times x to the beta power. Here we operate as in the normal world of normal multiplication.
• 11:00 - 11:30 Now we will translate. I know how to translate the usual multiplication into circled plus. So I have x to the alpha power, which is a positive number, circled plus another positive number. But wait, what is x to the alpha power? This translates to alpha circled times x, and the same goes for beta. So we got alpha times x circled plus beta circled times x. Please note that in axiom seven, we start from a normal addition
• 11:30 - 12:00 and end with a circled plus addition, why? Because this a term within V according to six, this a term within V according to six, we perform addition within V field. And nine, one circled times x, means x to the first power. And x to the first power, is x. One times V is equal to V. One times x equals x.
• 12:00 - 12:30 That's it, we got a vector space. That's right, you have a few axioms left to test on your own. Let's move on to the next non-standard example. This time we will take R two as group V, remember what R two is? x, y ordered pairs of real numbers. We will see that it will be a vector space over the field R with odd operations, not the usual operations. Remember how to perform addition within R two? Term by term respectively.
• 12:30 - 13:00 Remember how to multiply by scalar? Each of the components is multiplied by the scalar. Now we will define something totally crazy. Let's see what it looks like. We will take two vectors, u will be x, y, v will be a, b. We will take a real alpha scalar, and here are our new operations: u circled plus v, why circled plus again? Because it's not a normal addition. There is the normal addition in R two, term by term respectively,
• 13:00 - 13:30 and there is the strange addition we are about to define here. That is why it is marked with a circle. What have we done? In the second component we added normally, y plus b, the second component is completely standard. But what did we do up here? We took x minus two, plus a. In other words, we took x, added a, and subtracted two. x plus a, and subtracted two, we performed a shift.
• 13:30 - 14:00 Pay attention, we did u plus v normally, which is x plus a, y plus b, then we performed a shift, minus two, zero. We shifted it, we shifted the first component, the second component we did not touch. That's how we defined it. Now let's define scalar multiplication. Alpha, some real number, which we multiply by the u pair,
• 14:00 - 14:30 x, y will look like this. Notice what we did. Usually we do alpha x times alpha y. What did we do here? We multiplied x by alpha. Just don't forget that this time the x was x minus two, So instead of multiplying x, we multiplied x minus two, and added two. We have reversed the shift. And we have alpha y, in the second component everything behaves normally.
• 14:30 - 15:00 Well, this too can be split this way. I will leave it to you to do it by yourself. I will only say it verbally. This we will prove using u plus v minus two, zero. We will prove axioms of scalar multiplication using this notation. Your homework is to reverse this. Combinations, remember? You will prove this using what is written here as a column,
• 15:00 - 15:30 and the scalar multiplication you will prove using the notation of addition and subtraction within R two. How do we translate this? Pay attention. Here I have alpha x, alpha y. It's basically alpha times x, y. That is, what is written here is alpha times u, plus alpha times two, zero, because here I don't have alpha with a two...
• 15:30 - 16:00 Sorry, minus, right. Alpha minus two, zero. Plus two, zero, because this two has no alpha. Again, if we want to translate this into the notation of vectors within R two, which uses the addition and subtraction of R two, we will write that alpha times u equals alpha u, minus alpha times two, zero plus two, zero.
• 16:00 - 16:30 Write it down and try to prove the axioms of scalar multiplication using the other notation. I will demonstrate using this notation. I will demonstrate the addition using this notation, and you will try it with this notation by yourself. So let's get going. Addition to closure, it's simple. When I add u plus v, I get u plus v minus two, zero. All I have to show here is that I am within V.
• 16:30 - 17:00 What is within V? Ordered real pairs. An ordered real pair plus an ordered real pair minus an ordered real pair, is an ordered real pair. Therefore closure exists. Axioms two and three, again we use the addition of the regular R two for the strange addition in our group.
• 17:00 - 17:30 Let's do this. u circled plus v, plus w. Let's write this. u circled plus remains u circled plus. Now inside the brackets, we need to perform the operation in brackets first. Remember what that means? Add normally, and subtract two, zero. Well, v plus w, normal addition, and we subtract two, zero. Now I have to restart this, so what am I going to do?
• 17:30 - 18:00 Normal addition, u plus whatever is here in the brackets, minus two, zero. Let's write it. u plus whatever is in the brackets, v plus w minus two, zero, minus two, zero. The first plus the second minus two, zero. The first plus the second minus two, zero.
• 18:00 - 18:30 Now let's rearrange everything a bit. So I have it like this, I want to bring the brackets to u and v. And thus, the first plus u plus v minus two, zero. I take u, I take v, and minus two, zero, one to me. Here, what do we have left? This w that we haven't used yet, and minus two, zero that we haven't used yet. w, and minus two, zero.
