273 - הקדמה - מונחים מתחום הפונקציות

Summary

In this enlightening session, Dr. Aliza Malek guides us through an essential topic in linear algebra: understanding functions and their properties within vector spaces. After discussing invertible matrices and determinants, we turn our attention to exploring the fundamentals of functions between vector spaces. This episode delves into the definition of functions, examines conditions that define them, and explores domains, ranges, preimages, and images. With clear examples, Dr. Malek illustrates various function types, including injective and surjective functions, and explains how to identify them through structured equations and reasoning. Excitingly, she concludes with a hint of an intriguing connection to be revealed in the next lesson.

Highlights

• Understanding the basic terms related to functions is crucial for linear algebra. 🌟
• The definition of a function involves a precise correspondence between two groups, ensuring no element is left unmatched. 🔗
• Examples illustrate how to determine if a set is the domain or range and how the correspondence rule defines the function. 📊
• Injective functions map distinct values to distinct images, ensuring uniqueness in mapping. 🔑
• Surjective functions guarantee that every element in the range has a corresponding preimage. 🔄
• The graphing of functions using software like DESMOS provides a visual way of comprehending complex ideas. 📈
• The lecture ends with a hint of further mysteries to unravel, keeping the learning journey engaging. 🛸

Key Takeaways

• Functions are essential components in vector spaces, acting as correspondences between sets. 🧠
• A function must meet specific conditions: each element in the domain maps to only one element in the range. 🔗
• Understanding preimages and images is crucial for defining and evaluating functions. 🔍
• Functions have types: injective (one-to-one) and surjective (onto), each with distinct properties. 🧩
• Injective functions map different elements to different images, while surjective functions cover the entire range. 🌍
• Real-world examples and mathematical equations help cement understanding of theoretical concepts. 📚
• Despite technical concepts, the lecture maintains a light-hearted and intriguing atmosphere. 😊

Overview

In this intriguing episode, Dr. Aliza Malek returns to vector spaces after tackling the more technical concepts of invertible matrices and determinants. She opens by explaining the fundamental ideas surrounding functions in linear algebra, detailing how they serve as correspondences between sets. Key concepts such as domain, range, preimage, and image are redefined in relation to functions, providing learners the groundwork needed for deeper understanding.

Dr. Malek proceeds to clarify what it means for functions to be injective and surjective. Through clear and relatable examples, she demonstrates how injective functions (one-to-one) ensure unique mappings, while surjective functions (onto) cover every element in the range. Her discussion highlights how the setup of a function—including its domain and range—affects its classification, using simple equations to demonstrate these properties effectively.

The episode concludes with a promise of more engaging content, hinted by the mysterious appearance of a spaceship in the background. Dr. Malek maintains a captivating and accessible tone throughout, merging technical content with an enjoyable presentation style that keeps viewers curious for what comes next in their mathematical journey.

