 So, let us begin with a quick recap of what we have seen so far, we have tried to solve equations such as these and we saw that you can always take this A to a row reduced echelon form like so right and then we saw that if you want to solve for Ax is equal to 0 under certain situations precisely when you have a greater number of variables and the number of equations there will always be a solution to this a non-zero non-trivial solution to this, but how does that help us in solving the overall thing? Now of course, once you have gotten it to this row reduced echelon form right what you need to add on to the solution of this like this complementary part is essentially just this B hat right so for instance if you have written your x to be something like u1 plus something like let us say c1 c2 like so. So, there will be one part let us say some ck there will be one part that is precisely this part which will satisfy Rx is equal to 0 and therefore Ax is equal to 0 to that if you also add B hat that will be your solution right. So, this is precisely not the solution of this, but also the solution of this yeah so that part is hopefully clear so although I did not explicitly mention that you have to also solve for Ax is equal to B, but from the way we solved and worked it out this solution of course will of course only exist if the rank condition checks out right we showed you that that if you have this R and you check the row rank of this and if you check the row rank of R with B hat only the ranks of these two fellows are the same only then this checks out and you can get a solution we have also seen that right. So, otherwise it does not make sense so then the question still remains as to what do you do and no solution exists we will not be in a position to answer that question completely till we have addressed topics such as inner product spaces and for those we will deal with very specific type of number systems such as real and complex numbers and talking about number systems it is precisely this that led us to our next topic which was we started studying this group this object that we called a group right it is essentially nothing but a sort of a number system accounting system you are defining certain operation with certain properties why do we do that because we want to push our luck and see whatever we did in obtaining the row reduced echelon form can we extend it to other number systems as well right. So, first thing we understood was groups and then we also saw that with the additional property of commutativity you also get abelian groups right. Now we are going to look at something more something beyond just groups if you notice the way I wrote that solution over there it was something like summation c i v i at least a part of the solution was of this form. So, there are basically two operations that happened one which was addition of certain n tuples of numbers the other was scaling of those n tuples of numbers by some scalars, but so far when we describe groups we have only spoken about one kind of operation right. So, it means that if you want to get a fuller understanding of what is going on in an abstract sense we need to talk about not just a set with one operation one binary operation, but with two binary operations right. So, in addition to what we have defined so far with groups where we have a set and presumably an operation where of course this plus stands for any operation that you want any binary operation, but let us call it addition for the time being. So, this is quote unquote addition it can be anything that I wanted to be provided it satisfies those rules and let us also talk about a second operation which we will call as the multiplication again within quotes because this is not by any means to be read as only the usual multiplication that you are familiar with ok. So, we need a set with one binary operation that is addition and another binary operation that is called a multiplication to satisfy certain properties and if it does so then we give it a certain name. So, we are going to try and introduce that. So, suppose S with this addition is an abelian group that already tells you a lot about S with respect to this operation and the following hold. So, you can already guess that what we are going to say subsequently has got everything to do with S with respect to the second operation that we are going to now describe. So, what are the properties? First property there exists a unique one right when I say one what I mean is essentially a multiplicative identity yeah in S such that 1 times a is equal to a times 1 is equal to a for all a belonging to S ok. So, what do we have a set with the addition binary operation and the multiplication another binary operation such that there is an identity element with respect to the multiplication operation that I am going to define on that set ok. Of course, needless to say maybe I should just mention it in a different manner over which S is closed yeah. So, I am not even going to like make that a different point that closure property is a given right you take two fellows in the set you act using this binary operation of multiplication what you get should also be a part of that set right. So, that closure is already kind of taken for granted. So, then the first important property is that you have this a multiplicative identity. The second property for a b c belonging to S we have a operation b operation c is equal to a operation b operation c the order will not matter you know what this property is called it is called associativity just to refresh your memory it is called associativity right. Anything else you think is worth adding here to this? Distributivity. Distributivity. So, these are pertaining exclusively to the multiplication operation, but now when you have two operations how do those two operations combined with one another is what is described by the property of distributivity. So, we have as point three for a b c in S a acting on b plus c is equal to a acting on b plus a acting on c and b plus c being acted on by a is equal to b acted on by a plus c acted on by a. So, these two together give you distributivity of multiplication over addition. So, the multiplication operation distributes over the addition anything else we will not go into that with this itself we have the makings of a certain structure. Can you think of an example of something that you are familiar with which already meets this well that is maybe you know going to suppose you consider a set of matrices of size 5 cross 5, 10 cross 10, n cross n where n is a finite integer positive integer. What about matrices do they meet this these properties you have an identity with respect to addition. So, it is an it is