 Welcome back to our lecture series, Math 42-20, abstract algebra one for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In this lecture 35, I want to talk about the so-called classical matrix groups. Sometimes these are called the classical linear groups, or sometimes the classical league groups, or just sometimes the classical groups in general. And so what do we mean by those? So some of these matrix groups, classical matrix groups, we've seen already. For the main one, of course, is the so-called general linear group. The two classical versions, of course, are the real general linear group and the complex general linear group. We haven't talked so much about the complex matrices as much, but they kind of work by analog, the way we've talked about with real ones. So what is GLN of R mean again? So the notation here is to suggest that GL is short for general linear group. The idea of a linear group is that this is a group of matrices because matrices act as linear transformations on vector spaces. We'll talk a little bit more about that a little bit later. So sometimes people use the word linear group as a synonym for a matrix group of some kind. The adjective general linear group is to suggest this is the largest of all of the linear groups up to some dimensions of a matrix. There are other linear groups like the special linear group we'll see in a moment and the other classical groups amongst others that, again, we'll mention in this lecture. So the general linear group is all of them. We abbreviate it as GL. The little subscript in right here is to suggest the size of the matrix. We're looking at in-by-in matrices when we have that subscript of in right there. The letter right here suggests the type of scalars, the entries that appear inside the matrix. So the difference between the general linear group R and the general linear group C is that GLN of R has real scalars inside the matrix. GLN of C has complex scalars inside the matrix. So this shows you that there is some ability to change the scalars in play right here, real numbers versus complex numbers. What else could we do? We could talk about GLN of Q. We could take matrices with rational entries. Some people even talk about GLN of Z, right? What if we just take integer scalars there? We could also talk about GLN of, say, the quaternions, right? The Hamiltonian numbers. That is numbers that look like you have, let's say, a plus bi plus cj plus dk where a, b, c, d are just real numbers. So you get these Hamiltonian numbers which are these four-dimensional numbers whose units 1, i, j, and k have their interpretation from the quaternion group we've seen previously. So this is something we could talk about in a future lecture. We could interchange this set of scalars with any so-called ring. What is a ring? A ring which we will see later on in this lecture series. We'll talk a little bit more about it in Chapter 16. A ring is a set of numbers for which there are two binary operations which we always call addition and multiplication. So we add, we multiply, and we do expect some things about these numbers. So with respect to addition, we have an abelian group. It's added, addition is associative, commutative. There's an identity which we call zero. There are inverses which we call negative. Multiplication, we don't require as much. Basically, we're going to require, again, this kind of depends on different settings. But for the sake of this conversation, we'll say that multiplication has to be associative. It'll have an identity which we typically call the number one. We don't necessarily require inverses. So we don't always have quotients of things. Like for example, when it comes to the integer z, you don't have multiplicative inverses of like the number two. One half is not an integer. So we don't require inverses necessarily. And also with ring, we don't typically require the multiplication be commutative. Although you can, it's called a commutative ring. But we also do require that between addition and multiplication. There's some type of distributive law. So it turns out you could interchange the scalars of a matrix to any type of ring. And you get a sort of interesting structure. And we could talk about, and so then the general linear group would be the set of non-single, that is the set of invertible matrices with that scalar in there. So when it comes to gl, like let's say gl2 of z, for example, this would be two by two integer matrices with inverses. You have to be careful because if you look at something like this, or two, zero, zero, two or something like that, this matrix doesn't have an inverse inside gl2z because the inverse would be one half, zero, zero, one half. So this matrix would not belong to gl2z, but it would belong to like gl2q for which these are rational numbers. So the existence of these numbers depends on whether it's inverse also belongs to that family or not. So in the conversation of the general linear group, a particular importance is when we put in for our scalars here something called a field. A field is a ring, again addition subtract, addition subtraction multiplication division. Addition multiplication are both associative, commutative. They have identities, which we call zero and one. They both have inverses with the exception of zero. We don't require division by zero. So a field is a number system that kind of acts like the real numbers. We can add, we can add, subtract, multiply, divide by the usual rules of associativity, commutativity, distributive laws, things like that. We'll talk about more about fields later on in this series when we get to chapter 16, as an example. But to really capture the ideas of linear algebra, you just have to replace this scalar with any field. R and C, of course, are the classical fields. These, the R and C, the grilles and complex numbers, they're classical in the sense we've studied them for centuries. But they also have applications to, like say, physics and chemistry, the physical sciences. Because in addition to the algebraic structure, which is a field, these numbers systems, R and C, also have important geometric properties. And so when we talk about the classical groups, we do restrict our general linear group to GLN of R and GLN of C. Those are the classical ones, but we could modify the scalars to be other fields and rings. And in particular, when we study the simple groups, simple finite groups, we've seen a few of these already in our lecture series. So we've seen, for example, that cyclic groups of prime order are finite simple groups. We've seen that the alternating group, as long as the degree is five or bigger, is a finite simple group. And it turns out with the exception of, like, 27 so-called sporadic groups, every other finite simple group is a so-called group of lead type. So I want to give a little bit of explanation what that means, for example, for a moment. And in group theory, there's an important family of groups, which we