Chapters

• 00:00 - 00:30: Linear maps introduction In this chapter titled 'Linear maps introduction,' Dr. Aliza Malek welcomes back students after covering technical topics like invertible matrices and determinants. The focus returns to vector spaces, specifically on functions between vector spaces. Dr. Malek aims to refresh foundational terms from the functions field that will be relevant in this context.
• 00:30 - 01:00: Definition of a function A function is defined as a correspondence between two groups, A and B, where each element in group A is paired with one and only one element in group B. There are two essential conditions for this to be considered a function: 1) Every element from group A must be matched to a single element in group B, and 2) This matching must be exclusive—no element in A pairs with more than one element in B. Each element of A corresponds to a unique element in B, reinforcing the one-to-one relationship essential for defining a function.
• 01:00 - 01:30: Correspondence rule and examples This chapter introduces the concept of functions, highlighting two essential conditions. A function, denoted as f: A → B, involves a mapping from domain A to range B. The chapter emphasizes the notion of a 'correspondence rule,' which assigns each element in the domain to an element in the range.
• 01:30 - 02:00: Examples of functions The chapter titled 'Examples of functions' discusses the concept of functions in mathematics, focusing on the association between two sets, usually denoted as A and B. It explains that functions are defined by a rule or correspondence between these sets whereby each element in set A is paired with an element in set B, called the image. Any element in set A used in this relation is referred to as a preimage. It highlights that changes in the correspondence rule, domain (set A), or range (set B) will result in a change of the function itself.
• 02:00 - 02:30: Detailed function properties The chapter titled 'Detailed function properties' focuses on exploring specific properties of mathematical functions. An example function 'f' is defined from the set of real numbers (R) to itself, using the correspondence rule 2x + 6. The chapter delves into understanding why this formula constitutes a function and explores its characteristics through detailed explanations and examples.
• 02:30 - 03:00: Zeros of functions This chapter discusses the concept of functions, especially focusing on the zeros of functions. The text emphasizes that for every x in the domain of a function, there is a corresponding and unique element in the range, thereby fulfilling the conditions of a function. It mentions that for each x, only one result is obtained, which lies in the realm of real numbers. An example provided is f of 7 equals 20, illustrating how each input relates to a unique output.
• 03:00 - 03:30: Image of a function This chapter introduces the concept of functions in mathematics, focusing on the relationship between input values (preimages) and output values (images). It explains the terminology using examples, stating that if an input, like the number 7, results in an output, like 20, then 7 is called the preimage of 20, and 20 is the image of 7. The concept is further illustrated using the function f from R to R defined by f(x) = x^2, emphasizing that for every input (x), there is a single output value, reinforcing the fundamental property of functions.
• 03:30 - 04:00: Graph interpretation The chapter titled 'Graph Interpretation' explains the concept of functions in the context of real numbers. It illustrates how an element from the real numbers (R) can be substituted into a function to yield an image also in R. The example given shows that both 2 and -2 can be substituted into a function to get an image of 4, demonstrating the idea of preimages and images in function mapping.
• 04:00 - 04:30: Injective and surjective functions The chapter discusses injective and surjective functions. It provides examples to explain the concept. One example suggests that the formula f(x) = 1/x cannot have the domain as R (all real numbers) because applying f on element zero would be impossible, thus it would not qualify as a function. Instead, the domain must be R without zero, and other domains can also be considered.
• 04:30 - 05:00: Summary and example In this chapter, the concept of a function is discussed, particularly focusing on the domain of the function f. It explains that any set containing the number 0 cannot serve as the domain for the function because it would violate the definition of a function. The chapter suggests considering sets without 0 to form a valid domain for f, and emphasizes that for f to describe a function, its domain must be the real numbers excluding 0. The presence of 0 in the domain prevents the function from satisfying the first condition required to be considered a function.
• 05:00 - 05:30: Injectivity and surjectivity examples In this chapter on injectivity and surjectivity examples, the discussion revolves around the concept of functions, their domains, and images. A key point raised is about substituting numbers into a function, which results in a unique value. However, every element in the domain should be substituted, excluding certain elements like 0 in specific cases. An example given is how the number 2 serves as the preimage of 0.5, with 0.5 being the image of 2. This chapter aims to delve deeper into terminologies and concepts associated with functions, such as preimages and images, zeros of the function, and how they are related when mapping from set A to set B. This involves using function-specific terms and understanding the relationships within mappings.
• 05:30 - 06:00: Comparison of functions f and h In this chapter, the focus is on comparing the functions f and h. The transcript discusses the characteristics and workings of the function f. It identifies its domain as A and the range as B, and emphasizes an interest in finding the zeros of the function, which are the x-values in the domain A that map to 0 in the range B. This set of x-values is denoted as K.

