132 - פעולות בין תמ: סכום

Summary

The video discusses vector space operations, focusing specifically on the sum of subspaces. The lecture by Dr. Aliza Malek covers the definition of a sum between vector spaces, examples involving matrix operations, and proofs showing that sums of subspaces result in another subspace. Key examples include demonstrating the sum of upper and lower triangular matrices and symmetric versus antisymmetric matrices. The concept of a direct sum, where each vector in the sum can be written uniquely as a sum of vectors from each subspace, is also introduced. Throughout, it emphasizes the mathematical properties that ensure operations remain within defined subspaces.

Highlights

• Discover how sums in vector spaces work. ➕
• Sum of upper and lower triangular matrices equals all square matrices! 📐
• Axiomatic proof ensures U+W is always a subspace. 📚
• Direct sum provides a unique notation for sums. 🔢
• Explore examples with 2x2 matrices for clarity. 🔍

Key Takeaways

• Sum between vector spaces is all about addition! ➕
• Every matrix can be expressed as the sum of upper and lower triangular matrices. 📐
• Symmetric and antisymmetric matrices have a unique sum notation. 📏
• Direct sum means one unique way to express each element. 🔢
• Remember: every U+W is a subspace of V, always! 🧠

Overview

In this video, Dr. Aliza Malek from the Technion covers the fascinating world of sums in vector spaces. The lecture begins by defining what a sum between vector spaces entails, using concrete examples with matrices to illustrate the point. You'll learn how adding upper and lower triangular matrices can yield all square matrices, showing the elegance of linear algebra operations.

Dr. Malek breaks down why the sum of certain kinds of matrices - like symmetric and antisymmetric - result in particular unique notations. This concept, known as a "direct sum," signifies a unique representation of every element in the sum, providing a deeper understanding of vector space operations.

Through a series of proofs and examples, you will understand how every U+W results in another subspace. By working through specific matrix examples, you’re set to grasp the practical applications of these theoretical constructs. Dive into this concise yet comprehensive guide to subspace sums and see how pure math translates into beautiful structures!

Chapters

• 00:00 - 00:30: Vector Spaces Dr. Aliza Malek discusses operations in vector subspaces focusing on the sum of vector spaces. She begins by building on previously discussed concepts such as intersection, union, and difference of vector spaces. The current chapter delves into the definition and explanation of the sum of two subspaces within a vector space V over a field F. Two subspaces, U and W, are used as a basis for defining and exploring the concept.
• 00:30 - 01:00: Definition of the Sum of Subspaces This chapter explains how to define the sum of two subspaces in a vector space. In essence, the sum of two subspaces is the set of all vectors that can be expressed as the sum of a vector from each subspace. Specifically, the sum of subspaces U and W consists of all vectors of the form u + w, where u is a vector in U and w is a vector in W. This definition highlights the additive properties inherent in vector spaces, enabling combinations of elements from different subspaces.
• 01:00 - 01:30: Example with Matrix Addition The chapter titled 'Example with Matrix Addition' provides an example of how matrices can be added together using specific types of matrices. It explains that given an n by n square matrix V, an upper triangular matrix U, and a lower triangular matrix W, the sum of U and W results in matrices that are composed of both upper and lower triangular matrices. This illustrates the fundamental concept of matrix addition, showing how matrices of different forms can be combined to create new matrix structures.
• 01:30 - 02:00: Properties of Subspace Sum The chapter 'Properties of Subspace Sum' explores the concept of adding subspaces of matrices and resulting in vector spaces. It highlights that the sum of upper and lower triangular matrices still fits into a vector space. Through the example of a 2x2 matrix where 'A' is expressed as the sum of an upper triangular and a lower triangular matrix, the chapter demonstrates this property of matrix operations and subspace sums.
• 02:00 - 03:00: Proving the Sum is a Subspace In the chapter titled 'Proving the Sum is a Subspace,' the narrator demonstrates a process of decomposing and reconstructing a matrix. Starting with the elements a, b, 0, d, they describe how to arrive at the form a, b, c, d by strategically adding zeros to manipulate the elements. They are initially left with a partial decomposition expressed as 0, b, 0, 0, and note that this format represents an upper triangular matrix. The process illustrates a method for proving the sum of elements forms a subspace by manipulating components in a structured way to achieve a desired arrangement of the matrix.
• 03:00 - 04:00: Containment and Examples This chapter discusses the concept of containment and provides examples using matrices. It highlights how a lower triangular matrix can be derived and suggests there are potentially infinite ways to do so. The chapter emphasizes that any matrix is essentially a sum of its upper and lower triangular forms, demonstrating the equality between these components.
• 04:00 - 04:30: Direct Sum and Uniqueness The chapter discusses the properties of vector spaces, specifically focusing on the concept of direct sum and the conditions of uniqueness. It explains that given a vector space V over a field F, and two subspaces U and W, the sum of these subspaces U+W always forms a subspace. Unlike the intersection of two subspaces which is always a subspace, the union of two subspaces is only a subspace when one is contained within the other.
• 04:30 - 05:00: Matrix Examples The chapter titled 'Matrix Examples' delves into the properties of subspaces, focusing on the sum and difference of subspaces. The transcript begins by emphasizing that the difference of subspaces does not always result in a subspace, except for trivial cases. In contrast, the sum of subspaces consistently forms a subspace. The chapter proceeds to prove this claim by considering two subspaces U and W, and demonstrating that their sum, U plus W, retains the properties required to be considered a subspace of a larger vector space V. The proof is centered around establishing the three fundamental properties that define a subspace.
• 05:00 - 06:00: Conclusion on Direct Sum In this section, the concept of a direct sum is concluded by discussing that the sum of two subspaces U and W, represented as U+W, is not empty. It is emphasized that since 0 is included in both U and W because they are subspaces, the zero element can be expressed as the sum of an element from U and an element from W. This concept illustrates that even without specific elements from U and W, the zero element can still be expressed as a sum of representatives from each subspace.

