128.2 - דוגמאות לתמ של מטריצות

Summary

In this lecture, Dr. Aliza Malek delves into the concept of vector subspaces within the realm of n by n square matrices. She revisits the theorem necessary for identifying these subspaces, emphasizing the need for a set to be non-empty, closed under scalar addition and multiplication. As examples, Dr. Malek confirms that symmetric, antisymmetric, upper and lower triangular, diagonal, and scalar matrices all satisfy these conditions, thus qualifying as subspaces. However, ranked and canonical matrices don't fulfill these criteria because they lack closure to addition and scalar multiplication, respectively. The session concludes with a proof exercise involving 3x3 matrices with specific properties to validate their status as subspaces.

Highlights

• Dr. Aliza Malek explores the fascinating world of n by n matrix subspaces 🎓.
• Learn why symmetric matrices always make the cut as subspaces 📚.
• Discover how antisymmetric matrices extend to subspaces but ranked matrices don't 🚧.
• Explore the pitfalls of canonical matrices when it comes to being a subspace 🤔.
• Get hands-on with examples checking if trace-zero matrices form a subspace ✍️.

Key Takeaways

• Symmetric, antisymmetric, and triangular matrices are classic examples of subspaces 🧠.
• Ranked and canonical matrices fail to be subspaces due to lack of closure 🔍.
• Matrix subspaces need to be non-empty and closed under both addition and multiplication ✔️.
• Trace zero matrices form a subspace, holding true under specific conditions 🔄.
• Dr. Malek guides through intricate matrix proofs, boosting algebra insight 🧑‍🏫.

Overview

Dr. Aliza Malek kicks off her lecture by revisiting the core properties required for a set of matrices to qualify as a subspace within a vector space. She simplifies these by wrapping the three essential criteria into an engaging discussion, making the complex subject of linear algebra more accessible to learners.

Throughout the session, Dr. Malek uses engaging examples to illuminate the theoretical concepts, such as symmetric and antisymmetric matrices. Her exploration extends to illustrating the nuances, such as why certain matrices, like ranked and canonical ones, don’t quite make the cut as subspaces due to specific missing properties.

In a compelling wrap-up, Dr. Malek presents a practical exercise for students to prove whether certain 3x3 matrix sets are subspaces. Her methodical approach not only reinforces the lecture’s concepts but also encourages active problem-solving and critical thinking among students.

Chapters

• 00:00 - 00:30: Chapter 4 - Vector Spaces Chapter 4 introduces the concept of Vector Spaces and is part of a Linear Algebra course taught by Dr. Aliza Malek. The chapter focuses on examples of subspaces within standard vector spaces, specifically considering matrix spaces of order n by n. The lecture reviews a previous theorem concerning vector spaces V and their subspaces W, emphasizing that W is a subspace of V if it satisfies three key properties, referred to as A, B, and C.
• 00:30 - 01:00: Vector Space Subspace Criteria A subchapter of Vector Space focuses on the criteria for a subspace, namely not being empty and being closed under scalar addition and multiplication. The section uses an example involving n by n square matrices and examines sets like symmetric matrices to determine if they are subspaces.
• 01:00 - 02:00: Symmetric, Antisymmetric, and Triangular Matrices The chapter introduces and discusses the concepts of symmetric, antisymmetric, and triangular matrices. It begins by reinforcing the importance of writing down proofs for one's self, using antisymmetric matrices as a focal example. The chapter hints at considering antisymmetric matrices as a subspace. It concludes by suggesting that the validation process for antisymmetric matrices is quite similar to that of symmetric matrices, and it encourages careful note-taking throughout the exploration.
• 02:00 - 04:00: Ranked and Canonical Matrices The chapter explores various types of matrices, including upper and lower triangular matrices, diagonal matrices, and scalar matrices. It suggests that all these forms are subspaces, offering continuity in their properties. The chapter encourages readers to engage actively with the content by writing down proofs to solidify their comprehension.
• 04:00 - 05:00: Exercise: Matrices with Zero Trace This chapter discusses matrices with zero trace, focusing on their properties and behaviors. Initially, it revisits ranked matrices and addresses the limitations of a zero matrix when operations like scalar multiplication and addition are involved. Despite being inherently ranked, once other operations are applied, especially addition, the matrix does not maintain closure, meaning it loses certain mathematical properties and constraints that it initially had.
• 05:00 - 09:00: Exercise: Matrices Interchanging with Model The chapter discusses the concept of ranked matrices and their properties, focusing on the results of matrix addition. Examples of ranked matrices and their addition are provided to illustrate that the sum of ranked matrices may not necessarily result in another ranked matrix. The process demonstrates a decrease in the number of zeros and highlights the lack of closure under addition for ranked matrices. The discussion concludes by referencing a previous set or concept discussed earlier.
• 09:00 - 11:30: Non-Subspace Example: Matrices Squared to Zero The chapter discusses canonical matrices and their properties, focusing on the example of matrices that when squared result in a zero matrix. It begins by introducing the zero matrix, which is considered canonical. However, the chapter illustrates a challenge: when adding two canonical matrices, the result is not necessarily a canonical matrix. This is demonstrated with two canonical matrices, showing that their sum, for instance the combination 1,1,0,1, does not maintain the canonical form. The discussion emphasizes the lack of closure under addition in this context.