call lead groups. Which lead groups, it's named after Sophus Lee. I think that's how you spell his name. I hope so. Anyways, we're named after Lee, of course. But a lead group is a combination of, basically, you have algebra. You have algebraic structure, since it's a group. But you have also a calculus structure in play here. Or, without going to all the details, we'll just call this, there's a geometry. For lead groups, this should be a differential geometry. But again, I won't go into all the details of that. So when you look at your operations, you have your group, your group of multiplication. What we require for a lead group is that multiplication turns out to be a differentiable function. So what does it mean for it to be differentiable? It has something to do with the geometry. We also require that the inversion map be differentiable. And so these lead groups have been extremely important. For the most part, these lead groups, of course, are going to be infinite groups. The geometry of the calculus kind of requires that it be an infinite group. And so we want a group structure that's compatible with the geometry in play. And that's what makes these two groups, in particular, so important. These are our classical lead groups. These are sort of like the prototypes, I should say, the archetype of all of the classical lead groups out there. And their applications to the physical sciences, I cannot even begin to emphasize how important they are as group structures. But to give you a little bit more idea what I mean by this lead group structure, if we take, for example, the set R to the 2 by 2. So by this, I mean 2 by 2 real matrices. And I don't require they be non-singular. This is an additive group right here. If we think of it as just as an additive group, this is identical. This is isomorphic with the group of column vectors R4 with respect to addition. Now, this right here is something you probably have seen or something similar to it, like in linear algebra or multibabric calculus. Again, multivariable calculus usually focuses on R3, but you could do four-dimensional space and talk about the geometry there. In terms of addition, these two groups are identical with each other, okay? But unlike R4, R2 by 2, when you think of it as matrices and not as column vectors, in addition to addition, you also have multiplication. This is getting back to that ring structure I was mentioning before. R2 by 2 has something more. You have addition and multiplication. And so if we focus just on the invertible matrices, let me take a step back. So in these settings, we don't just have algebra, we also have geometry. Addition is differentiable, multiplication is differentiable. You've done things like this in calculus, particularly multivariable calculus. You spend a lot of time talking about derivatives of vector functions and the like, right? We can take derivatives of these vector, or in this case, matrix addition and multiplication. So if we focus on those matrices, which have this differentiable multiplication function, what if we focus on those ones which are invertible, that then gets us back to this idea of a Lie group? Well, the simple finite groups, like I said, with the exception of cyclic groups, alternating groups, and the sporadic groups, every other simple finite group is so-called of Lie type, which basically means it looks, it's a finite group analog of these differentiable groups, these so-called Lie groups. That is, in some regard, the finite simple groups behave kind of like subgroups of the general linear group for the finite analogs. For example, we could put a finite field into here. And again, that's a little bit of a stretch because you have like the projective linear, the projective special linear groups and such. But like I said, in some regard, the finite simple groups of Lie type are kind of like linear groups in some regard. But again, I don't wanna say too much more about that right now. Just to emphasize here, the general linear groups are pretty important. Now, when it comes to the general linear group, a very important homomorphism is the determinant map, right? If we take the general linear group GLN, let's just say R for the sake of example here, and this sends us to the non-zero scalars, right? The determinant does this because we can prove that a matrix is non-singular if and only if its determinant is non-zero. This gives us a homomorphism because the determinant of A times B is equal to the determinant of A times the determinant of B. Since this is a homomorphism, its kernel will be a normal subgroup of the general linear group. Now, that normal subgroup is called the special linear group. So we call that SLN of R. So it's important to realize here that SLN of R is going to be a normal subgroup of GLN of R. So the special linear group is normal inside of the general linear group. And that's also true for the special linear group with complex coefficients. It'll be a normal subgroup of the general linear group with complex coefficients. And so whenever you see the adjective special put in front of one of these linear groups, that typically will mean that we're saying the determinant is equal to 1 because after all the kernel of the previous map, the kernel can be those matrices whose determinant is 1. So that's what special typically means in this situation. And again, you could modify the real numbers with basically your favorite field or your favorite ring and you could then talk about the special linear group over that ring, over that field for which you would take the set of all matrices, non-singular matrices whose determinant is 1 where 1 is the multiplicative unit of set ring. Now in the classical sense, we only talk about the real numbers and the complex numbers and maybe in some situations we can talk about quaternions, but I don't want to say too much more about that right now. So these, so the general linear group and the special linear group, these are two sort of like the two main of the two main families of the classical classical matrix groups. In subsequent videos, we'll talk about the orthogonal group which is an important matrix subgroup of the general linear group over real numbers. We'll talk about the special orthogonal group. We'll also talk about the Hermitian group, excuse me, the unitary group, which is an important subgroup of the general linear group with complex coefficients. We can talk about the special unitary group and we'll see those in the other two videos for this lecture. The other classical matrix group, which we're not going to say much about is called the symplectic group. The symplectic group's a little bit more complicated and again that kind of goes in the directions of quaternions a little bit. And so although it's an interesting group, because it's a little more complicated for the sake of time, we won't talk about that classic classical group, but we'll see the orthogonal and unitary groups in the next two videos.