126 - תת מרחב וקטורי: הגדרה

Summary

The transcript from the Technion Teaches video covers the concept of vector subspaces in linear algebra. Dr. Aliza Malek explains when a subset W of a vector space V can be considered a vector subspace. A key focus is on checking four specific properties to determine this: 1, 4, 5, and 6. The video further distinguishes between different spaces and gives examples like R2 vs. R3 and R+ vs. R to clarify these distinctions. The latter part promises a more efficient method in future lessons, achieving the same validation with fewer checks.

Highlights

• Vector subspaces are all about identifying the magic mini spaces within big vector spaces! ✨
• Only four properties need verification to confirm a vector subspace – so much easier than starting from scratch! ⚡
• R2 isn't a subspace of R3. It's like trying to fit a flat pizza into a round skillet! 🍕
• Mixing fields like R and C means 'no subspace love here!' Different rules, different grounds. 🚫
• The operation matters too, y'all. Just like different recipes for the same dish! 🍳

Key Takeaways

• Vector subspaces are derived from existing vector spaces, simplifying the process of validation by focusing on fewer properties. 🌌
• Examples like R2 vs. R3 help clarify why not all spaces fit neatly within each other. ✖️
• The difference in fields (like R vs. C) remains crucial in determining subspace relationships. 🔍
• Understanding subspaces can pave the way for efficient problem-solving in linear algebra. Ready to cut those checks down? ✂️

Overview

In this lesson, Dr. Aliza Malek dives into the concept of vector subspaces in linear algebra, defining them as subsets of vector spaces. A vector subspace retains certain properties of the vector space it's part of, without requiring a check of all ten properties that define vector spaces initially. Instead, only four need to be verified – properties 1, 4, 5, and 6.

Through examples, Dr. Malek clarifies misconceptions, such as R2 not being a subspace of R3 due to differences in dimension and order. She also explains that R2 and C2 cannot be subspaces of one another because they're defined over different fields. Importantly, the addition operation in spaces like R+ differs from R, highlighting why not all subspaces are apparent at first glance.

Viewers are encouraged to understand why this narrowing down to fewer checks matters by looking at antisymmetric and symmetric matrices. The teaser for the next lesson hints at even more efficient methods to identify vector spaces by reducing the checks from four to two, signaling significant effort savings.

Chapters

• 00:00 - 00:30: Vector Spaces In Chapter 4 titled 'Vector Spaces', Dr. Aliza Malek introduces the concept of vector subspaces within the larger vector spaces. The lesson explains the relationships and properties of a space that exists within another space, and presents a theorem that simplifies the process of determining the necessary conditions for a subset to be considered a subspace. The discussion begins with a reminder that if V is a vector space over a field F, then W, a subset of V, is studied for its subspace characteristics.
• 00:30 - 01:00: Vector Subspace - Definition The chapter focuses on defining a vector subspace and highlights how to determine if a subset W is a vector space within a larger vector space V. Of the ten properties required to be a vector space, only four need verification for W: properties 1, 4, 5, and 6. The rest are inherently valid, being inherited from V. The chapter ensures clear guidance on which properties necessitate checking through color-coded categorization.
• 01:00 - 01:30: Naming the Vector Subspace This chapter discusses the concept of a vector subspace. It introduces the idea of naming a subset of a vector space. Specifically, if V is a vector space over a field F, and W is a subset of V, then W is referred to as a vector subspace. This chapter provides a definition and context for understanding this concept within the broader framework of vector spaces.
• 01:30 - 02:00: Vector Subspace Properties The chapter focuses on the concept of vector subspaces, which can also be referred to simply as 'subspaces.' It discusses conditions under which a subset of a vector space can itself be considered a vector space, or subspace. The text emphasizes the importance of using the same operations and field for the subspace as those defined for the original vector space. Changing operations or fields would disqualify the subset from being a subspace.
• 02:00 - 03:30: Examples of Vector Subspaces The chapter begins by discussing the concept of subspaces within the realm of linear algebra, focusing initially on 2 by 2 matrices. The instructor reminds the reader that they have already learned that real symmetric 2 by 2 matrices form a subspace within the space of all 2 by 2 matrices. Through this, the reader is encouraged to understand the hierarchical relationship between different types of matrix spaces within linear algebra, focusing on symmetric matrices as an initial example. The chapter sets the stage for exploring more instances and examples of vector subspaces with the aim to provide a clearer understanding of the topic at hand.
• 03:30 - 05:00: Difference Between Vector Space and Subspace The chapter discusses the concept of vector spaces and subspaces, focusing on whether R2 is a subspace of R3. It clarifies that R2, a plane, is not a subspace of R3, the space, because R2 is not a subset of R3. This is due to the difference in dimensions: R2 consists of ordered pairs, while R3 consists of triples.
• 05:00 - 07:00: Checking Properties of Subspaces The chapter discusses the criteria for determining if one mathematical space is a subspace of another. It emphasizes the importance of fields and specifies that spaces must be over the same field to be considered subspaces. An example is given with R2 and C2, highlighting that since they are over different fields, one cannot be a subspace of the other.
• 07:00 - 07:30: Efficient Checking of Subspace Properties This chapter discusses the concepts of vector spaces and subspaces in the context of different fields. It explains that R2, a vector space over the real numbers R, and C2, a vector space over the complex numbers C, are distinct and not subsets of each other. Furthermore, it examines whether R+, the set of all positive real numbers, can be considered a subspace of the real numbers R. The distinction lies in the fact that R+ is a portion of R and consists only of positive values, highlighting the criteria for a subspace within vector spaces.

