133 - פעולות בין תמ: סכום ישר

Summary

This lesson in Dr. Aliza Malek's Linear Algebra course at the Technion focuses on vector spaces and specifically on the concept of direct sum. She explains the definitions and provides the method to check if the sum of subspaces is a direct sum, highlighting the uniqueness of representation in direct sums. Examples are given using matrices, demonstrating how the intersection of subspaces affecting the direct sum. The lesson concludes with the theoretical proof of the direct sum condition and an exercise for students to explore further.

Highlights

• The concept of direct sum involves unique representation of vector spaces. 🎯
• Understanding the intersection of subspaces helps in identifying direct sums. 🔍
• Practical examples using matrices make learning engaging and clear. 📊

Key Takeaways

• Direct sum ensures unique representation of vectors! 🤓
• Understanding subspace intersections is key to determining direct sums. 🧐
• Matrix examples easily illustrate concepts of direct and non-direct sums. 📚

Overview

In the session, Dr. Malek refreshes our understanding of vector spaces, focusing particularly on direct sums and their importance in linear algebra. The direct sum allows each vector in the sum to be represented uniquely by elements from each subspace, a crucial property in vector space theory. This is illustrated using detailed examples involving matrix subspaces.

The presentation covers the procedure to determine if a sum of subspaces is a direct sum, which hinges on the intersection of these subspaces. If the intersection only contains the zero vector, then the sum is a direct sum, ensuring unique representation. Dr. Malek engages students with an equation-solving approach to solidify this concept.

Finishing off, the lesson delves deep into examples involving antisymmetric and diagonal matrices, challenging students to apply the learned principles. It emphasizes the need for understanding direct sums beyond just definitions, inviting students to test and prove these concepts in other matrix spaces, enhancing critical thinking and problem-solving in linear algebra.

Chapters

• 00:00 - 00:30: Vector Spaces The chapter titled "Vector Spaces" includes a section on vector subspace operations, specifically focusing on the concept of a direct sum in linear algebra. The lecturer, Dr. Aliza Malek, revisits definitions related to the sum of vector spaces and introduces a theorem useful for confirming when a sum of vector spaces can be deemed a direct sum. The setting elaborated on involves a vector space V over a field F, with U and W being two subspaces. This chapter aims to deepen the understanding of these fundamental aspects of vector spaces, including operations between them.
• 00:30 - 01:30: Definitions and Direct Sum Concept The chapter delves into the concept of writing a sum from two subspaces, U and W, defining the terms U and W respectively. It explores the scenarios where this sum can be expressed uniquely and situations where it can be represented in multiple ways. Additionally, the chapter introduces the definition of a 'direct sum' focusing on its unique notation.
• 01:30 - 04:30: Examples of Direct Sum The chapter titled 'Examples of Direct Sum' explains the concept of direct sum in the context of vector spaces. It describes that when two subspaces, U and W, combine to form a sum space, this space is recognized as a direct sum, denoted by U circled plus W. This chapter emphasizes the uniqueness and importance of direct sum in mathematics.
• 07:00 - 14:00: Theorem on Direct Sum Theorem on Direct Sum: The chapter explores the concept of a direct sum, emphasizing its uniqueness property. Each vector in the direct sum, represented as U + W, can be uniquely expressed as a combination of one vector from U and one from W, with no other possible representation. The text highlights matrices of order n by n as common examples, with U1 representing the space of upper triangular matrices.
• 08:00 - 10:00: Proof of Theorem Part 1 The chapter 'Proof of Theorem Part 1' delves into the properties of different matrix spaces. It begins by defining the space U2 as lower triangular matrices of order n by n. It then introduces W1 as symmetric matrices and W2 as antisymmetric matrices. The transcript emphasizes that the set of all matrices of order n by n can be decomposed into the sum of U1 and U2, specifically referring to upper triangular matrices. This serves as a foundation for the proof of the theorem discussed in this part.
• 10:00 - 13:00: Proof of Theorem Part 2 In this chapter, the focus is on expressing matrices in a specific form. It addresses the notation differences in expressing matrices as a sum of upper triangular matrices versus symmetric plus antisymmetric forms. While the former lacks a unique notation, the latter has a unique representation. The chapter explains the concept of presenting matrices as Rnn being equal to W1 circled plus W2, indicating a combination of methodologies to achieve a unique notation.
• 13:00 - 14:30: Example: Antisymmetric and Diagonal Matrices This chapter discusses the concept of antisymmetric and diagonal matrices. It emphasizes the unique way in which every matrix can be written, acknowledging that although any matrix can be expressed, there are multiple methods to approach this expression. An example is provided with V being R3, and a subspace U, defined by all a, b, c in R3, with the condition that b minus c equals 0, implying that b equals c.
• 14:00 - 14:30: Conclusion The conclusion discusses the appearance of terms in a sequence. It explains the condition where certain terms need to be equal for the sequence rules to be satisfied, such as 'b = c' leading to the sequence term 'a,b,b'. Another example given is setting 'x = z' to achieve a general term in another context, resulting in the sequence 'x,y,x' with specified variables belonging to R. The overall theme focuses on adjusting terms to maintain sequence consistency.

