121 - מרחב וקטורי: הגדרה

Summary

This lecture introduces the concept of vector spaces in linear algebra, emphasizing their foundational role in the field. Dr. Aliza Malek, from the Technion, begins with a basic definition of vector spaces over a field, illustrating with examples such as real matrices of order 2 by 3. She outlines essential properties, including vector operations like addition and scalar multiplication. The lecture stresses understanding vector spaces as frameworks for combining vectors and scalars, differentiating them from familiar arithmetic, and highlights the abstract nature of vectors beyond the common perception of arrows signifying direction and magnitude.

Highlights

• Dr. Aliza Malek introduces vector spaces with a touch of humor, likening field concepts to cows in a pasture. 🐄
• Real matrices, commonly used in examples, help in visualizing and understanding vector spaces. 🧮
• Dr. Malek chaotically outlines ten properties essential to vector spaces, akin to laws in physics, that strings together the universe of vectors. 📏
• Matrix addition and scalar multiplication, fundamental operations, intertwine to fortify the structure of a vector space. 🔗
• This lesson demystifies the idea of vectors – they’re more than arrows and directions, they’re part of an abstract universe. 🎇

Key Takeaways

• Vector spaces are fundamental in linear algebra, forming the environment where it thrives. 🌟
• Understanding the properties of vector spaces is crucial for grasping higher-level concepts in mathematics. 📚
• Vectors in these spaces are not limited to our usual notion of arrows; they can be matrices or other constructs. 🧩
• Addition and scalar multiplication make the operations within a vector space robust. ➕✖️
• Fields provide the scalars necessary for operations within vector spaces, emphasizing the interconnectedness of mathematical concepts. 🌐

Overview

In this session, Dr. Aliza Malek from Technion kicks off Chapter 4 with a dynamic introduction to vector spaces. She paints a picturesque analogy, humorously drawing comparisons between field concepts in mathematics and cows grazing, setting a jovial tone for the lesson. The lecture transitions into defining a vector space over a field, with real matrices serving as tangible examples. These provide clarity on complex topics, breaking down the abstract notion of vector spaces into more palatable segments.

Dr. Malek meticulously examines the layers of vector spaces, diving into ten distinct properties that frame the operations within these mathematical frameworks. These properties, akin to the laws of physics, govern how vectors interact within their spaces, particularly through operations like addition and scalar multiplication. The lesson underscores the duality of these operations, showing how they underpin the broader structure and functionality of vector spaces.

Throughout the lecture, Dr. Malek challenges conventional understanding of vectors, moving beyond the simplistic view of them as mere arrows. This session encourages viewers to appreciate the abstract nature of vectors as members of vector spaces, paving the way for future exploration into sophisticated mathematical theories. The conclusion ties up with a promise of more examples, teasing both standard and novel approaches to vector spaces, nurturing a sense of curiosity among students.

Chapters

• 00:00 - 00:30: Chapter 4 - Vector Spaces Chapter 4 introduces the concept of vector spaces over the field F, as explained by Dr. Aliza Malek in a Linear Algebra Course. The chapter begins with a light-hearted analogy comparing fields to cows, setting a friendly tone for the complex subject of vector spaces. Dr. Malek poses the fundamental question, 'What is a vector space?' hinting at the in-depth exploration that will follow.
• 00:30 - 02:00: Defining a Vector Space The chapter introduces the concept of a vector space, which is central to linear algebra. It begins by establishing a vector space as a foundational world of study, closely relating it to our real world. The chapter then breaks down the fundamental components crucial for understanding vector spaces: defining a set 'V' and a field 'F'. While 'V' can be any set, 'F' refers to a field, which must be a predefined mathematical field. This sets the stage for deeper exploration into the properties and applications of vector spaces in linear algebra.
• 02:00 - 05:30: Properties of a Vector Space In this chapter, the properties of a vector space are explored, focusing on the concepts of fields, particularly infinite fields. The chapter distinguishes between two main operations within a vector space: addition of terms within set V, and scalar multiplication involving terms from both set V and field F. There is a mention of different types of fields, such as real and composite fields for infinite fields, while finite fields are noted as a separate consideration.
• 05:30 - 15:00: Examples and Illustrations The chapter "Examples and Illustrations" delves into the relationship between sets and fields, highlighting that fields provide sets with scalars. It introduces the concept through a running example utilizing real matrices of order 2 by 3, intended to concretely demonstrate the definition in practice. This example facilitates understanding by providing a tangible reference while exploring theoretical definitions.

