128.3 - דוגמאות לתמ של פולינומים

Summary

In this lecture from the Technion titled "Examples of Polynomial Vector Subspaces," Dr. Aliza Malek explores the properties and examples of subspaces within vector spaces, focusing specifically on polynomials of degree less than or equal to n, designated as Fn[x]. She discusses the essential conditions for a subset to qualify as a subspace: non-emptiness, closure under addition, and closure under scalar multiplication. Polynomials with a common root are examined, and the class delves into various examples illustrating these properties, including polynomials of a specific degree and those with their third derivative equal to zero. Dr. Malek provides practical scenarios and mathematical proofs, engaging students in thoughtful reflection on these concepts. The lecture concludes with a discussion on how these principles apply to known vector spaces like R2[x], setting the stage for further exploration in future lessons.

Highlights

• Exploring Fn[x], polynomials of degree less than or equal to n, with unknown X and coefficients in field F 🔄.
• Subspaces must meet criteria: not empty, closed under addition, and scalar multiplication ✔️.
• Polynomials with a root in common, such as 1, form a vector subspace when meeting these criteria 🌟.
• Example with alpha as root shows challenges in maintaining closure under addition ❌.
• Investigating polynomials with third derivative zero reveals structures like R2[x] 🌈.

Key Takeaways

• Polynomials can form vector subspaces if certain criteria, like common roots, are met 🌱.
• Understanding the importance of closure in addition and scalar multiplication is key to defining subspaces 🔑.
• Polynomials with the same root or a specific degree can create interesting subspaces 📚.
• If a polynomial's third derivative is zero, it's part of a known vector space like R2[x] 🚀.
• Exploring these examples helps grasp broader algebraic concepts in vector spaces 🔍.

Overview

In this enlightening lecture, Dr. Aliza Malek takes us through the fascinating subject of vector spaces and their subspaces, particularly focusing on polynomials designated as Fn[x]. She begins by revisiting the foundational criteria that define a subspace: non-empty, closure under addition, and closure under scalar multiplication. With these essentials in mind, Dr. Malek encourages us to delve into the examples of Fn[x] and examine which sets qualify as subspaces and why.

Throughout the lecture, Dr. Malek offers practical examples, such as polynomials having a common root, to illustrate these abstract concepts. For instance, when polynomials in a set have 1 as a root, they meet the vector subspace conditions. She unpacks the significance of having a common root and how that influences a polynomial's role in a vector space. Further examples showcase polynomials of specific degrees and highlight instances where closure fails, sparking engaging discussions on algebraic structures.

Concluding with a profound example, she examines polynomials whose third derivative is zero, establishing their place within well-known vector spaces like R2[x]. This investigation not only clarifies complex algebraic ideas but also sets up a foundation for upcoming lessons. Dr. Malek’s approach combines theoretical exploration with practical application, engaging her audience in a deeper understanding of vector spaces.

Chapters

• 00:00 - 00:30: Chapter 4 - Vector Spaces In this chapter, Dr. Aliza Malek introduces the concept of vector subspaces within the context of vector spaces, specifically focusing on Fn[x] - polynomials of degree less than, or equal to n, with an unknown X, and coefficients in the F field. The chapter begins with a review of the definition and theorems related to vector spaces.
• 00:30 - 02:00: Properties of Subspaces This chapter delves into the properties of subspaces within the context of vector spaces. It begins by defining a vector space V and then introduces the concept of a subset W being a subspace of V. For W to qualify as a subspace, it must satisfy three key properties: it should not be empty, and it must be closed under both addition and scalar multiplication. The chapter introduces a combined property, termed B-tag, indicating that any linear combination of elements from W results in an element that also belongs to W. It concludes with examples pertaining to the polynomials of a certain order.
• 02:00 - 05:30: Exploring Subspaces in Fn[x] The chapter discusses the identification of subspaces within the polynomial space Fn[x]. Specifically, it examines the set W1, which includes all polynomials in V with a degree equal to or less than n, and the condition that p(1) equals 0, meaning all these polynomials have 1 as a root.
• 05:30 - 10:00: Examples and Non-Examples The chapter discusses the concept of a subspace, emphasizing the importance of the zero polynomial as an example of such a subspace. It highlights that the zero polynomial turns every x into 0 and remains unchanged upon addition of other functions. The chapter also mentions utilizing property B-tag for efficiency in certain scenarios.
• 10:00 - 15:00: Understanding W5 with Derivatives The chapter titled 'Understanding W5 with Derivatives' covers the concept of using scalars (alpha and beta) to multiply polynomials or functions. It discusses the condition that when 1 is the root of both p(x) and q(x), multiplying by scalars results in a new polynomial, alpha p plus beta q, which operates on 1. The chapter explains this using the operations of scalar addition and multiplication, leading to the expression alpha p1 plus beta q1.
• 15:00 - 16:30: Conclusion and Next Steps The chapter "Conclusion and Next Steps" summarizes the logical conclusions derived from the presented mathematical operations. It establishes that linear combinations of the functions p and q, denoted as alpha*p + beta*q, adhere to the ROOT condition when both individual functions have a common root. Specifically, if 1 is a root for both functions p and q, then 1 will also be a root for any linear combination of these functions. The explanation reinforces the concepts with mathematical expressions and ensures the reader understands that these principles are crucial for consequent analyses or applications.

