 Hi, I'm Zor. Welcome to Unizor Education. I would like to talk about linear functions today. It seems to be like a relatively simple topic, however. There are many nuances which you probably should know, and they're quite interesting actually, even from a historical perspective. Well, what can be simpler than linear functions, something like this, right? Well, obviously. It is a simple function. However, I would like to point out certain peculiarities and properties of these functions, graphical representation, etc. So let me start from certain definitions. Well, there is a certain duality in the concept of what linear function actually is. There is an abstract, more, I would say, purely mathematical aspect of linearity. It means basically the following. We can consider function to be linear in this abstract sense. Let's say you have a function f of x. Now, let's assume x is just a number, let's say a real number. So the concept of linearity in its abstract sense says the following. That the function is supposed to be considered linear if the following is true. So these are just numbers. So u and v are numbers where the function is defined. Now, if you will multiply u by some factor and v by some factor and combine them together in this expression, then the function, if it behaves this way, then it can be called a linear function. Now, for example, obviously, e is something like this. Well, indeed, if you will combine 2 of a u plus b v, obviously it's equal to 2 a u plus 2 a b v, which is equal to a 2 u plus b 2 v, right? So this becomes f of u and this becomes f of v. So this property of linearity, abstract linearity, is preserved in this particular function. Now, how about a different function which we also used to call linear, at least in high school, something like this. Is this function linear in this abstract sense? Well, let's just calculate. 2 times a u plus v v plus 1, right? Now, that's on the left. On the right, you have 2 times the function of u, which is, the function of u is 2 x plus 1 plus b, sorry, it's not x, it's u, plus b 2 v plus 1. Now, are these two equal to each other? Well, let's consider this is, let's open the parenthesis, b v plus 1. Here, if we will open the parenthesis, it will be 2 a u plus a plus 2 v v plus b. Obviously, it's not exactly the same as this one. So this function in this abstract sense is not linear. What I would like to say right now is the following. Usually, this type of function is considered to be linear in high school and in some other places. And I will talk about why. The abstract linearity concept. Well, you just probably should remember it if you want to, but usually we will not be talking about linearity in this more strict sense. We will be talking about linearity in this sense, which basically means that this is some kind of a polynomial and the x, the highest degree of the polynomial is 1. And then quadratic functions will be those polynomial functions where the highest degree of the argument will be 2. So, to get about abstract linearity quality, and let's concentrate on this step. So now I can define a linear function in this, I would say more elementary if you wish sense. I don't know how else to qualify this particular thing. But basically what I would like to say is that the function of type A x plus B where A is not equal to 0. This is a mandatory requirement. A is not equal to 0. It's also called linear function. Now, a little later I will talk about why A should not be equal to 0. But basically understand that the function is not really the same if A is equal to 0, which means it doesn't really depend on argument, it's constant B. So, it's still a function from the strict definition of the function, but it's not an interesting function to research or to analyze. It doesn't have many properties with which this function where A is not equal to 0 does actually have. And we will talk about these properties. Now, if B is equal to 0, by the way, let me just for one second return back to abstract linearity concept. If B is equal to 0, then function A is equal to A x is linear in this abstract sense. Well, indeed, if Y is equal to A x, then A times A u plus B v is equal to, let's open the parenthesis, it will be A u A A u plus A B v, which is the same thing as A A u plus B A b. So, these are exactly, if you remember, f of the function f of u and f of v. So, basically, we can open these parenthesis and put it outside of the function, and that actually means that the function Y equals A times x is linear in this abstract definition. However, again, let's forget about abstract. I always promise to forget about this for a while and then I'll return to it, right? All right, so anyway, this is the functions which we will be considering. A can be anything which is not equal to 0, B is anything. Now, anything means what? Well, it can be an integer, it can be an irrational, it can be an irrational or whatever. Now, traditionally, and that's exactly what we will be considering. I will consider A, x and B to be real numbers, real, not integer, not rational, real. It can be complex, I mean, in theory there is nothing to prohibit us from this, but this is not what we will be dealing with. We will be dealing with real functions. So, real linear function