an abelian group with respect to addition commutativity commutativity is there. So, with respect to multiplication we are not demanding commutativity in which case it is fine right if you are taking square matrices the products need not commute, but you do not ask for commutativity here what you ask for is this associativity which is true what you ask for is distributivity which is also true left distributivity right distributivity there is no issues with this right. So, already you see the set of matrices satisfies this now this has a name we call it a ring with identity. This is called a ring with identity and you already see that the set of matrices the familiar ones that you are familiar with the square matrices they already meet this condition all of these conditions. So, they qualify as a ring with identity under the usual matrix addition and multiplications right what if I now impose the condition that let us add a condition ok. Multiplication actually I should probably also ok this I have already written the multiplication is also commutative all right then what we get is ok I have not got space here. So, let me write it here a commutative ring with identity any example comes to mind of a commutative ring with identity think of yes set again rational numbers have something more not just this they actually that is an overkill you are right. So, see there is a set with certain rule basis the more the number of rule basis you add you are actually shrinking that set, but I am more interested in getting an example of a set that just about meets this, but nothing extra in addition because if they meet something extra then they probably got a different name. So, I do not want to go super specialized into that. So, integers, but this again something more we will add and then integers will qualify we will sharpen this a little more and then integers will fit in perfectly right now integers fit in, but it is a loose fitting it is like you bought clothes for a child for the next 2, 3 years allowing him or to grow into the clothes right it is still got more room integers have something more not just this think of polynomials right. So, they commute yeah if you did not have the commutativity property would they still be part of this they would because commutativity is. So, something that is part of a subset is obviously, part of the super set. So, if you are given me an example of a polynomial as a ring with identity yeah it is valid sure, but it is got some more property, but now whenever you find it you see the matrices that had described just a while back they do not form a commutative ring the square matrices right under the usual addition and multiplication, but now the polynomials just fit in with this. See we are not asking for inverses yet right if we did then we would have problems with polynomials because the inverse of a polynomial in the way we understand divisions and all would be a rational function not a polynomial. So, it does not belong to the set of polynomials the inverse of a polynomial unless it is just a constant polynomial right. So, therefore, the polynomials give you an example of a commutative ring with identity ok. Now I am going to add another line here commutative ring with identity which satisfies the property that whenever a into b is equal to 0 either a is equal to 0 or b is equal to 0 of course a and b belong to s yeah either a is equal to 0 or b is equal to 0 or both is called anyone has heard of this yes it is called an integral domain. So, I am deliberately making a grammatical mistake by using capital letters in the middle of a sentence, but that is because it is the name. So, it is important now if you talk about integers of course, they are an integral domain because the integral domain gets its name from integers everything that behaves like integers can you think of a place where you have used this from a very young age you have been using this property every now and then. When you solve equations quadratic equations cubic equations you factor out a polynomial and the next step you say is either x is equal to this or x is equal to this or x is equal to this and so on and so forth right. It never occurs to you that it is quite possible that the product of two numbers that are non-zero can turn out to be 0. However, let me give you an example of a situation where that may be the case kind of like a counter example. So, consider the system of numbers 0, 1, 2, 3, 4, 5 and the operations this is s and the operations are plus modulo 6 and product modulo 6 all right. Now look at the product of 3 and 2 is 3 0 modulo 6 is 2 0 modulo 6, but what is 3 into 2 modulo 6 right. So, you can never be sure unless you are dealing in integral domain. So, this fellow clearly is not an integral domain right yeah. So, I have clearly given you an example of some operation some set with two operations where the condition that we often take for granted need not necessarily be true. So, as I said you have to forget all that you have learned and look at only the operation in and of itself right. So, this is not an integral domain. If you are solving for equations in this you will have to be very careful right you cannot always say that you know things that we are so used to believing in. If you have a fifth degree polynomial equation you say there are 5 roots we take those things for granted nothing is for granted you can have more than 5 roots you can have less than 5 roots you can have probably no roots depending on the number system in which you are living you may not have any roots right. So, funny things happen for example, you know this typical thing you have 0 and 1 right sorry I should give it a set notation 0 and 1 and you take addition and multiplication with respect to modulo 2 that is the binary number system basically right. So, if you take x plus y times x plus y in this usually we write this is as x squared plus 2 x y plus y squared, but now if you take modulo 2 this is of multiple of 2 always an even number. So, this vanishes. So, indeed that 2 x y becomes an extra right. So, this becomes equal to x squared plus y squared. So, for people who think in binaries that is an extra right. So, these things have to be kept in mind what is the set and what is the operation under which you are acting the rules of the operation must be respected ok, but we are not done yet. So, this is an integral domain ok this is clear. So, maybe I can erase this part because now we will go deeper with that or rather we will take a sort of a digression into defining another sort of objects which are not necessarily a subspecialization of a commutative ring with identity rather we will now talk about a division ring ok. Now from all of these properties