132 - פעולות בין תמ: סכום Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Subspace Operations - Sum Linear Algebra Course Dr. Aliza Malek Hello, thanks for coming back. After seeing the intersection, the union, and the difference, It is now the turn of the sum, so let's get going. A sum between vector spaces, here is the definition. I have V a vector space over the field F, I have U and W, two subspaces of V. Remember that in order to define a sum,
• 00:30 - 01:00 we need to know how to add terms between the groups. How lucky that in vector space, we know how to add terms. The sum between two subspaces is the collection of all vectors in V, which can be written as the sum of a term in U, and a term in W. U plus W equals all terms of the form u+w, when u belongs to U, and also, w belongs to W.
• 01:00 - 01:30 We add one from here with one from here, one from here, one from here. Let's see an example right away. I take V to be the n by n square matrices. We will take U to be upper triangular matrices of order n. As you remember, W is the space of lower triangular matrices of order n by n. So what would U plus W be? These are all the matrices that are the sum of an upper triangular, plus the lower triangular.
• 01:30 - 02:00 Upper triangular of order n of course. But these are in fact are all square matrices. Right, so if we did U plus W and got all the matrices, we got a vector space. Of course we will soon see that this is not by accident, but let's see why all the matrices truly turn out. Here's an example, we will demonstrate this for n=2, that's enough. Here is a matrix A, which is 2 by 2, a, b, c, d. I will write it as the sum of an upper triangular and a lower triangular.
• 02:00 - 02:30 Here, a, b, 0, d. Now I want to get a, b, c, d, therefore, to a we will add 0, to b we will add 0, we will add 0 to c, to get c, after that I have d here, so I'm left with 0, I was able to decompose. But wait, I can write it a little differently. I can write it as 0, b, 0, 0, it's an upper triangular matrix. And to complete this to a,b,c,d,
• 02:30 - 03:00 we will take a,0,c,d as the lower triangular. We were able to do this in two different ways. And do you recall that if we have two ways, there are probably infinite ways. But still, each matrix is a sum of upper triangular and lower triangular, and vice versa, an upper triangular plus a lower one, is an n by n matrix, So thus we got equality between the groups. The sum of upper triangular and lower triangular
• 03:00 - 03:30 is all the matrices of order n by n. So here is our axiom: V is a vector space over F, U and W are two subspaces. The axiom states that the sum U+W is always a subspace. Remember what we had? The intersection always came out a subspace, the union came out a subspace only in the case of a containment,
• 03:30 - 04:00 and the difference always came out not a subspace, unless these were the trivial cases. And now we have the sum, and the sum always turns out to be a subspace. Let's prove it, here is the proof. U and W are two subspaces, I want to show that U plus W, this new group of U plus W is also a subspace of V. For this we will prove the three properties of a subspace.
• 04:00 - 04:30 Let's start from number 1, U+W is not an empty group. I know for sure that 0 is in U and in W, because they are subspaces, so we will take 0, and write it as 0+0. I took 0 and wrote it using a representative from U, 0, plus a representative from W, 0. Maybe I could have written it differently, but without you telling me what is U and what is W, I would not know. But as 0 plus 0, it is certainly possible to write this.
• 04:30 - 05:00 That's it, so I have at least one vector that is a sum of one of U, and one of W, which is 0, and our group is not empty. Now we will show that U plus W are closed for addition. So I take two vectors, v1 and v2, that are in the sum. I want to show that this new vector, v1+v2, also belongs to the sum. So if I know that v1 plus v2 are in the sum,
• 05:00 - 05:30 what does that mean according to the definition of the sum? This means that each of them is the sum of one from here, and one from here. Officially, there exist u1 and u2 in U, w1 and w2 in W, so that v1 will be u1 plus w1, I write v1 as one from here and one from here, and v2 is u2 plus w2, one from here and one from here. Now let's do the math.
• 05:30 - 06:00 v1 plus v2, what is it equal to? We will place... We will get u1 plus w1, which is v1, to which we add u2 plus w2, which was v2. Now we will use properties two and three of vector space. Remember what that means? I have the commutative law, and the group law, I am allowed to change the order as I wish, and group and add them as I want. And here is what I get out of all this.
