 Welcome back to our lecture series linear algebra done openly. As usual, I'll be your professor today, Dr. Andrew Missildine. This is the first video for section 1.2. And I just wanted to recap what we saw in the previous section, 1.1, about systems of linear equations. We really just recapped topics that we've seen before from other algebra classes as are related to these linear systems and solving systems of linear equations and such. What we're going to do now in section 1.2 is kind of take a different path than what you've probably seen before. And in fact, we're going to take a slightly different path than what many linear algebra classes do. So to give you some understanding of what we're planning to do right now, I just want to point out that the way that our course has been structured is that chapter 1, it's going to try to expose us to essentially all of the main players in the topic of linear algebra. That is, when we study something called linear algebra, well, algebra we're quite familiar with, but what does linear mean? Linear kind of means we're describing something to do with a line, whatever. But why are linear equations called linear equations? Well, you know, it's probably because their graphs are lines and things like that. What we want to do in chapter 1 is make the student become more aware of what does linear actually mean. So we're going to expose ourselves to many of the main characters we will meet in this course. And then we're not going to go super in depth on all of these topics, not in chapter 1. Chapter 1 is all of focusing on exposure. And later chapters of our course will get deeper and deeper and deeper because you'll actually notice from the previous lecture 1.1, although we learned about linear systems, we never actually talked about solving them. We talked about what a solution was. We talked about the terminology and such. And we're going to be playing around with just our prior knowledge of solving systems of linear equations like substitution elimination, but be aware that that prior knowledge is intentionally left with you somewhat limited. We'll add to it later on. Now the topic of this section, 1.2 called fields, is actually, we're going to talk about what is a number? Right, what does it even mean to be a number? Like what is a two? What is a pi? What is a square root of seven? This discussion about what is a number is kind of like a mathematical philosophy in some regard. And this discussion is often postponed to more advanced mathematics courses where abstraction is actually king. But for linear algebra students, abstraction is a very important part of understanding linear algebra. But for many of you, your ability to think abstractly is just a budding flower. It hasn't yet bloomed and that's okay. This course, linear algebra, will primarily be a computational introduction, but we will have some select proofs and applications we'll see along the way. In spite of this goal, the abstract notion of a field will not hinder us. In fact, the introduction of various fields I think will strengthen your understanding of the concepts and computations of linear algebra that we're gonna see. And in fact, what you will see, I should say is actually, what we're gonna see right now is really necessary to deepen understanding. And by allowing the field in play to be variable, it'll help us understand how to do computations with real numbers better. This will also strengthen the realization that when the discussion of complex vectors comes into play or real vectors place come to play that these aren't two separate topics that they're actually one in the same story. So let's get back to the question, right? What is a number? What does one mean by a number? Well, although numbers are useful for counting, they are much more useful than that, right? A simple answer to this question would be to say that a number is simply what, a number is what a mathematician plays with, right? A number as well be studied in mathematics. This is like saying, what is a living creature? Well, it's what biologists are interested in, right? What's a rock? Well, it's what a geologist studies, right? It's kind of circular and crude in that regard, but it does kind of help us think about it more abstractly in that regard. A number is simply just something we're gonna do mathematics on. That's all the number is. And so when we talk about real numbers or complex numbers, adding, subtracting, dividing complex numbers is really no different than adding vectors, scaling vectors, adding, multiplying matrices. In some essence, a vector is a number. It's a vector quantity as opposed to a scalar quantity. A matrix is a number. It's just a two-dimensional array as opposed to a one-dimensional array, which a vector is in a zero-dimensional array, which is what a scalar is. And so we can go deeper, deeper into this philosophy some other day if we want to, but a number is really just something we're gonna do mathematics on. We're gonna do algebra on numbers. And so we'll see lots of different types of numbers in this course. So a little bit of formal definitions I do wanna say in this video here. Let's talk about the idea of a set. A set to us will mean a collection of objects. So a set is a collection of some type of object, a collection of objects, and these objects are gonna be typically called the elements of the set. So these are called the elements of the set. And the term element here is a chemistry parable. The elements are these things in terms of molecules that don't break down any smaller, right? So the elements of a set are what makes the set a set. Now, those elements could be numbers. They could be colors. They could be Pokemon. They could be people. They could be other sets. They could be whatever they want. Now, typically sets will be denoted by a capital letter such as capital A, capital B, capital C. And this is to try to help us distinguish sets from other types of numbers. The elements, so the sets are typically denoted with capital letters. The elements themselves are gonna be typically denoted by lowercase element letters like little A, little B, little C. We might use other alphabets like the Greek alphabet if necessary. The sets will often denote, typically, we will start a set with some type of curly brace of some kind like so, and then we'll close it with some curly brace. And then we put things in between the two braces to indicate that we have some type of set in play here. So for example, we could say the set A is the collection of the numbers one, two, three, four. And so if one has a finite set, we might just list all of the elements of the set, separating individual elements with commas, and it will start and end with these curly braces to describe we have a set. Now, sometimes the sets, the membership of the set can be much more complicated, right? And so we use a so-called set builder notation. We might say A, the set is the collection of all elements X. So X is like this generic element inside of the set. Maybe it belongs to another set. It's in some type of universe. And then we say like, okay, we want all elements X such that X satisfies some rule. X satisfies some rule. And this will be more clear later on