128.3 - דוגמאות לתמ של פולינומים Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Examples of Fn[x] Vector Subspace Linear Algebra Course Dr. Aliza Malek Hello, thank you for coming back. We have reached the final standard vector space, where we will see examples of vector subspace. And this time, Fn[x], polynomials of degree less than, or equal to n, with an unknown X, and coefficients in the F field. Let's get going. As usual, we will first start by remembering the definition and the theorem.
• 00:30 - 01:00 V is a vector space, W is a subset, W is a subspace of V, if and only if, the following three properties hold: It is not the empty set, it is closed to addition, and it is closed to scalar multiplication. Properties B and C together become B-tag, alpha U plus beta V, belongs to W. This is the theorem, these are the properties we will use. Now, as mentioned, we look at examples of Fn[x]. We will take subsets of the polynomials from an order of at most n.
• 01:00 - 01:30 So let's take a look, and see which of the following sets will be subspaces of Fn[x]? W1 is all the polynomials in V, of a degree lower, or equal to n, such that p1 equals 0. In other words, all the polynomials for which the number 1, is their root. They all have at least one common root, which is 1.
• 01:30 - 02:00 Let's check that it's really a subspace. It is not an empty set, because the zero polynomial, P0(x) equals 0, if you recall, this is the zero polynomial. It's the polynomial, or the function, that turns every x to a 0, and it is indifferent to addition of functions. It will fulfill the condition: Every x we place we get 0, especially if we place the number 1. Now we will work with property B-tag, because it works, and when it does, sometimes B-tag will end up being shorter.
• 02:00 - 02:30 p(x), q(x), and W1, meaning that 1 is the root of both, we multiply by scalars, we chooses alpha,beta scalars to multiply, and here is the condition: Alpha p plus beta q, this polynomial, or this function, operates on 1, is equal: According to the definition of the operations of scalar addition and multiplication of functions, this thing is equal to, alpha p1, plus beta q1.
• 02:30 - 03:00 This is how the operations are defined. Wait, p1, I know what it's equal to, if p is in W1, p of 1, is 0, q of 1, is also 0. Therefore, if we look at the figure and place it, we will get, alpha times 0, plus beta times 0, is equal to 0. We got that alpha p, plus beta q, also satisfy the condition that 1 is a root, that is, if 1 is a root of p, and 1 is a root of q, 1 will also be a root of alpha p, plus beta q, for every alpha,beta,
• 03:00 - 03:30 and therefore, is in W1. We got that properties A and B-tag exist, and therefore, W1 is truly a vector subspace of V. If there was another root, a good question... Again, the 1 does not play a role. If 2 is the root of all, also a subspace, if 7 is the root of all, also a subspace.
• 03:30 - 04:00 What is important is that they all have the same common root. Why is a common root so important? Let's look at the following example: Here it says, there exists an alpha in F, such that p of alpha equals 0. There is some kind of root, it doesn't matter which one, it doesn't say here what alpha is. Each polynomial can have a different alpha. This is the meaning of what is written here. In example A, they all had the same root, 1.
• 04:00 - 04:30 Here, there is an alpha that is a root, and it doesn't matter to us which alpha. So W2 will not be a subspace. Why? Let's choose the field F to be equal to R, as a counterexample. When choosing a counterexample, we have to take a particular case that contradicts. In our particular case, the field will be the real field.
• 04:30 - 05:00 p(x), which is x plus 1, has some alpha scalar which is a root, minus 1, therefore is found in W2. q, which is x squared minus x, also has some real root, it has a real alpha root. Which one? Zero. There are other roots, but the main thing is for the root to be real. What happens to the sum? p plus q, equals x squared plus 1.
• 05:00 - 05:30 This time, there is no alpha to be the root. Therefore, there is no closure to addition. Here is section C: All polynomials in V, whose degree equals exactly to 2. Remember, we at Fn[x], it is a degree of at most n. From n, and down. Here, we want a degree of exactly 2. Well, W3 is really not a subspace.