• 06:00 - 06:30 I can write this amount as u1 plus u2, together, plus w1 plus w2, together. But wait, remember? u1 and u2 are in U, therefore the sum is also in U, because it is a vector space and it is closed for addition. The same goes for the w's. w1 and w2 are in W, W is a space closed for addition. Therefore, from the definition of the sum and the closure for addition of U and W, what will I get?
• 06:30 - 07:00 That v1 plus v2 are basically, some vector from U plus some vector from W. And therefore they are also in U plus W, because I was able to write it as the sum of one from here and one from here. Which one is the one of U? u1 plus u2. Which one is the one of W? w1 plus w2. Three... We will show that U plus W is closed to multiplication by a scalar. Let's take v which is in U+W, let's take a scalar,
• 07:00 - 07:30 we want to prove that αv is also in U plus W. I multiplied by a scalar, I stayed within the group. v is in U+W, so v can be written as some vector from U, and a vector from W. Let's take one as such, and get v equals to u+w. Now let's see where αv is. αv equals... Again we place, we get α times u+w.
• 07:30 - 08:00 Now we will use property ten of a vector space that tells me that I can open the brackets. We will get αu+αw. Again, u is in U, U is a vector space closed for multiplying by a scalar, therefore αu is also in U. In the same way, αw is in W. Therefore, from the definition of the sum, and the fact that u and w are closed for multiplication by a scalar, we get that αv is in U+W.
• 08:00 - 08:30 That's it, this completes the proof for us. Of course we will now notice that U and W are contained in U+W. Meaning, what does it mean that U plus W are contained in U+W? This means that U and W are subspaces of U+W. Why is this true? Let me show you. Every u that is in a U, that is in the space, I can write as u+0,
• 08:30 - 09:00 because 0 is indifferent to addition. What is 0? This will be the representative I choose in W. That is, I can write every term from U as a term from U, and a term from W. The representative from U is U itself, the representative from W is 0. I will do the same with vectors in W. Every w in W, I will write it as: The representative from U will be 0, and the representative from W will be W itself.
• 09:00 - 09:30 Since I can write this notation, it automatically follows that both U and W are subspaces of U+W. Now let's see examples. Before that I want to show you something very nice. U intersecting W is contained in U, and in W. We just said that U and W are contained in U+W,
• 09:30 - 10:00 and according to axiom, U+W is a subspace of V, meaning U+W is contained in V, look at this sequence. I am being asked where the union is, where has it gone to? The union is not a subspace, so it is of less interest to us. But we must remember that U+W sits inside the sum U+W, because U is there and W is there, therefore the union is there, but it sits there as a subgroup, and not as a subspace,
• 10:00 - 10:30 because union is not a subspace. Good question, thank you very much. That's it, now it is time for the examples. So I take V to be R4. Here is the first space I choose, U will be all terms, a,b,c,d, in R4, so that: c+d equals 0, and both a and b are 0. Basically, the terms there look like this:
• 10:30 - 11:00 0,0, because we said that a and b are 0, d, -d, why? Because we said that c+d is equal to 0, so one is d, and the other is minus d. Let's take another space. W will be every a,b,c,d in R4, such that a equals b, equals d. What will this look like? Here... a is equal to b, therefore a,a, and d is also equal to a, therefore I wrote a instead of d as well.
• 11:00 - 11:30 a and c of course can be any number within R. And now we ask ourselves... Of course we will not show that these are subspaces, you will do that yourself. We will ask ourselves, what will a vector look like in the sum? So, a vector in the sum can be written as a vector of the form 0,0,d,-d, plus a vector of the form a,a,c,a. Let's see an example.
• 11:30 - 12:00 I want to take a vector and write it as a pattern like this plus a pattern like that, this will tell me that it is in the sum. Here, 4,4,7,13. I ask myself, is it in U+W? Let's try to write this down. I see that there is a 0,0 here, so there is no choice, 0,0 must be written over here.
• 12:00 - 12:30 But I want to get to to a,a, for me it is 4,4. Therefore, there is no choice, these two places must be 4. But in W the last place is also 4, so 4 is also written over here, no choice here. Now over here, it says 4, I have 13, therefore we have no choice, over here there must be a 9. But, over here it says d,-d.
• 12:30 - 13:00 So, if there is a 9 here, there is no choice, what must come here? Minus 9. I want to get to 7, so there is no choice, over here there must be a 16. And now, pay attention, 0,0,-9,9 is the pattern of U. 4,4,16,4 is the pattern of W, a,a,c,a. I was able to write 4,4,7,13 as a sum of a term from U, 0,0,-9,9,