or just plain linear function as I probably would, you know, skip the word real means that argument is any real number and coefficients, A and B are any real numbers except A is not equal to 0. That's so much about definition. Now, so domain of this particular function with any real A and B, A not equal to 0, obviously domain can be any real, all real numbers because all real numbers can be multiplied by some real number and added another real number which means the function is defined on all real numbers. Now, that's the definition of the function. That's the domain where it is defined. How about the range? How about the codomain? Well, obviously the codomain in general is also all real numbers, but are we reaching every real number using this function? So, what we do is for any real number we can have an argument which would be converted into this real number with this function. Well, let's just think about this way. Let's take any real number R and let's assume that X is equal to R minus B divided by A. By the way, this is one of the reasons why A should not be equal to 0 because otherwise I cannot really construct this. Now, let's take this particular X and substitute it into the function. So, what do we see? Y is equal to A times this X which is R minus B divided by A plus B. What is it equal to? Well, obviously A and A are reduced so it will be R minus B plus B which is B and B reduced which is R. So, if X is equal to this where R is any number, any real number and A and B, they define our function then the function will take the value R which means we can reach using this function any real number. So, if these are real numbers and these are real numbers first the full function can be defined for any one of those. That's obvious, right? What also I'm saying is that for any real number R I can find the prototype. It's this one. So, any real number can be reached which means that the range of this function all the values of this function they cover entire space of real numbers. So, the main is entire space of real numbers and the co-domain is entire space of real numbers and the range of the function is entire space of real numbers which means every number can be reached using this function from some kind of a prototype. So, that's important. Now, let's talk about a little bit more interesting property of this which is very much related. What we are saying right now is that if this and this are sets of all real numbers then the function AX plus B with A not equal to zero now this function is defined for every number here and its values cover an entire space of real numbers Question is, is it a one-to-one correspondence? Can we establish one-to-one correspondence between this set of real numbers and this same set of real numbers using this particular function? I mean, we know that from here you can get there and from any there we can get some prototype argument here but is it one-to-one correspondence? Well, since we can always define these we can say that any number from here corresponds to some number here. All we have to do right now to establish one-to-one correspondence is to prove that this is impossible. We cannot take two different numbers from the set of real numbers using this function arrive to the same value. If I will prove that we cannot arrive to the same value it means all of these guys are different. Whenever these are different these are different and since I already know that they cover an entire space and for every one of them there is a prototype that would prove that this is a one-to-one correspondence. So all I have to do is I have to prove that I cannot take two different real numbers apply my linear function and get the same value. Well, let's prove it. Let's assume there is a pair of real numbers u and v which substituting them into this function where a is not equal to zero obviously would result in the same number. a u plus b equals to a v plus b. Is it possible if u and v are different? Well, let's just think about it. Now, these are equals, right? So from equal if we subtract equal we will get equal. Remember this, right? So let's subtract minus b from both sides. What I will have is a u is equal to a v. Now, if equal values divide by the same not equal to zero value we will get equal. So let's divide by a and I will get u is equal to v. But I have assumed that these are different. Now I've got it equal. So it's a contradiction which means we cannot get into the same value from two different arguments. Which means we have established a one-to-one correspondence between sets of real numbers and itself, the same set of real numbers using this function. For every real number argument there is one and only one value. For every value there is one and only one argument which can be used as an argument to this function to get that value. This is a one-to-one correspondence. Okay, that's fine. Next, have this plan on the left. What's left? One-to-one correspondence. Ah, monotonic. Another property of linear function is that it is monotonic. I'm sure you understand what it means. Basically what it means let me just repeat it. It's always increasing or always decreasing, basically. More precisely. First, case a greater than zero. Then, if u less than v, then a u plus