273 - הקדמה - מונחים מתחום הפונקציות Transcription

• 00:00 - 00:30 Chapter 8- Linear maps Introduction- Terms from the functions field Linear algebra course Dr. Aliza Malek Hello, thank you for returning after two somewhat technical chapters, invertible matrices and determinants, we're going back to our favorite vector spaces, and this time in this chapter we'll deal with functions between vector spaces. So let's first recall the terms related to the functions that will also be used by us, let's begin. So here's what we know about functions.
• 00:30 - 01:00 This is correspondence f, between group A and group B, called a function if each element of A is matched with one and only element of group B. And note that there are two conditions here that must be met in order for us to say we have a function. For each element from A, one and only from B. Each element of A matched one and only element from B, to each A- one from B.
• 01:00 - 01:30 Two conditions that must be met for us to say we have a function here. We denote f : A arrow B. The function f takes elements from A and matches them elements from B. Group A we call domain and group B we call range, f we call correspondence rule. Correspondence rule, takes an element in the domain
• 01:30 - 02:00 and matches it with an element in the range. Each function is determined by these three. A change in the correspondence rule or the field or range will change our function. For each a in A we define f of a (small) equals b (small) when b belongs to B. The element a of A is called preimage, and the element b of group B is called image.
• 02:00 - 02:30 Let's see a few examples with the terms we just mentioned. Let's look at f from R to R, which is defined by the correspondence rule- 2x plus 6. f is a function that takes an element from the domain R to the real range R and the correspondence rule is 2x plus 6. This formula describes a function, why?
• 02:30 - 03:00 Because it is defined for every x in the domain, it can be applied to every element in the real numbers, and we get a result that is also a real number. For each x in the domain, only one element of the range is matched. Both conditions of a function are met, each element in R is matched to only one element in the range. For example f of 7 equals 20,
• 03:00 - 03:30 if instead of x we write 7, 2 times 7 plus 6 we get 20. So we say that 7 is a preimage of 20, 7 is a preimage of 20, or 20 is an image of 7. Here is another example. f from R to R defined by f of x is equal to x squared, this also describes a function. Every number that we put here, we will only get one value,
• 03:30 - 04:00 and it's possible to substitute any element of R, and the image will also be in R. Therefore, this is a function from the real numbers to the real numbers. In this case, 2 and minus 2 are both preimages of 4, why? Because if we substitute 2 here we get 4, and if minus 2 we also get 4. Or 4 is the image of both 2 and minus 2.
• 04:00 - 04:30 And here is another example. If we want to say that the formula f of x is equal to 1 divided by x describes a function, we can no longer say that f goes from R to R, because if we take R to be the domain it will be impossible to apply f on element zero, and therefore it will not describe a function. To describe a function we must choose the domain to be R, real numbers, without zero. Of course, it's possible to choose another domain as well,
• 04:30 - 05:00 but any group that contains the 0 cannot be a domain for f, because then it would not be a function. If we take a group without 0 and we can substitute the numbers instead of x, then we will get a function. So f will describe a function when our domain is the real numbers without 0. Of course the 0 prevents the first condition of a function to exist.
• 05:00 - 05:30 Indeed, when we substitute numbers we'll get only one value, but we want to substitute every a in the domain, so the domain must not include the element 0. Here, for example, 2 is the preimage of 0.5, or 0.5 is the image of 2, exactly the same terms. Here are more terms related to functions we will want to use. For example, if 0 is in group B, then the zeros of the function f from A to B are,
• 05:30 - 06:00 that is, notice what is written here, I already know that f is a function, it's written here, A is the domain B is the range, but in the range I have 0. So, I'm interested in the zeros of the function, what are the zeros? The group of zeros will be denoted by K, it's all the x's that are in A, so that when I apply f on them, I will get the 0 that is in B.
• 06:00 - 06:30 Of course this requires that 0 be in group B. Let's look at an example, in polynomials are all the roots of the polynomial, because I want to take f of x which is a polynomial, compare it to 0, that's exactly how to find the roots of f. For instance, in our example f of x is equal to 2x plus 6, it's a polynomial of the first degree. The set of zeros K will be minus 3, this is the only element that holds 2 times minus 3 plus 6 is equal to 0.
• 06:30 - 07:00 If we look at our second example, in the function f of x equals x squared, K equals 0, the only number that will give us 0 is when x equals 0. In the example f of x is equal to 1 divided by x, the set of zeros is empty, why? Because the equation 1 divided by x equals 0 has no solution, I don't have x's that if I place them here I will get 0.