• 13:00 - 13:30 plus a term from W, 4,4,16,4. So, is it within U+W? The answer is yes. On the other hand, 2,1,3,4 is surely not in U+W. There's no way I will be able to decompose into a pattern of the form 0,0, because this will require that the same number be written over here,
• 13:30 - 14:00 and if I have 0,0, there is no way to add the same number and get to 2,1. So, 2,1,3,4 is in R4, but it is not in U+W. It is not there, it will be impossible to decompose it according to these patterns. Of course we don't work like that - by guessing, this is the first example, here I showed you how it works, and how to do it. After that we will see how to solve equations that lead us to the correct decompositions.
• 14:00 - 14:30 Note that unlike the previous example with the triangular matrices, here we got only one way, to write 4,4,7,13 as a sum of one from U, and one from W. We had only one solution, we could not choose anything here, all the numbers were imposed on us. Because here was a 0, here was a 4, and this 4 led to a 9, and the 9 led to the minus 9, and the minus 9 to a 16.
• 14:30 - 15:00 We couldn't choose anything at will, we only had one decomposition. Let's see another example, now let's take the matrices again, we really like the n by n matrices. We choose U to be symmetric matrices, W will be antisymmetric matrices of order n, so U+W is all the matrices that are: the sum of a symmetric matrix and an antisymmetric matrix.
• 15:00 - 15:30 I really hope you remember that these are all the matrices, remember that? Here is a matrix of order n by n. I write it as half A plus At, plus half A minus At, opening the brackets, it really turns out an A. So, half A + At is symmetrical, half A - At is antisymmetric, and we got that V is equal to U+W. Every U+W matrix is in V,
• 15:30 - 16:00 every matrix in a V is a sum of a symmetric one plus an antisymmetric one. And so here, unlike the previous example, where we found a vector that is not in the sum, here, every matrix in the sum. Therefore, this time the subspace U+W came out as the whole space, the whole V. This time too, we found that this notation is unique. We see to show this phenomenon in an official way.
• 16:00 - 16:30 Let's define a unique notation versus a non-unique notation. A direct sum, the definition states that given that V will is a vector space over the field F, U,W will be two subspaces. The sum space between two subspaces is called a direct sum, and is marked U circled plus W, if every vector v in U+W, every vector in there,
• 16:30 - 17:00 can be written uniquely as the sum of a term from U, and a term from W. That is, if a vector that is in the sum, which is a term of U, and a term of W, there is only one way to write it, we say that the sum is direct, and we write U circled plus W. If there is more than one way to write it, then we simply write U+W.
• 17:00 - 17:30 This is how we tell the world whether the notation is unique, or not. Whether I have only one way to do it, or several ways to do it. Let's see an example. I take V to be real square 2 by 2 matrices, I take U1 to be upper triangular matrices, and I take W1 to be lower triangular matrices, of order n, of course for us n is equal to 2.
• 17:30 - 18:00 And now I write, I take the matrix A, And we have already seen that when it comes to triangular matrices, I can write A in at least two different ways. Here's one way, and here's the other way we have already seen, and I'm sure you will find many more ways to do it. In fact, an infinity of ways. But there is more than one way. Therefore, in addition we also found that every matrix can be written like this, so we can say that the n by n matrices, or in our case the 2 by 2 matrices,
• 18:00 - 18:30 it is possible to do it n by n, are equal to U1+W1. And we don't put a circle around the plus sign, why? Because we have many, many ways to do it. But, when we took U and W to be symmetric and antisymmetric, we found that there is only one way to do it. And so in this case, the sum here is not direct,
• 18:30 - 19:00 but if we take symmetric and antisymmetric ones, there, there is only one way to write a matrix as the sum of one from here and one from here, therefore in that case, we will get that Rnn is equal to U circled plus W. Then, a side observer, who sees that here it says U circled plus W, will know that someone has already checked that every term in the sum, can be written uniquely as one from here, and one from here. And if we see U1+W1 without a circle around the plus sign,
• 19:00 - 19:30 this automatically tells us that there are many ways to do that. That's it, in the next lesson we will repeat the definition, and define some kind of an axiom that will allow us to check in a very simple way, whether the sum is direct, or not. That's it, thank you very much.