121 - מרחב וקטורי: הגדרה Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Space - Definition Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. Today we start a new chapter. The chapter will be called vector spaces over the field F. Vector distances, V D, over the field F, a field, really reminds me of cows. These are the cows you will see in the background both at the opening and later. And what is a vector space, you ask?
• 00:30 - 01:00 This is the world in which linear algebra lives. Wait, this is also the world we live in. Indeed, let's see how it all comes together. So let's get started by defining what a vector space is. Let's start. So we have V, a set, and we have F, a field. A set can be anything we want. A field, must be something we already know is a field,
• 01:00 - 01:30 because we proved it to be a field. For us in this course, infinite fields are real, or composite, and for those who study it, there are also finite fields, but we will do that separately. We will define operations of addition between terms of V. We add within the V set. And we have another operation, scalar multiplication, that's this point, between a term from V and a term from F. Let's summarize this. We have a set, we have a field.
• 01:30 - 02:00 In a set, we perform addition. What is the connection between a set and a field? The field provides the terms of the set with scalars. And now... We will do the whole definition with a running example. What is a running example? We will look at the definition, and think about this example, where V is the set of real matrices of order 2 by 3. This is what we will have in mind when we run the definition.
• 02:00 - 02:30 We also need to select a field. So the field will be the real field. Real matrices, a real field. It's not the same thing, I have a set of matrices, and I have a field. Now operations are needed. What will be the operations? Matrices we add, we learned to do it, and multiply by a scalar, we learned to do that too. So these are the two operations we choose.
• 02:30 - 03:00 2 by 3 matrices, a real field, scalar addition and multiplication, which we have already learned. Let's see what the definition says. The set V is called a vector space over the field F, and the terms of V are called vectors. If for every u, v, w in V, that is, three terms that we will take in the V set, And for every alpha,beta in F,
• 03:00 - 03:30 two terms which we will take in the field F, all ten of the following properties are met. Now pay attention, I'm going to write ten properties here. Very similar to what we did at the time in a field. But this time, we have terms that come from V, and we have terms that come from F. To help you differentiate between them, we will place an external separation. The terms in V will be called u, v, w, the letters that exist in the V world, and in field F the scalars will be marked with Greek letters, alpha and beta.
• 03:30 - 04:00 This will help us make a visual separation between vectors, u, v, w, and scalars, alpha, beta. Now let's see what are the ten properties that must be met. One, closure to addition. Meaning, we have already seen it, u plus v are added together, and stay in the set. Now, let's recall our example. That's right, when we add two matrices of order 2 by 3,
• 04:00 - 04:30 what do you get as a result? A 2 by 3 matrix. 2 by 3 matrices are truly closed to matrix addition. Let's see the second property. Addition associativity, that is, u+(v+w) will be the same as (u+v)+w. When matrices are added together, the associative law applies. That is, adding matrices is associative. We have this law, we have seen it a long time ago.
• 04:30 - 05:00 Here is law number three, addition commutativity. u plus v equals v plus u. Wait, what do we know about addition of matrices? We know that the commutative law applies there. Addition of matrices is commutative. There exists in V a term indifferent to addition, marked with 0, A term indifferent to addition we mark with 0, remember? What is true about a term indifferent to addition? 0 plus v, equals v plus 0, equals v.
• 05:00 - 05:30 When we add it, nothing happens, that's why it's called that. Does our set V have a term indifferent to addition? Of course, we already know it. The 0 matrix of order 2 by 3, is indifferent to addition. Here is property five, V has an inverse term marked minus v, and what is true about it, how do we recognize that we have an inverse? Minus v plus v is like v plus minus v, equals 0. What is 0? Indeed, it is the term indifferent to addition we found in property four.
• 05:30 - 06:00 Do we have one of these in matrices? For sure. The minus A matrix is the inverse of the A matrix. Pay attention, in the five properties we did, one to five, deal only with the addition operation in the V set. We haven't even touched scalar multiplication yet. Now let's start looking at properties six through ten,
• 06:00 - 06:30 and that's where the scalars will come too. In the meantime, we haven't touched the scalars yet. Let's continue. Here is property six, closure to scalar multiplication. Wait, closure scalar multiplication, remember? Closure means no going out, and no going in. But the scalar comes from the field, it comes from outside. As we said, we will call it closure to scalar multiplication , because it's very convenient to say that, and shortens our articulation a lot.
• 06:30 - 07:00 Indeed, we know that when we take a 2 by 3 matrix, multiply it by the alpha scalar, we get a 2 by 3 matrix. Alpha plus beta times v, is equal to alpha times v, plus beta times v. It is allowed to open brackets when I have scalar addition, and I multiply by the vector v. It will be a vector only after we finish showing all the properties, of course.
• 07:00 - 07:30 We know that in matrices this property holds. If we have a sum of scalars, when we multiply by a matrix, we are allowed to open brackets. But pay attention, the addition that is written here, is not the same as the sum written here. We have already seen these things. This is an addition of scalars. this is a addition between terms in V. How do I know? Alpha v, as written in six, is found in V.