• 07:00 - 07:30 Although that 0 belongs to B where B is all the real numbers. Remember that in this function, 0 was not in the domain, it was not in A, but the range remained all of R. On the other hand, we cannot reach the number 0 here, therefore the set of zeros of this function is empty. Next, if we choose, for example, the function f of x is equal to the sine of x,
• 07:30 - 08:00 we haven't talked about it yet, but if we choose the f of x to be sinx, we'll get that the set of zeros is all the elements of the form pi T, where T is an integer. These are all the elements that when you calculate their sine you get 0. So notice that sometimes in set K, the set of zeros, we have a finite amount of zeros. Sometimes we have infinite zeros,
• 08:00 - 08:30 and sometimes we have no zeros at all, the set changes according to our function. Let's see more terms we need. The image of f from A to B, of the function f, note that from this point on they will all be functions. We denote it by Im(f), which comes from the word Image of f, and it's all the y's in B, so that for them there is an x in A
• 08:30 - 09:00 that if we apply f to it, we'll get the y. The collection of all elements in B that have a preimage by f in the set A, this is the image. For instance, in the example f of x is equal to 2x plus 6 the image is R, meaning all the numbers. Any real number you take, it will be possible to find x for it, so that 2x plus 6 will give the number we want.
• 09:00 - 09:30 In the example f of x is equal to x squared, the image is R plus unity 0, that is, all positive numbers together with 0, or in other words, the non-negative. In the example f of x is equal to 1 divided by x, the image is R without 0. 0 is in the range but we will not reach it,
• 09:30 - 10:00 thus it's not in the image, it's in the range but not in the image of the function. And in the example f of x is equal to sinx, what will the image be? The image will be the entire closed segment minus 1, 1, because any real number on which we apply the sine x function we will get some number between minus 1 and 1, and therefore the image is the closed segment minus 1, 1.
• 10:00 - 10:30 Real functions from R to R we usually illustrate using a graph, so we see how the function behaves. We will use the DESMOS software here to show the graph of the functions we just discussed. The graph is actually a collection of all the ordered points of the form x, f of x, which we describe in a plane and get a graph, let's see our examples. f of x is equal to 2x plus 6, that's the graph, a straight line
• 10:30 - 11:00 that intersects the x-axis and the y-axis at the corresponding points. We won't go into that, it's not related to our topic. And here f of x is equal to 1 divided by x, how the graph of the function look like, meaning the collection of all points x, f of x where x is in the domain, therefore x will not be 0, here. This line and this line, this is what happens in the real part, and this is what happens to the negative x's. Next, f of x is equal to sinx, here's the graph of the function.
• 11:00 - 11:30 I assume you already know these things. And last but not least, f of x is equal to x squared, here is the graph of the function. This is how we describe the real functions whose domain and range is R, or some subset of R into the real numbers. Here are some more terms we want, a function f from A to B is called injective,
• 11:30 - 12:00 when? Let's see what it means that a function is injective. f is called injective if for every x1 different from x2 in A exists f of x1 is different from f of x2, of course in B, because f of x1 and f of x2 are already in B. Another possibility to define it, if f of x1 is equal to f of x2 in B, then x1 is equal to x2 in A.
• 12:00 - 12:30 If we reached the same point we left the same point. Let's see our examples. The function f of x equals 2x plus 6 is injective, why? If f of x1 is equal to f of x2, if f of x1 is equal to x2, let's see that x1 is really equal to x2. Let's substitute, we'll get- two x1 plus 6 that is equal to two x2 plus 6, we'll reduce 6, two x1 equals two x2, divide by 2, we'll get x1 equal to x2.
• 12:30 - 13:00 We got that f of x1 equals f of x2, which implies x1 is equal to x2, thus our function is injective. This is also the case with the function f of x is equal to 1 divided by x, because if I take two elements in the domain which is a real number without 0, and if we take x1 and x2 which are not 0, and f of x1, that is 1 divided by x1, is equal to f of x2, which is 1 divided by x2 exists, what do we get?
• 13:00 - 13:30 that x1 must be equal to x2, just multiply diagonally and that's what you get. But if we look at the function f of x is equal to x squared, it's no longer injective, why? Because I can take two different numbers, 2 squared, minus 2 squared, both give me 4, that is, f of x1 is equal to f of x2, however, x1 and x2 are not equal, one is 2 and the other is minus 2.
• 13:30 - 14:00 Therefore this function is not injective. And also the function f of x is equal to sinx is not injective, because for example, I can take 0 and pi which are two different x's, apply f on them and get the same result. f of x1 is equal to f of x2 but x1 is not equal to x2. Here's another term "surjective function", f from A to B, that is onto.