we get rid of this blue property. So, we are back to dealing with a ring with identity. So, what is a division ring? A division ring in addition to satisfying all the properties of a ring with identity it is a ring with identity such that for all a belonging to S sans the 0 element yeah there exists a inverse such that a inverse times a is equal to 1 and a times a inverse is equal to 1. So, every non 0 element yeah if it has a multiplicative inverse in a ring with identity then such a ring is called a division ring. Now unfortunately not all division rings are commutative if you have ever come across things like quaternions how many of you have heard of quaternions ok they lost a bit of popularity, but they are again back in business in you know last century or so ok. So, these quaternions is a classic example of forming a division ring yeah, but they are not a commutative division ring ok because the quaternion multiple. So, every non 0 element every non 0 quaternion has an inverse yeah, but the operation the multiplication with quaternions is not commutative ok. So, you do have ring which is the basic structure then if you add the commutativity property you get a commutative ring with a commutative ring if you impose the other property that is product of two non 0s that is non additive identities can only be 0 if at least one of them is 0 otherwise they can never be 0. So, product of two non 0s is never 0 then it is an integral domain ok. Now, you get rid of the commutativity again forget about the integral domain bit go back to the basic structure of a ring with identity just impose the condition that apart from all of this there must also be a multiplicative inverse. If it is so, if there is a multiplicative inverse for every non 0 entry in the set then it is a division ring. So, these concepts are clear there is I hope that you do not suffer from any confusion between which is a subset which is contained in this division ring is quite a part right it is not contained in anything it does not have to meet commutativity, but if it does meet the property of commutativity. So, I hope this part is clear right. So, at each stage it is important maybe to remember some examples as I said because that will immediately help you recall what property is coming in what property is going out that often helps it is a memory 8 in some sense. So, here is a special name for commutative division ring. So, each of those terms clearly tells you what we need is called a maybe I should write it as a capital letter field ok. The most important object that we will come across is the starting point of everything in formal linear algebra is this field right. A commutative division ring is called a field this is where our scalars will come from this is the scalars on which we will build the structures of more sophisticated objects such as vector spaces which probably will be able to cover today or at least start discussing ok. So, this is a field. So, hereafter we will see that everything that you did while trying to solve for those systems of equations A x is equal to b and got getting to that row reduced echelon form is well and truly extendable provided your number system satisfies the properties of a field all right. You might still be able to do quite a lot if it is a commutative ring many of those things can still be done yeah, but a field makes life easy because it is exactly replicating whatever you did with real numbers in getting from any arbitrary matrix to its row reduced echelon form ok. So, let us see some interesting features or properties of what we are calling this as field right ok. So, this definition is kind of clear I hope because you know normally people would start a description of a field by writing out all properties, but I have already done the hard work of writing those properties I have described what an Abellion group is. So, you know with respect to addition it is an Abellion group. In fact, you can also think of it like if you look at all the elements apart from the 0 element then even multiplication is an Abellion group for a field yeah it is a commutative the multiplication is commutative there is an identity there is associativity and apart from that. So, it is a Abellion group with respect to addition without the 0 it is an Abellion group with respect to multiplication and then there is this distributivity that is the interplay of those two operations which is encapsulated by the distributivity property. So, that is a simple way of remembering instead of like oh there are 10 points I remember 9 what is the 10th it is not history it is mathematics right. So, you do not have to recall things from memory that is the way you reason out. So, this is an example. So, of course, the real numbers naturally are fields of course, the complex numbers are also fields yeah. So, that brings us to this interesting notion of what is a sub field you know what is a subset given a set if you take some of the elements of that set and cook up another set that is a subset. So, does it mean that R is of course, a subset of C that is R is sub field of C in this case it is so alright, but in general can we always say that we have a number system say for example, 0 1 2 yeah let us just stop there with addition modulo 3 and multiplication modulo 3. So, this sorry right this is you can check an example of a field let us call this z 3 let us take another z 2 which have just written a while back that is the binary now of course, this set is a subset of this, but is this a sub field of this what do you think why not it is a sub field how many of you think it is a sub field z 2 is a sub field of z 3 what is your best guess exactly. So, when you are talking about sub fields you cannot just afford to look at only the sets you see you have to also look at whether the operation is inherited from the so called super super field or whatever you call it right. So, this operation and this operation are different. So, even if the sets look like they are subsets of one another that does not make it a sub field the operation must be respected only then it is a sub field. So, just thinking about it in this manner you can already figure that sub field will demand that you have certain elements in it. For example, you leave the 0 and 1 without that you cannot make it a sub field right. So, we will hopefully have by now we have defined what is sub field here and fields at least we have given you an idea what is a sub field. So, that brings kind of this segment to a close, but now next we shall look at some further interesting properties and the relations between these integral domains and fields and what not and we shall study them in some closer detail. We will do a few proofs.