• 18:30 - 19:00 So what do I have here now? In brackets I have a term, plus the next one minus two, zero. This translates to u circled plus v. Then again, I have a term, plus the next one minus two, zero. This will translate to the term circled plus the next one, and we succeeded. If it's a bit hard to follow, you can stop it on this calculation here,
• 19:00 - 19:30 calculate this separately, and show that it came out the same. I like to flow from the beginning of the equation to the end of the equation. Let's see number three, u plus v equals the first plus the second minus two, zero. Now it's much simpler. I simply need to write v plus u instead of u plus v, I'm allowed to do so. Because in a normal addition in R two there is an exchange principle, here. What does is it equate to in our world?
• 19:30 - 20:00 Firstly the first, circled plus the second. We started with u plus v, we ended with v plus u. And here is number four. It exists in an indifferent to addition V. Which is it? Two, zero. Why? Remember? I made a shift, I need to cancel it, that's why I add two, zero. Let's see it. If I take v and add two, zero, where will it take me? Remember?
• 20:00 - 20:30 First plus the second, minus two, zero. First plus the second, minus two, zero. We made a shift, we canceled it. so what happened? Nothing, we are back to v. And therefore, v plus two, zero, is v. Indifferent to addition, that is its definition. Now we want an opposite. We made a shift and then we have to cancel, so we have to take it twice, let's see.
• 20:30 - 21:00 I start from the minus v, I want to reach two, zero. That's why I take minus v, to cancel v, plus four, zero, why? Because I will remove two, zero once, and eventually I want to reach two, zero which is the indifferent to addition of ours. Let's see that it really works. v, to which we will add minus v plus four, zero, what is it equal to? First plus second minus two, zero.
• 21:00 - 21:30 First plus second minus two, zero. Minus v cancels v, luckily we moved by a four here, now we made a shift of a two. Four minus two is two, zero with a zero, what do we have left in the end? Two, zero, which is the indifferent to addition of ours. It's a bit confusing, you have to work slowly and carefully. I leave it to you to write it up at home. Let's move on to scalar multiplication, This time I will work with the definition written in this way:
• 21:30 - 22:00 In order to show you a bit how to work with the arithmetic in R two, and with our regular arithmetic. So here we have alpha times u equals, I remind you, alpha times x minus two, the one that we moved, plus two, normally in the second component, alpha times y. I want to show that it belongs to V, what does it mean to be within V? All in all, an ordered real pair.
• 22:00 - 22:30 All these numbers are real, these numbers are real, we got a real pair. That's it, we are guaranteed a closure. In seven to ten we of course use the axioms of the scalar addition and multiplication within R two. Let's demonstrate seven, it's a bit long, so I will only do one example this time. I will show you number seven. Alpha plus beta circled times u. Remember where I want to get to?
• 22:30 - 23:00 To alpha circled times u circled plus beta circled times u, remember? This plus is a regular plus in the field R, and after that it will be translated to a circled plus in group V. So let's translate. How do we multiply by a scalar? Here it's written. Take the scalar, multiply by x minus two, add two. In the second component, normal multiplication. So I will get...
• 23:00 - 23:30 Instead of alpha, my scalar is called alpha plus beta, here it is. Times x minus two, plus two, and normal down here, alpha plus beta instead of alpha. Now the calculations are a bit complicated, so we'll stop here. We will have what needs to come out at the end of the calculations. As we said, alpha circled times u, circled plus beta circled times u. Let's calculate it separately and see that we reached what we wanted to reach, which we have here.
• 23:30 - 24:00 Because the calculation is a bit long, the advantage this time is that I can see right in front of my eyes where I want to get to, and that will give me a clue of what I should do. So I place. First of all, if I want to add a pair, I need to have two pairs. So I first need to translate the circled times. So let's place what is alpha circled times u, and what is beta circled times u. I place these brackets once with alpha, and once with beta. Between them will be circled plus, and then I will have two brackets,
• 24:00 - 24:30 two terms of ordered pairs, which I will know how to add. First plus second, and so on. So here it is. Alpha with x minus two plus two, and beta with x minus two plus two, Now let's implement the addition. The first plus the second minus two, zero. That is, I add as normal the first and remove two,
• 24:30 - 25:00 and the second I add as normal, and do not touch. Let's write it up. Here. The first plus the second minus two. Down there a normal addition. What will I do with all this? It's written right here. I want to get x minus two multiplied by alpha plus beta. Very simple. Here is x minus two, x minus two which we take it out of the brackets
• 25:00 - 25:30 and see what is left. Hopefully we will only have a plus two left. Indeed, here is plus two, here is another plus two, but here is a minus two. From all of this calculation we will be left with exactly alpha plus beta x minus two, and a two once. It looks the same in the second component, because there we add and multiply normally. And here is what we got, alpha plus beta, x minus two plus two, alpha plus beta times y.
• 25:30 - 26:00 That's it, we got a vector space. A little weird. Some axioms that we have not done yet you will complete on your own, but we have non-standard examples here. Of course, the work is much more confusing and complicated. Therefore, later on we will not demonstrate using these, we will only demonstrate with the standard examples. But it is very important to know that there are such an examples as well. That's it, thank you very much. We will meet in the next class and start talking a little about theory in vector spaces.
• 26:00 - 26:30 Thanks.