• 14:00 - 14:30 What does it mean that a function is surjective? Let's see. The function f is called surjective if for every y in B there is an x in A such that f of x equals y. In other words, what is written here? That the image of f is indeed all B. Let's look in our examples, 2x plus 6 is a surjective, why? Because every y you take, no matter which y in R, which is our range,
• 14:30 - 15:00 you'll always have the possibility to find x that will reach it, how do we find it? We will look for 2 x1 plus 6 equal to the y you gave me, we will change sides and get that x1 is equal to y minus 6 divided by 2. It doesn't matter what y you give me, I find the corresponding x, so all of y is in the image. In the function f of x that is equal to x squared, this is no longer a surjective function, why? Because for example minus 4 has no corresponding x.
• 15:00 - 15:30 We've already said that it's possible to reach only non-negative numbers. That is, if I look for x, that f of x is equal to minus 4, I will not find it, thus, as long as my range is R, I cannot say that my function is surjective. Of course, if I change the range to non-negative numbers, then it will be surjective, this of course changes the function. The function f of x is equal to 1 divided by x is also not surjective because as we have already said,
• 15:30 - 16:00 0 is in the range but has no preimage, there is no x on which if we apply the function 1 divided by x we will reach 0, and therefore it is not surjective. But as we said before here too, let's define a new function, call it g from A to A, which is A? A will be the set R, real numbers, without 0. Now if I define g to be g of x equal to 1 divided by x, now g is indeed a surjective function.
• 16:00 - 16:30 But f that goes from A, which is basically R without 0, to R, is not a surjective function. So, f and g are not the same function, one will be surjective and the other will not. Of course the conclusion is that one is injective and surjective depend on the domain and range of the function. Let's look at a summary example, we will create a table, we have criteria for comparison here, and notice which functions we will compare.
• 16:30 - 17:00 The first function f of x will be from R to R, defined by f of x equal to x squared plus 1. And here is the next function, let's call it h of x, it will be from the complex numbers to the complex numbers. h of x will be equal to x squared plus 1. These are different functions, let's see what is similar and what is different between them. So the first thing we check is domain and range.
• 17:00 - 17:30 Here both the domain and the range are R, the real numbers, while here both the domain and the range are C. You can apply the function on all R, and this function on all the complex numbers. The correspondence rule, what is the correspondence rule? Here we call it f and it will be x squared plus 1, and here we call it h and it's the same rule, x squared plus 1.
• 17:30 - 18:00 But these are different functions, different domain and range, these are different functions, even though it's the same correspondence rule. Next, let's see what the set of zeros is, in the case of f, an empty set, I don't have a solution to the equation x squared plus 1 equals 0. Therefore, the set of zeros is empty, even though R has 0, not one x reaches there, so the set of zeros is empty. What is going on here? Oh, here there are i and minus i.
• 18:00 - 18:30 A set where we have the 0. What are the elements in C, in the domain that reach the 0 of C in the range? It's I and minus I. Next image, what's the image of f? All x's in R, so that x is greater than or equal to 1. For example, if we want to reach 0.5, even though it's in the range, we won't succeed because x squared plus 1 is equal to 0.5 means that x squared is equal to minus 0.5. Well, we don't have real x's like that, and so, in the image there are only the x's
• 18:30 - 19:00 that are greater than or equal to 1, that's what in the image. In contrast, what's the image here? Oh, all C, every complex number you write here, move 1 to the other side, you'll get a new complex number, take a square root, and we already know that you can take a square root from any complex number, we'll get the result, so function h is,
• 19:00 - 19:30 its image is all of C. Let's see if they are injective. f of 1 is equal to f of minus 1 is equal to 2, therefore it is not injective, I have different x's whose image is the same. Of course, the same example will also be used here, h of 1 will be the same as h of minus 1, so here we get 2 as well, therefore h is also not injective. What about surjective? Well surjective,
• 19:30 - 20:00 of course f is not surjective because we've just seen that the image is not the entire range R, it's only a part of it. Therefore, it's not surjective but, h is surjective because the image is all of C, therefore h is surjective while f is not surjective. That's it guys, we're done going over all the terms we'd like to use, and if you're wondering what that spaceship was doing behind me, in the next lesson we'll reveal the mystery, thank you very much.