• 05:30 - 06:00 The 0 polynomial will not hold that its degree is exactly 2. We said that for the 0 polynomial, either the degree is undefined, so it is not 2, or it's minus infinity, so it is not 2. It's not found there, 0 is gone, not a subspace. Let's get all smart a bit, let's add the 0 polynomial. In other words, we will look at the new set W4, a degree of exactly 2, or the 0 polynomial.
• 06:00 - 06:30 Now, it is impossible to say that the 0 polynomial does not exist. We placed it in the set. Does it work now? Did we manage to fix what didn't work before? No, there still won't be a subspace, why? Pay attention, we have no problem with scalar multiplication. We take a polynomial of degree 2, multiply it by a scalar, either the degree remains 2, the same, or, if the scalar is 0, we get the 0 polynomial, which is also inside.
• 06:30 - 07:00 We have no problem with scalar multiplication. Where does the trouble come from here? That's right, in the addition. Here is an example: We have no closure to addition. Here is a polynomial q, which is minus x squared, plus x, plus 3. its degree is 2, and therefore it is W4. The second polynomial will be p(x), x squared, plus x, it is also of the 2nd degree, it is also in W4. What happens when we add them? p plus q will be equal to
• 07:00 - 07:30 x squared plus x, plus minus x squared, plus x, plus 3, so we get 2x plus 3, a polynomial of degree 1. But we don't want 1, we want exactly 2. Therefore, there is no closure to addition. Section five: Now we want all the polynomials in V, so that triple-tag p of x, is equal to 0. Pay attention, this section we would like to solve it in two different ways.
• 07:30 - 08:00 For it to work nicely, like in this example, we will assume that n equals 3, that is, our polynomials of degree less than, or equal to 3. It doesn't matter that much, because the first way in which we will solve, will be correct for every n. But, in the second way, we will want n equals 3. So what do you care if n equals 3, in this example? So what do we do? In the first way, we will prove normally, sections A and B-tag.
• 08:00 - 08:30 W5 is not an empty set. Let's see that there are such polynomials, whose third derivative is 0. Well, obviously that for every constant polynomial, its first derivative is 0. So in particular, the third derivative is 0, therefore, all constant polynomials are definitely present in W5. The 0 polynomial is also inside, because its derivative is 0. We have no problem, this is not an empty set.
• 08:30 - 09:00 Let's take two polynomials, p and q in W5, whose third derivative is 0. Let's take two scalars, alpha,beta in F, and let's check what happens with alpha p plus beta q? Alpha p plus beta q, third derivative, equals... We will use derivative properties, and we know that when we derive, the scalars are not affected by the derivation, and thus, it's alpha triple-tag p, plus beta triple-tag q,
• 09:00 - 09:30 let's place the given data: Triple-tag p is 0, triple-tag q is 0, according to the given data, 0 plus 0, is equal to 0. And here we got that alpha p plus beta q, belongs to W5. And n being equal to 3, wasn't that relevant. But let's look at this example a little differently. It will be interesting to see things a little differently. Here is the second way: I remind you again what W5 is... Now let's take a polynomial,
• 09:30 - 10:00 remember n equals 3, therefore polynomial p in V, looks like this: ax to the third, plus bx squared, plus cx, plus d. Now we know that it is in W5. So the third derivative is 0. Let's calculate it. Here is the first derivative, 3ax squared, plus 2bx, plus c. Here is the second derivative, 6ax plus 2b, and here is the third derivative, 6a.
• 10:00 - 10:30 That's it, we are done deriving. If our polynomial is in W5, triple-tag p equals 0, that is: 6a equals 0, in other words, a equals 0. Wait, if my polynomial is ax to the third, plus bx squared, plus cx, plus d, and my a is 0, then the polynomial found in W5, actually looks like this: bx squared, plus cx, plus d.
• 10:30 - 11:00 That is, p belonging to W5 means that its degree is less than, or equal to 2. Because a is 0, we have no x to the third. Therefore, W5 is all polynomials of degree less than, or equal to 2, it is R2[x]. But R2[x] is a well-known, standard vector space, so I don't have to prove anymore. I recognized that W5 is another name for R2[x]. R2[x] is a known space, and we are done.
• 11:00 - 11:30 That's it for this time, we are done with the examples. In the following lessons, we will continue with the theory of vector spaces and subspaces. Thank you very much.