125.1 - תת קבוצה של מו שהיא מו בפני עצמה

Summary

This lecture delves into understanding when a subset of a vector space can itself be a vector space. Dr. Aliza Malek guides us through the intricacies of vector spaces, specifically focusing on subsets like symmetric matrices. The exploration involves verifying several vector space properties such as addition, scalar multiplication, and zero element existence within the subset W compared to the larger space V. The lecture emphasizes which properties need checking and demonstrates this with symmetric matrices as well as an example where a subset doesn't fulfill vector space criteria.

Highlights

• Vector spaces consist of specific properties that must be satisfied. 🌌
• We verify subsets like symmetric matrices for vector space properties. ✅
• Some properties are automatically true for subsets; others need checking. ✔️
• Example of 2x2 matrices shows which properties hold in subsets. 🧮
• Key properties: addition, scalar multiplication, identity, and opposites. ⚖️

Key Takeaways

• Understanding subspace as a vector space! 🎓
• Symmetric matrices are a key focus in vector spaces. 🔍
• Ten vector space properties need validation; some are granted as a "gift"! 🎁
• Not all subsets are vector spaces—additional validation is needed! 🚦
• Closure, commutation, and identity properties are vital for vector space credentials. 📜

Overview

In this detailed lecture, Dr. Aliza Malek takes us through the fascinating world of vector spaces, focusing on understanding when a subset itself qualifies as a vector space. The session opens with a refresher on what defines a vector space, including properties of addition, scalar multiplication, and zero element existence. Dr. Malek uses symmetric 2x2 matrices as an example, demonstrating how they fit within the broader space of matrices and satisfy necessary vector space properties.

The exploration involves confirming various properties within subsets, mainly looking at 2x2 symmetric matrices. Certain properties, such as closure and commutation, are emphasized as inherently true for these matrices, while others still require validation within the subset W. This understanding is crucial for anyone diving into linear algebra as it solidifies the concept of subspaces.

Finally, Dr. Malek provides an example where a subset does not fully adhere to vector space properties, illustrating the importance of thorough validation. Through this careful examination, the lecture instills a deeper understanding of when a subset is indeed a vector space. The session promises rich insights into the structural backbone of linear algebra while making the complex notions accessible and engaging.

Chapters

• 00:00 - 00:30: Chapter 4 - Vector Spaces Chapter 4 introduces the concept of vector spaces and explores their properties. The instructor Dr. Aliza Malek explains what vector spaces are, provides examples of standard vector spaces, and discusses five additional properties that exist within them. The focus of this chapter is to determine when a subset of a known vector space is itself a vector space.
• 00:30 - 10:30: Checking Properties of Subset and Symmetric Matrices In this chapter, the focus is on verifying the properties of subset and symmetric matrices, specifically within the context of vector spaces. The chapter starts by revisiting the definition: given a set with an addition operation and a field, V is a vector space if certain properties hold true for any three terms in V and two scalars. The standard example discussed involves matrices of order m by n as vector spaces over a field F, incorporating scalar addition and multiplication of matrices.
• 10:30 - 15:00: Example of Non-closed Set U In this chapter, we explore a specific case involving a subset of real 2 by 2 matrices over the field R. The focus is on symmetric matrices, which are those matrices satisfying the condition At = A. This condition defines what it means for a matrix to be symmetric.
• 15:00 - 18:00: Summary and Next Lesson In this chapter titled 'Summary and Next Lesson', the process of determining whether a particular set qualifies as a vector space is discussed. The lecture delves into the verification of ten vector space properties, a comprehensive task aimed at ensuring the set adheres to all necessary mathematical criteria. The instructor begins by questioning whether the sum of two terms, A and B, from the set W, continues to reside within the set, thus starting the evaluation of the vector space properties.