133 - פעולות בין תמ: סכום ישר Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Subspace Operations - Direct Sum Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. In the previous lesson we saw what a sum of vector spaces is, as well as a direct sum of vector spaces. We will recall these two definitions, and see a theorem on how to check that a sum is a direct sum, let's begin. So first we recall the definitions. V is a vector space over F, U and W are two subspaces. The sum between two subspaces is the collection of all vectors in V,
• 00:30 - 01:00 which can be written as a sum of a term from U, and a term from W, here. u plus w, when u is in U, and w is in W. We saw that sometimes it is possible to write this sum in one and only way, and we also saw that sometimes it can be done in many ways. So let's define what a direct sum is. A direct sum means unique notation, let's see the definition:
• 01:00 - 01:30 V is a vector space over F, U and W are two subspaces. The sum space between two subspaces is called a direct sum. The sum space, we already know it is a vector space, we proved it. So this space is called the sum space, and is called a direct sum, and it is marked U circled plus W, a special notation, so that the whole world will know that it is a direct sum.
• 01:30 - 02:00 What is special about a direct sum? That each vector in the sum, U plus W, can be written in a unique way as a term from U, and a term from W. There is only one way to write this notation. In the examples we have seen, we have taken V to be matrices of order n by n. This is simply a very, very typical example that is easy to remember, and internalize very quickly. The n by n matrices space, we took U1 to be the upper triangular matrices,
• 02:00 - 02:30 U2 is the space of the lower triangular matrices, everything is of order n by n, of course. W1 will be the symmetric matrices, and not surprisingly, W2 will be the antisymmetric matrices. We have already seen that the collection of all matrices of order n by n, is equal to U1 plus U2. That is, every matrix can be written as an upper triangular matrix
• 02:30 - 03:00 plus lower triangular matrix. And we saw that this notation is not unique, and thus we write the matrices as U1+U2, but when writing it as symmetric plus antisymmetric, one from W1, and one from W2, there, the notation is unique, there is only one way to do it. How do we present this to the world? We write that Rnn is equal to W1 circled plus W2.
• 03:00 - 03:30 When we see this, we know, every matrix can be written uniquely. When we see this, what do we see? Any matrix can be written, but there are many ways to do it. Let's see another example. We take V to be R3, and here is some subspace U, all of a,b,c, in R3, so that b minus c is equal to 0. Wait, b minus c equals 0 means b equals c.
• 03:30 - 04:00 So what do terms in a U look like? They look like a,b,b. b minus c = 0? We will place b equals c, in the term a,b,c, and we will get a,b,b. Here's another one. All of x,y,z, so that x equals z. We will place in there also, and we will get a general term in W. x,y,x, where x and y belong to R. We simply placed x and z to be equal.
• 04:00 - 04:30 Now we will take a vector in sum, and try to write it in different ways, or in one way. We will see if it can be done this way, or that way, why? Because we want to know whether the sum is a direct sum, or not. We will take 1,2,3, for example, and try to write it as a term from U, meaning, of the form a,b,b, plus term from W, of the form x,y,x. If we add the two terms, we get that 1,2,3 should be written as,
• 04:30 - 05:00 a+x, b+y, b+x, and we get a system of equations. Note that we got three equations, four unknowns, a non-homogeneous system, and we already know that either there is no solution, or there are infinite solutions. What does it mean there is no solution? No solution means that 1,2,3, is not found in the sum space, So it cannot be written as the sum of one from here, and one from there.
• 05:00 - 05:30 After all, we don't know if the sum is the entire space, or not. We just took some vector, and we try to write it down. If we get that there is no solution, then it is simply not in the sum. If we get that there is a solution, meaning that it is in the sum, we already know that there will be an infinity of solutions. Let's see what really comes out. I already solved it, check me. a,b,x,y is equals, minus y, 2 minus y, 1 and y, y, where y belongs to R. In other words, we got infinite solutions.
• 05:30 - 06:00 That is, there are infinite ways to write 1,2,3, as a term from U, and a term from W. This of course immediately tells us that the sum is not direct. Here is one way to write it, we choose y to be -1, here we will place y equals to -1, then we will know what a,b,x, and y are, which will give us 1,2,3. We get that it is possible to write as 1,3,3, plus 0,-1,0.
• 06:00 - 06:30 And you will see that it is indeed a term that is in the U, right? a,b,b, 1,3,3, plus x,y,x, 0,-1,0, and we are done writing this. But this is not the only way, because I can also choose y to be 1, for instance. If we choose y to be 1, we will get a different notation. -1,1,1, a,b,b, plus 2,1,2, x,y,x.
• 06:30 - 07:00 And so on, and so forth. Once we see that the sum is not direct, there are an infinity of ways... When there is an infinity of ways, the sum is not direct. Is there a simpler way to determine if the sum is direct, or not? Each time I want to know whether the sum is direct, or not, should I try to write, and check if I have a unique solution, or infinite solutions? Or is there an easier way to do it?