here. And so we have this like set builder notation that we're gonna build a set based upon some type of condition, some type of criteria, some rule that will decide who's in the set and who's not in the set. And so membership of a set is gonna be king here. So we'll say things like X, you'll see this little symbol looks like an E, A here. And what this means is that X is in A. And so this little symbol right here will just be a shorthand for membership here. So, oh, X is an element of the set A. In converse, if you ever see a mathematical symbol with a slash through it, that actually means the negation of that symbol. So if you see X membership slash A, you would actually read that as X is not in the set A. And so we wanna mention this idea of sets because we're gonna be talking about sets all the time. So the fields, which we'll define in just a moment, themselves are a type of set. And so we can talk about X belongs to A, X doesn't belong to A. And if A is in fact a set, there should be a clear rule that tells us whether X belongs to A or X doesn't belong to A. There should be no ambiguity there. We should be able to decide definitely whether X belongs to A or X doesn't belong to A. It either belongs or it doesn't. It can't be both, it can't be neither. One of those statements must be, in fact, true if A is a set. Now, when I say that there must be a clear rule, I don't mean that rule is necessarily easy. It could be a three hour computation to decide whether X belongs to A or not. And of course I exaggerate there, but who knows? I mean, the rule could be very, very difficult to decide. It could be a hard decision problem, but to be a set, there has to be a definite answer. Now, another thing I wanna mention is that in our discussion, sometimes the set that we describe might be an empty set. The empty set meaning it's a set that contains no elements. So sometimes this is denoted as a curly brace and then another curly brace. There's no elements between it or we'll often abbreviate it as a circle with a slash through it. We actually saw this symbol in the previous lecture when we talked about inconsistent systems. We said the solution set was empty. That's just one example of this empty set because the rule one uses to describe the set might actually lead to including nothing. Like I could be like, oh, let's take the set Q to be the set of all purple unicorns you see on the screen right now. Well, I don't see any purple unicorns. Well, that would be an empty set, right? It could be that there's a rule, but no one actually gets included in the set because the rule is too restrictive in some way or another. So sets are gonna be a big piece of conversation for us in this course, right? Another bit of vocabulary I wanna mention is the idea of a function. A function, I'll write out the full word here, a function, typically we'll call this F. We usually like to use lower case letters to describe functions, maybe like F, G, H, things that are, you know, things that are close in the alphabet to F. F is a new monochrome function there. A function is a relationship on sets. Say, and we have a set A and we have a set B. And so we have like a one set A that's gonna be pointed to the other set B. And so the function is a relationship on these sets A and B such that each element of A is gonna be assigned exactly to one element of B. And so we would write something like A is gonna be associated to some element in B which is called F of A. Now, A is an, little A is an element that belongs to, I guess I can just write it here. Little A is gonna be an element that belongs to capital A, the set. And then F of A is this element associated to A but it belongs to the set B. And so this is what we mean by a function. There's some rule that assigns to each element in the first set to some element in the second set. Now, a little bit of vocabulary here. This set on the left that we are gonna assign values to, this is called the domain of the function. And this second set B is commonly referred to as the co-domain of the function. Co here is just short for complimentary. It's the compliment of the domain. And so what a function does is it assigns to each, it assigns to each domain element, something, it assigns it a co-domain element. And that's it. Now, be aware that not everything in the co-domain necessarily gets somewhat assigned to it, right? So like if we think abstractly here, we have this set that contains these three objects over here and then we have a set over here. So this was A, this is our B, we might have some objects over here. And so what the function does is an assignment. So it's like, okay, the first object we're gonna assign right here, the second object we're gonna assign right here and the third object will also assign right here. So some things to note here is that for a function could assign two different elements in the domain to the same element in the co-domain that's perfectly kosher for a function. We will talk about functions in the future which are one-to-one which don't have that property but that's a topic for another day. And what's also should be mentioned is that with a function, not everything in the co-domain necessarily has something assigned to it. There might be things that are missed. If we don't like that, we can actually have what's called an onto function or a surjective function which makes sure it hits everything in the co-domain. But again, that's getting ahead of the game right now. The function is just a rule that assigns to each element in the domain something in the co-domain. All right, now we are very, I mean careful with this word co-domain. I'm not using the word range because although that's the word you're probably used to, that word is probably a little ambiguous to what you think that means and we'll be more precise about on another date. Well, we won't worry about that right now because I wanna introduce this vocabulary so we can actually define the following term which will help us prepare for our discussion of fields in the next video here. And so I wanna talk about what we mean by an operation. So an operation is a type of function. It's a function which actually take two input. So we have A cross B. So we have two numbers, two sets in the domain and this will map to a third set. And so we often think of it, we have this ordered pair A comma B and this will then assign to A B some third element F of A comma B. Now an example of an operation that you're probably familiar with is the operation of say addition. Addition could take something like a real number. You take two real numbers and then it produces a real number. And so this addition, what this means is you take the ordered pair A comma B, I will take specific numbers like two and three here and then it maps that to the sum two plus three which we define that to be five. And so an operation is just a function which will take two input and it'll produce a third output from these sets. These sets of course do not have to be the same sets but oftentimes as we think about these operations they will be in fact the same set over and over and over again. So this will help preface the definition of fields which we'll talk about in the next video.