• 07:30 - 08:00 Beta v, as written in six, is found in V. That's why here I am adding terms in V. On the other hand, alpha and beta are in F. And so here I am adding terms in F. We won't mark it in red every time, we have to get used to it, understand what it says from the context. We do it all the time, and we will continue doing it in the future. Sometimes I help you a little bit being alert to these differences. Let's move on to property eight.
• 08:00 - 08:30 Alpha times beta, times v, is the same as alpha times, beta times v, we have already seen this, remember? I simply moved the brackets, but I turned a multiplication of two scalars to a multiplication of a scalar with a term from V, according to six, beta v is found in V. And it's the same as beta times alpha, times v, or multiplying v, and bringing all the scalars to the right side. We already know that in matrices these laws hold,
• 08:30 - 09:00 we saw this as properties of matrix addition, and matrix scalar multiplication, and so everything works here too. And note over here as well. A multiplication between two scalars, a multiplication between two scalars, and all the rest, via scalar multiplication with a term from V, very soon will be called a vector. Here is property nine. For 1 in F, F is a field, remember?
• 09:00 - 09:30 In a field we have a term called 1, a typical term indifferent to multiplication. When I take the 1 in the field, and I multiply the scalar 1 by the term v, in the V set, nothing happens. 1 is supposedly indifferent to scalar multiplication. We take a term in V, multiply by it 1, nothing happens to it. We really know that when we take matrix A, and multiply it by the scalar 1,
• 09:30 - 10:00 we get the matrix A. Everything is going very well for us. Ten, please note that this is the final property. What characterizes it? Here, we finally have a property that links between adding terms in V, and scalar multiplication. Alpha times u, plus v, equals alpha times u, plus alpha times v. We are allowed to open brackets, or, note that if we read it backwards, to take a scalar out of the brackets.
• 10:00 - 10:30 This is also a property that we have already seen in matrices, we have seen all these properties in matrices, indeed. And so, we get that 2 by 3 matrices are a vector space over the field F, that was R. Let's see some more examples. First of all there is the example we have already seen, such as the running example. We take the set to be 2 by 3 matrices, we take the field to be R,
• 10:30 - 11:00 and we have a vector space. Out mascot tells us and reminds us, that we cannot multiply matrices of order 2 by 3. It is Okay. Note that in vector spaces, we do not multiply, we multiply only in scalars. Within set V itself, we do not perform multiplication. Hence, the fact that 2 by 3 matrices are not multiplied, as a rule, does not matter to the fact that the 2 by 3 matrices are a vector space.
• 11:00 - 11:30 Here is another example: Instead of taking 2 by 3 matrices, after all, the order of 2 by 3 didn't really have any meaning here. We can take matrices of any m by n order we want. Real matrices. And we will have to choose the field to be real. We choose the field to be real, so we can actually perform the multiplication and stay in the set. Wait, this can also be generalized a bit further.
• 11:30 - 12:00 Note that real m by n matrices, and the field F, are to be real. I can generalize a bit further, look. Now I have matrices over some field of order m by n. No matter which field I choose, it can be composite, it could even be the rational field we already know, although we don't work with it here.
• 12:00 - 12:30 But then it must be above the F field, the same field. So example two shows us that matrices are vectors. Because they exist in vector space. We have a good question here: How can a matrix be a vector if it has no length and direction? I assume that everyone who has already heard the term vector, said to themselves, we know what a vector is, it's an arrow that has a length and direction.
• 12:30 - 13:00 So how can a matrix be a vector if it has no length and direction? So a vector is not something with length and direction. A vector is a term in a vector space. An arrow with length and direction is just a private case. That's right, the model is a 3 by 3 matrix, she is located in the set of 3 by 3 matrices, which is a vector space, and therefore, she is also a vector.
• 13:00 - 13:30 Nothing to do about it... Everything in a vector space is a vector. Let's try to illustrate this. So first of all, we have a field. In our case, the field F will be R, the field of real numbers. A lot of clouds are hovering above it. Every such cloud will be a vector space over the field R.
• 13:30 - 14:00 What spaces have we seen so far in our examples? We have seen 1 by 6 matrices, 6 by 6 matrices, 2 by 3 matrices in the running example, 8 by 1, 5 by 3 matrices, no matter which order we choose, each set of matrices you choose is a vector space in itself, a different cloud hovering above the R field. And what does R do, you ask? R has scalars.
• 14:00 - 14:30 The role of the field is to provide the scalars, so that within the vector space it will be possible, not just to perform addition, but also to multiply by a scalar. But remember, we do not multiply within the set itself. We will not multiply 2 by 3 matrices, therefore the fact that it is not defined, does not matter to us. But 6 by 6 matrices can be multiplied. It is irrelevant, we do not multiply. In a vector space we perform addition and multiply by scalars,
• 14:30 - 15:00 which are provided by the field F. That's it friends, later we will see more examples, standard examples, like what we just saw, and slightly less standard examples. Thank you very much.