125.1 - תת קבוצה של מו שהיא מו בפני עצמה Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Vector Space - A Subspace that is a Vector Space Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. By now, we have seen what a vector space is, we have seen examples of standard vector spaces, and we even saw five more properties that exist in a vector space. What would we like to do today? We want to check when a subset of a known vector space, is itself a vector space? That's what we want to check, so let's begin.
• 00:30 - 01:00 As usual, I will first remind you of the definition: We have a set with addition, we have a field, V is a vector space. If for every three terms in V, two scalars, all the following ten properties are true, to which we are already used to. As mentioned, what we want to do today, we are looking at the standard example of matrices of order m by n, a vector space over F, with scalar addition and multiplication, of matrices.
• 01:00 - 01:30 We want to look at a private case. Let's take a subset of real 2 by 2 matrices, not totally critical, but for now we will take real matrices, over the R field, remember? We don't mix fields. And we want to check out a subset, symmetric matrices. Let's recall, W is equal to all matrices of order 2 by 2, that satisfy At=A, We said that this is the definition of a symmetric matrix.
• 01:30 - 02:00 Let's check whether the set is a vector space? Indeed, we need to check ten properties of a vector space, from 1 to 10, a lot of work. So let's get going. This is the set, I'm writing it down over here so we don't forget, and let's start with the first set, is A plus B belong to W? I take two terms in W, and I ask myself, does A plus B belong to W?
• 02:00 - 02:30 We have already seen this, symmetric matrices are closed to addition. If I take a symmetric one, and another symmetric one, and I add them together, I will get a symmetric matrix. We have already seen this as a property of this set. Now we want these two properties, commutation and transposition. But wait, we know that these properties hold in the collection of 2 by 2 matrices, so what does it matter if our matrices just so happen to be symmetric? If both of these proprieties are true within the whole set,
• 02:30 - 03:00 they will also be true in part of the set. Closure does not have to exist in W, But properties of commutation and transposition, what does it matter where we took the matrices from? Let's continue the checking. There exists a zero term in W that holds: zero plus v, is equal to v plus zero. In the W set, do we have a term indifferent to addition? In the V set, we do, it's the zero matrix. It's not enough, we want the zero matrix to be in the W set.
• 03:00 - 03:30 Would this happen? Certainly, we have seen that the zero matrix is symmetric. Because if we take the zero matrix and transpose it, nothing will happen. A matrix that is transposed, and nothing happens, is a symmetric matrix. The 2 by 2 zero matrix is symmetric, so it's in W. So in W we have a term indifferent to addition. Is there an opposite element in W? I remind you what an opposite element, -v plus v, or v plus -v, must give the zero of section four.
• 03:30 - 04:00 So we ask ourselves, is the opposite of every symmetric matrix, also a symmetric matrix, and is therefore in W? Let's check it out. So I will show you that it is true. Here, I take a matrix A in W, meaning an antisymmetric one, and let's check out -A, put a small t on at the top. We will do calculations and check, whether we will get a -A, again in the end. Because if we a small t at the top, and nothing happened,
• 04:00 - 04:30 it means that we have a symmetric matrix. So (-A)t equals... Remember the law that stated that a term with a minus, means -1 times that term? Therefore, -A means, -1 times A. Now we have a transposition, we can now apply the law of transposition. A scalar times a matrix, we transpose, do you remember what happens then? Indeed, the scalar is not affected, and the transposition affects only on the matrix A. Wait, but A is a symmetric matrix, so what is At equal to?
• 04:30 - 05:00 Indeed, to A, so we will get -1 times A. We will use the same property once again, -1 times A means, -A. Here, we got a symmetric matrix. We started from (-A)t, and by the end of the calculation what did we get? -A, and so, -A is in W, and it is also a symmetric matrix. Next, we will continue checking. At=A, and we move on to check number six.
• 05:00 - 05:30 Scalar multiplication, meaning that when I take a symmetric matrix, and I multiply it by a scalar, will I get a symmetric matrix? Of course, we have already seen that this is one of the properties of symmetric matrices. Look at that: 7,8,9,10, all of these properties are true for the entire V space of the 2 by 2 matrices, so they will also be true in particular in the W set.
• 05:30 - 06:00 Opening the brackets, what does it matter if the matrix is symmetric, or not? If it is allowed to open brackets, then we will open brackets. The same with the multiplication, if we are allowed to move the brackets in multiplication, what does it matter if the matrix is symmetric, or not? It is something that is related to the property of matrix scalar multiplication, and is not related to the W set. In property 9 as well, 1 times A, we get A, this is true for all matrices, in particular to the symmetric ones. And the same goes for the scalar times the sum of the matrices.
• 06:00 - 06:30 Whether these matrices are symmetric, or not, it is allowed to open brackets, and so, all of these properties automatically hold in W as well. Properties 7 to 10 are known properties, and will therefore hold in a subset as well. That's it, all ten of our properties hold, and so we got that W is a vector space. Let's summarize.
• 06:30 - 07:00 Because the W set is a subset of a known vector space, we only needed to check some of the properties. Let's check, what it is that we did indeed in fact checked? I will now show you all ten properties, here you go: We are already used to to the order in which they appear. Now, let's mark them. One color will mark what we checked, and another color will mark what we did not check,
• 07:00 - 07:30 what we got as a gift from V. That's it, we painted everything. In red painted are 1,4,5,6, these are the properties we checked. 2,3,7,8,9,10, are painted in green, that will do, no need to check. So when there is a space within a space, we will check four properties.
• 07:30 - 08:00 Indeed, that's what happened right now. I will soon show you a theorem that states, that we can do with even less than four properties, but for now, this is what happened, we checked four. We got all the other six properties as a gift. But be careful, subset U of vector space V, will not automatically hold the properties 1,4,5,6,
• 08:00 - 08:30 we will always have to check those. Only for 2,3,7, to 10, it will happen automatically. Let's see an example. Here is an example, let's take the following set: We have 2 by 2 matrices, a,b,c,d, This time U will be the set a*b=0, we multiply. Properties 4,5,6, hold, but property 1, does not.
• 08:30 - 09:00 Why do 4,5,6, hold? Let's check. 4 means, we have a term indifferent to addition. Does the zero hold a*b=0? For sure. 0,0,0,0, multiply zero by zero, you get zero. Property five speaks of the opposite. If I take the opposite, instead of a and b, there will be -a and -b, but if a*b=0, then (-a)*(-b)=0 is also true,
• 09:00 - 09:30 so the opposite will also hold the condition. Property six talks about scalar multiplication. If I multiply the matrix by the scalar alpha, instead of a and b, it will be alpha a, alpha b. What will happen when I multiply alpha a by alpha b, if a*b=0? I will get a zero as well. But the addition won't work for us, let's see why. Here is matrix A: a=1, b=0, it is in U, because 1 times 0, equals zero, a*b=0.
• 09:30 - 10:00 Here's another one as such, B: 0,1,0,0, belongs to U, why? Because zero times 1 , equals zero, a*b=0. What happens when we add? a+b is 1,1,0,0. This time, a*b is 1 times 1, it is no longer zero. Meaning, we took a matrix in U, plus a matrix in U, we added them together, and got a matrix outside of U.
• 10:00 - 10:30 Indeed, the sum is in the space V, but it is not in U set. Therefore, the U set is not closed to addition. We must not give up this check, we must check. Because closure to addition, or closure to scalar multiplication, or existence of indifferent and opposite terms, these are things that are guaranteed to me in the set, in the space V, and not in the subset U. That's it guys, we are done for now. In the next lesson we will summarize this example,
• 10:30 - 11:00 and bring up the properties that need to be proven, and those which we don't have to prove. Thank you very much.