b less than a b plus b. So, the value of the function would be greater if the value of argument is greater. Well, how can we prove it? Well, it's basically very easy. Let's think about it. If u is less than v and then you multiply a smaller number by some positive number, then the result will be smaller, which means a u will be less than a b. Now, if you have two numbers, one is less than another, and both are, two both are adding the same number b, well, obviously the result would also be smaller. So, starting from the smaller argument than another, we multiply by a positive real number a, the result also would be smaller, and then we add some number, it doesn't really matter by the way, positive or negative. If one number is smaller than another, we add the same value to both of them, the result will go so smaller than this one. Now, what does it mean? It means that if a is greater than zero, function is monotonically increasing its value. As the argument is increasing, the function is increasing. The argument is increasing, the function is increasing. Now, very, very similar. Let's consider a is less than zero. Well, from this, we multiply both sides of this inequality by a negative number, and that actually changes the sign of inequality. If you have a smaller number and a bigger number, but then you multiply, let's say, by minus one, then this number becomes bigger and this number becomes smaller. They are positioned on a different size of the axis, of the numeric axis. Then again, you add the same number to both sides. From this, next is a u is greater than a v, and then a u plus b would be greater than a v plus b. What's important here is that if a is negative, then the inequality is flipped whenever we multiply it by the same number a. Now, in this case, function is also monotonic, but it's not monotonically increasing. In this case, it's monotonically decreasing, which means the greater argument corresponds to a smaller function. So, arguments are increasing, but the function is decreasing. The only thing which I actually wanted to talk about is why we so adamantly insist on a is not equal to zero. And I gave you already a couple of examples of the properties which are important for linear function. For instance, the linear function establishes one correspondence between the real numbers and itself. Monotonic property, like when argument is increasing, the function is either increasing or decreasing. Now, there is one more, again, more abstract consideration. Mathematicians are introducing sometimes new concepts, new types of functions, etc. So, one type of a function is a polynomial. A polynomial function is the function which is expressed as a polynomial. y is equal to a0 x to the n degree, a1 x to the n minus 1, etc., plus a n minus 1 x to the first degree, plus a n. Now, a0, a1, etc., a n are coefficients. This x is an argument. Now, what is this polynomial degree? Well, it's n, right, in this particular case. But to really be the polynomial function of the nth degree means that a0 should not be equal to zero, right? Because if it's equal to zero, then it would be probably n minus first degree, etc. So, if we want to really qualify a polynomial function as the function of certain degree, then the maximum degree x is raised into should have a nonzero coefficient. Otherwise, we can't really say that this is nth degree polynomial function. Well, from this perspective, linear functions are polynomial functions of the first degree. These coefficients are equal to zero, but this one should not be equal to zero. Now, what is the function if a is equal to zero? Well, it's just the constant. From this perspective, all these are zeros, and only a n actually is left, which is x to the zero's degree, if you wish, right? So, whenever you have a function which is equal to constant always for any argument, we can actually talk about this function as a polynomial function of the zero's degree, not the first zero's degree. So, that's another difference. So, from the polynomial function's specification, our linear functions should not have the coefficient a equal to zero, because then it would be just a different type of polynomial function. This is just yet another consideration. The consideration before, like, for instance, that we have established this one-to-one correspondence, are probably more important. And then, obviously, whenever you're talking about certain equations which we have solved this particular equation, we chose the number r as a result, a real number we would like to get if we will have to find whatever the argument is. Now, I basically told you that x supposed to be equal to r minus b divided by a, right? How did I get? Well, I've solved this equation, obviously. So, whenever we will be talking about equations, obviously, again, this coefficient should not be equal to zero. So, there are all these considerations and that's why let's just assume that whenever we're talking about linear functions, a is not equal to, in this expression for linear function, a is not equal to zero. Well, that's it for today. This lecture is just some kind of an introduction to linear functions. I will be talking about graphs and some other properties, but that will be in the next lecture. Thank you.