128.2 - דוגמאות לתמ של מטריצות Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Fnxn Vector Subspace Examples Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. We continue with the examples of subspaces of standard vector spaces. And this time, matrix space of order n by n, let's begin. I remind you of the theorem again: We have a vector space V, we have a subset W, W is a subspace of V, if the three properties A, B, and C, hold:
• 00:30 - 01:00 Is not an empty set, is closed to scalar addition and multiplication, and of course, B and C together become B-tag, where the checking is a little shorter. So as we said, this time, we start with subsets of n by n square matrices. So let's take the space of n by n square matrices, and let's look at the following sets. Which of the following sets will be a subspace of V? Symmetric matrices, remember?
• 01:00 - 01:30 This was the first example we started with, it works. Even then I told you, write down all of the proof for yourself, or, actually, I did it. Antisymmetric matrices, remember? We talked also about them, as a private case we would like to check, we did not check, but we hinted that it would probably be a subspace. So what do you say? Indeed, that will work too. The check here is very similar to the checking of symmetric matrices, take notes.
• 01:30 - 02:00 Upper triangular matrices, what do you think will turn out? It will turn out that this is also a subspace, and if true for upper triangulars, then also for the lower triangulars. What about the diagonal matrices? That also would work. What about scalar matrices? Nothing new there, it works as well. I suggest you have a seat, and write down the proofs in full.
• 02:00 - 02:30 Let's move on to the next set of the special matrices we saw, remember? We had ranked matrices, what will happen this time? Well, that's it, the party is over. This time it won't work anymore. Although, the zero matrix is ranked, a square matrix, non-square, a matrix of zeros is always ranked. If we multiply it by a scalar, it will remain ranked, but as far as addition goes, we don't have a closure to addition.
• 02:30 - 03:00 Here is an example of a ranked matrix, 1,0,0,0, no zeros, two zeros. Let's add this ranked matrix to it: -1,0,0,1, which is also ranked, what do we get? 0,0,0,1, not ranked. Two zeros, one zero. The number of zeros is getting smaller and smaller, not ranked. There is no closure to addition. Remember the last set we talked about?
• 03:00 - 03:30 Canonical matrices. What do you think will happen here? The zero matrix is canonical, it is inside. But, it won't work, why? This time too we have no closure to addition, pay attention: Here are two canonical matrices, each leading term is 1, everything above and below is zeros, same goes for the second matrix, but when we add them together: 1,1,0,1, is not canonical matrix anymore, why?
• 03:30 - 04:00 1 leads under the zero, no problem. 1 leads, 1 is over it, not good. This is no longer canonical matrix, there are no closure to addition. In fact, there is also no closure to scalar multiplication either, because if we take a canonical matrix, and multiply by a scalar, we will not have leading terms that are 1 left. Therefore, the canonical and the ranked matrices, are two sets that do not constitute vector spaces.
• 04:00 - 04:30 Here is a new exercise. This time, we will take V to be square matrices, though 3 by 3. In this case, n=3. For each of the sets we want to check whether we will have a subspace, or not. Here is the first set: All matrices in V, 3 by 3, whose trace is zero. Now we need to check, and show, that it is indeed a vector space. Note that if we want a trace, the matrices must be square.
• 04:30 - 05:00 We don't have an empty set, why? The 3 by 3 square zero matrix, its trace is zero. Zero plus zero, plus zero, equals zero. Let's now take two matrices in W1, let's take two scalars in R. Remember that this time our field is R, and we will check condition B-tag. What is the trace of alpha A plus beta B?
• 05:00 - 05:30 We will use the properties of a trace, remember? The trace of a sum is the sum of the trace. The trace of alpha A is alpha times the trace of A. We will use these two properties together, and from the trace properties we will get that it is alpha trace A plus beta trace B. I am allowed to take the scalar outside, and I am allowed to open up the sum, this is due to the properties. Now, we will use the given data, trace A is zero, because it is in W1. Trace B is also zero because it is in W1.