126 - תת מרחב וקטורי: הגדרה Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Subspace - Definition Linear Algebra Course Dr. Aliza Malek Hello, I'm really glad you came back, because in the last lesson we saw what happens when we have a space within a space. Today we will give this thing a name, and after that we will formulate the theorem that tells us how many checks really need to be done, so let's get started. I remind you, if I have V, a vector space over field F, and I have W, which is a subset of the known space V,
• 00:30 - 01:00 and we want to know whether W is a vector space in itself, then, out of all ten properties, it would be enough to check just four. Let's recall those that should be checked, and those that should not be checked. Here all the properties are already marked with the correct colors. So out of all properties, 1 to 10, we got it all as a gift from V, except for properties 1,4,5,6, which we had to check.
• 01:00 - 01:30 Now, we will give this thing a name. This case of the space W, within the known space V, we shall give it a name. Here is the definition: Let V be a vector space over field F, and let W be a subset of V. I took only a portion, not all. W is called a vector subspace, which can be abbreviated as seen here:
• 01:30 - 02:00 Sometimes we will write like this, sometimes we will write like that. Vector subspace, or simply subspace, but when? If W is a vector space by itself, but it is not enough. Using the same operations, and over the same field. We are not allowed to change the operations,
• 02:00 - 02:30 and we are not allowed to change the field. We already saw in the previous lesson that real symmetric matrices of order 2 by 2, are a space by itself, within the space of 2 by 2 matrices, therefore, symmetric 2 by 2 matrices are a subspace of the space of 2 by 2 matrices. Now, we will see a few more examples that will help us refine well
• 02:30 - 03:00 the definition we have seen here just now. Here is the first example, is R2 a subspace of R3? Think about it for a second, R2, is it a subspace of R3? R2 is the plane, and R3 is the space, remember? So the answer is no, because we are told that W should be a subset of V. R2 is not a subset of R3. In R2, we have ordered pairs, in R3 we have triples, it's not the same thing.
• 03:00 - 03:30 The one is not a subset of the other, so it will not be a subspace. Here is the second example, is R2 a subspace of C2? We do not mix fields, remember? R2 is a space over R, and C2 is a space over C. But we are told that it should be over the same field, and so, the one is not a subspace of the other.
• 03:30 - 04:00 R2 is a space over R, C2 is a space over C, and these are two completely different spaces, they are not one inside the other. And a third question, is R+ a subspace of R? After all, both were vector spaces over R, we did not mix fields. Remember what R+ was? We took a portion of R, only the positive ones,
• 04:00 - 04:30 we defined operations, and got a vector space. We also saw that every field is a vector space over itself, R is a vector space over R. So is R+ a subspace of R, over R? The answer of course, is no, because pay attention that it states here, "using the same operations". The way we perform addition in R, is not the way we defined the addition in R+. I remind you that in R, 2 plus 3, is 5.
• 04:30 - 05:00 In R+, 2 plus 3 was, 2 times 3, that is 6. Not the same addition, not the same operations, is not the same subspace. This is a space, and this is a space, they are not a subspace of one another. So what is the difference between a vector space and a subspace, if this is a space, and this is a space? In the end, they both are spaces really, so what is the main difference? The main difference is in the number of checks that need to be performed.
• 05:00 - 05:30 Suppose we want to check whether W is a vector space. For example, we have seen symmetric matrices, and now I want to check antisymmetric matrices, for example. I ask myself, is this a vector space? If we manage to identify that our set is a subset of a known vector space... Antisymmetric matrices are a subset of square matrices.
• 05:30 - 06:00 Square matrices are a well-known vector space. Using the same operations, and over the same field. Indeed, in antisymmetric matrices we didn't change operations, we didn't change fields, so we get to check fewer properties. If I want to know whether antisymmetric matrices are also a vector space in itself, I will no longer have to check ten properties, because W inherits a large part of the properties from V.
• 06:00 - 06:30 Antisymmetric matrices will inherit many properties from the square matrices. Only four need to be checked, that's right, that is exactly what we saw. Remember what properties we had to check? 1,4,5,6, just four. In the next lesson we will formulate a theorem that states: That instead of checking four, we can check three,
• 06:30 - 07:00 and we will even formulate these three properties as two properties. That is right, we will save eighty percent of the checks, that is a lot of work we will save. If we would like to know whether antisymmetric matrices are a vector space, instead of checking ten, and instead of checking four, we will settle for just two checks. Wow, that is a lot of effort savings. So we will see these things in the next lesson.
• 07:00 - 07:30 So, in the meantime, thank you very much.