• 07:00 - 07:30 The next theorem will give us an answer. Let's see what the theorem says: V is a vector space over F, U and W are two subspaces, and now pay attention... The sum, U plus W is a direct sum, if and only if, wow, this, we really like, the intersection between U and W is equal to 0. An intersection between spaces is always a subspace, remember? What does it say here?
• 07:30 - 08:00 That this intersection space, is just the zero space. Then, there is a direct sum. And if there is a direct sum, the intersection will be equal to 0, let's prove it. First direction, we have if and only if, so we need to prove both directions. It is given that the sum U plus W is a direct sum, do you remember what the definition said? There is a unique notation. What do we want to prove? We want to prove that the intersection is 0. So let's see how to do it.
• 08:00 - 08:30 Let's take some v in the intersection, and we will show that when we pull a random v from the intersection, that it will have to be equal to 0. This means that the intersection has only the zero vector. I know v is in the intersection, because that's where we took it from. This means that v is in U, it is in both, currently, I am interested in it being in U. So if it is in U, I can write it as v=v+0. That is, the representative of U will be v itself, because it lives there,
• 08:30 - 09:00 and the representative of W will be 0. So we were able to write this as one of U, and one of W. But wait, if it's in the intersection, then it's also in W. Then I can now take v as the representative of W, and write v as 0+v, the representative of U this time will be 0, and the representative of W this time will be v.
• 09:00 - 09:30 But it is given that the notation is a unique, a direct sum, it is a unique notation. I can't possibly have managed to break it down in two different ways. And therefore, the two representatives of U are equal, what does it mean? That v from here, is equal to the 0 from here, that means that v is equal to 0. Indeed, the two representatives of W are also equal, that is, the 0 from here, must be equal to the v from here. No matter how we look at it, we get that v is equal to 0.
• 09:30 - 10:00 Now let's see the other direction. I know the intersection is 0. What do we want to prove? That the sum is direct. What does it mean to prove that the sum is direct? That the notation is unique. So what are we going to do? Let's take some v in the sum, And we will assume by negation, that it can be presented in two different ways, so let's do it. On the one hand, I will write v as u1+w1, when u1 is the representative of U, and w1 is the representative of W.
• 10:00 - 10:30 I will write it as u2+w2, when u2 is the representative of U, and w2 is the representative of W. And now, in fact, we will see that these two representations are one and the same. So what do we get? v is equal to something, that same v is equal to something else. So these two things must be equal. This means that u1+w1 must be equal to u2+w2, it is the same v.
• 10:30 - 11:00 Let's switch equation sides. But how do we do that? We will combine similar terms. What similar terms do we have here, you ask me? It's really simple. I know the u's are in U, so they are similar terms, they live in the same world. w1 and w2 are both in W, so they are similar terms, I will put them together. Then we get... We move u2, and move w1,
• 11:00 - 11:30 so that u1-u2 is equal to w2-w1. What did we gain from it? Pay attention. We know that U is a vector space, therefore, this difference, a sum of vectors in U, must belong to U. On the other hand, the other side of the equation, lives in W, because here I have a sum of vectors in W. Wait, so if something from U is equal to something from W,
• 11:30 - 12:00 then this vector that is in U is also in W, and vice versa, this vector from W is equal to something in U, so it's in U. In other words, they are in both U and W, that is, in the intersection. But our intersection is 0, so what do we get? That u1-u2 lives in the intersection, where there is only 0, therefore it must be 0. What is the meaning of this?
• 12:00 - 12:30 Indeed, that u1 is equal to u2. In the same way of course, w1 is also equal to w2, because w2-w1 is also in the intersection. And that's it, we got that the representation is unique. We thought that we had two representations, and we only got one. Indeed, we have finished the proof. Let's see another example. We take matrices of order n by n. We will take U to be the antisymmetric matrices of order n.
• 12:30 - 13:00 No, don't worry, this time we will not take the symmetric ones, we will take W to be diagonal matrices of order n, over R. Now we have antisymmetric ones, and diagonal ones, these are two known subspaces, and we are wondering regarding the sum... Is the sum, U plus W, which we will call Z, is this space, the sum space, which is Z, is a direct sum? We don't need to check the representation to be either unique, or not,
• 13:00 - 13:30 we will check the intersection. In real antisymmetric matrices, the diagonal is always equal to 0. So if I take a diagonal in an antisymmetric matrix, and make an intersection with diagonal matrices, then the entire diagonal must be equal to 0. Which means that U intersecting W, is equal to 0. And thus, it can be said that Z is the direct sum of U with W.
• 13:30 - 14:00 But pay attention, I leave it as an exercise for you to check, that U circled plus W is indeed a direct sum, but it does not cover the entire space for us. That is, not every matrix can be written as a sum of an antisymmetric one, and a diagonal one. But, the ones that can be written, can be written in one unique way, which is the meaning of a direct sum.
• 14:00 - 14:30 That's it guys, we are done for now. Thank you very much.