• 05:30 - 06:00 We will get alpha times zero, plus beta times zero, which is zero. Therefore, alpha A plus beta B, belong to W1, B-tag holds, and we indeed got that W1 is a vector subspace. Here is section B. All matrices A in V, that hold: that this matrix, 1,2,3, to 9, times A, is equal to A times 1,2,3, to 9. We want to see that W2 is a subspace.
• 06:00 - 06:30 Do you recognize the matrix over there? We haven't met her in a long time, this is the model. In other words, which is W2? The model times A, is equal to A times the model. All the matrices that interchange in multiplication with the model. Let's mark it with an M, so that it will be easier for us to write, and much less to. I don't mind, I write with the push of a button,
• 06:30 - 07:00 but when you write, it's a lot of writing. So let's just mark it with an M, and let's check. W1 is not an empty set. That's right, zero belongs... Sorry, the model times zero, equals zero times the model. But I want to show you another matrix that is in W2. I want to show you that M, meaning the model, is in W2.
• 07:00 - 07:30 Here is something we have seen before. Look, the model times the model, I took A to be M, equals the second model times the first model. They interchange in multiplication, so not only the zero is inside, the model herself is also inside. Okay, now let's do the proof. Let's take two matrices in W1 that interchange in multiplication with M,
• 07:30 - 08:00 Let's take two scalars, and check that alpha A plus beta B, are also in W1. M times alpha A, plus beta B... We want to achieve that for M, instead of it being on the left side, it will be found on the right side. Let's do the math. According to properties of matrix multiplication, if I open the brackets, and have M, alpha, A, the scalar can move from place to place. And so, I can write alpha MA plus beta MB.
• 08:00 - 08:30 Now I will use the given data, what does the given data tell me? MA is AM, so I can replace MA with AM, and MB with BM, let's do it. Here, I replaced them, now look what happened: In these two added terms, M is on the right side, therefore, we can take it out to the right side, and here we got alpha A plus beta B, times M.
• 08:30 - 09:00 M times the sum, equals to the sum times M. And so, alpha A plus beta B, really belong to W2. So we got that B-tag is true, it is not an empty set, and so W2 really is a vector subspace of V. You ask what would happen if there was a different matrix? Please note that we called the model M.
• 09:00 - 09:30 That means it doesn't really matter what the content is of the numbers that are written here, whether it is the model, or another matrix. What happens, is that if I take a set of the collection of all square matrices which interchange in multiplication with some given matrix M, that collection is a vector subspace of the space of square matrices. It must be must square, remember?
• 09:30 - 10:00 For interchanging in multiplication, it's only with square matrices. Let's see section C: W3 is all the matrices in V, that is 3 by 3, that when I square them, I will get zero. W3 is not a subspace because it is not closed to addition. Pay attention, indeed zero is here, because zero squared will give us a zero, but there is no closure to addition.
• 10:00 - 10:30 Let's see an example: Here are two matrices, I leave it to you to check that if you square A, you will get zero, if B is squared, you will also get zero. Remember how to do it fast? Think that I'm multiplying A times A, so I will have: I fetch a zeros column, a zeros column. I fetch the first column, it is a zeros column, and once again a zeros column, again a zeros column. So A squared will give zero. Same way, B squared will give zero. What happens when we do A plus B?
• 10:30 - 11:00 A plus B is the matrix 0,1,0, 1,0,0, and a row of zeros. Now we will take it, and square it, and we get, 1,0,0, 0,1,0, 0,0,0. We didn't get a zero. We took two matrices, that when squared, we get zero, but, we squared their sum, and we didn't get zero, and so, there are no closure for addition , and W3 is not a vector subspace.
• 11:00 - 11:30 That's it guys, we are done for now. We have one more standard space left, and we will finish the examples. Thank you very much.