 Let us view AX is equal to B as promised because too long have we dwelt on this AX is equal to 0. Let us see what is the benefit of looking at this or why did we even focus on this. How does this help us in figuring out when AX is equal to B has a solution or not or when it has is the solution unique and so on and so forth those important questions. So we know that a recipe for cooking up a simpler looking system of equations is to get to RX is equal to B hat. Now if you have done again a preliminary course on matrix theory let us say in your undergrad first year courses or wherever you have encountered this sort of thing you might recall that somebody gave you a condition like you know you have already heard of this rank of a matrix probably here we will define more general notions of rank but nonetheless. So you have already heard about this so you might recall that somebody told you to check for the rank of A and the rank of A augmented with B. Do you recall something like this some condition like this and what was the assertion there what exactly was the assertion there so rank of both have to be equal for a solution what is the ranks are not equal what do you have multiple solutions no solutions what what exactly can it be the case can you have rank of A more than the rank of so which one is always greater than or equal to which one can this be greater than this rank of A more than this so this rank is less than this so then what happens multiple solutions is that multiple solutions sure yes no so then how so then does it does it does it really fit in with the case when rank of this is less than rank of this so I am asking the question when this rank happens to be less than this rank what is the assertion multiple solutions no solutions unique solution what is it no solutions right but let us try to demystify this now based on what we understand so instead of looking at this we have already convinced ourselves that it suffices to look at the equivalent condition on the equivalent system of equations yeah so let us try to reformulate this condition in terms of the row reduced echelon form versions are and B hat what does it immediately tell you what is the rank of A or the rank of R they are the same because it is a definition right the rank of A is equal to indeed the number of nonzero rows in the row reduced echelon form and the rank of R is of course it is itself its own row reduced echelon form there is nothing more to do into this right so therefore the ranks of these two the row ranks at least are the same let us just say I mean for the time being we will just carry on with this row because we have not yet proved that row rank and column rank are going to be the same so let us just carry on with this legacy of using the term row rank because we have defined the row rank in that fashion now what does it mean when you say that the rank of this fellow got bloated because of the addition of this column it must mean that when you had this R here you had a bunch of nonzero rows here and then you had a bunch of zero rows here so this was R and when you augmented it by R and B hat if the rank has to bloat then indeed what has to happen you must have at least one more leading one or one more nonzero row but where can that come from that nonzero row cannot come from any of these entries here that are already there so that must come from this R remaining intact as it is but that nonzero row must be perhaps some singleton here the moment you have that what sort of a condition are you trying to meet you are trying to meet something times zero is equal to something that is nonzero an absurd proposition is it not you follow the argument what I am saying is that when you bloated when you augmented it by B hat the only rank of this fellow increases from this is when the number of nonzero rows of this fellow exceeds that of this the only number of nonzero rows of this fellow increases from this is if there is a nonzero B here B hat entry here corresponding to which all the other preceding entries are zero that is the only way to generate a new row in this augmented matrix which happened to be a zero in the preceding matrix but that essentially means that you are asking for a condition where something times all the excess and that something is zero precisely so zero times all the excess is equal to something nonzero you can never meet such a condition. So, therefore, of course no solution unless the ranks are equal it is very transparent once you look at the row reduced echelon form may not have been so when you look at this might have been something mystifying over what is this check and all right. Once more you now notice this is not contingent upon having a square matrix A because everything we have done up until this point this row reduced echelon form we have not gone via the route of taking an inverse or a determinant or anything of the sort we have dealt with the general m cross n matrix. So, m equations and n variables so this idea of checking for whether a solution exists or not through this test holds even for rectangular systems and now you know why because it is explicit here in this particular form right. So, this is an important observation that is immediately apparent from what we have claimed so far right. Next when you have an undetermined system if a solution exists can it ever be unique when I say undetermined systems few or number of equations greater number of variables resulting in an A matrix that is fat what have we just seen about undetermined systems sorry there will be free variables and therefore, the equation A x is equal to 0 will always have a non zero solution right. So, suppose for A in m cross n with m being less than n we have zeta in R n such that A zeta is equal to b ok. Now look for xi not equal to 0 such that A xi is equal to 0 clearly be very doubtful when you say clearly as I always said, but I hope you are not very suspicious of this because we have explained clearly what such a xi exists why we have proved it it is already a fat matrix. So, you will have free variables and therefore, you will have non zero vectors xi of course, in R n such that A xi is equal to 0. Now what happens if you for instance take A of zeta plus alpha times xi what is this going to be equal to A zeta plus alpha times where of course, let us say alpha is just a scalar alpha times A xi what is this this is equal to b because this part is just 0. So, therefore, this is b then what can we say about undetermined systems of equations if a solution exists that solution can never be unique either the solution does not exist because of the inconsistency the possibility that we have seen, but if the solution does exist you can give up all hopes of having a unique solution that is not necessarily bad because often times when you conduct fewer experiments then the number of variables you want to determine for whatever be the reasons may be the cost of the experiment is prohibitive. You want to determine say a million parameters by performing a 100 experiments. So, you will come up with probably 100 equations in a million parameters and you still want to better get a best idea best case idea about what these parameters ought to be like you do not want to conduct a million or more experiments right because they may be very costly. So, you might have a whole plethora of solutions out of them which is your best possible solution when I say best possible there may be different ways of describing what a best possible solution is if it is a if it is an optimal path that you are trying to figure out yeah based on some polynomials you might want to optimize certain things and choose the best possible set of solutions from all these infinite solutions that you have yeah. If it is some error that you want to minimize again some sort of an optimization problem can be cast or posed in that manner. So, it is not a bad thing at all having non unique solutions in case of these fat matrices is not a bad thing it is actually a blessing yeah unlike what we have been talked to think about like in mathematics oh it is a unique solution it exists it is unique that is the best thing no in many applications these things may be more desirable because it gives you more room to play around with and figure out what is best for your particular application maybe what is best from one particular metric may not be the best from another particular metric right. So, you get a whole playing field to tinker around with these right. So, that is an important observation from this that the solution if it exists is definitely not unique. So, next we will try to understand this solution and all this in a bit of a from a bit of a geometric perspective although we will not take it too far because this is a this is slightly you know on slippery slope when you are on this geometric thing and you try to build things on intuition often you run into trouble and that also I will illustrate. But in 2D at least in a plane at least with 2 variables we understand the geometry of when solutions exist when they do not and so on and so forth. So, there are different ways of viewing this one is of course you have A x plus B y is equal to alpha C x plus D y is equal to beta right. So, when you take this A x plus B y is equal to alpha and you want to sketch this for instance let us say it is a straight line like this. So, you take any 2 points on this take any 2 points on this suppose this is x 1 y 1 and this is x 2 y 2 right. So, you have A x 1 plus B y 1 is equal to alpha A x 2 plus B y 2 is equal to alpha and you subtract them and what you end up with is A B the vector times x 1 minus x 2 y 1 minus y 2 is equal to alpha minus alpha which is 0 even without going into things like inner products and all in 2 dimensions you understand what this is right what is this it is a dot product a dot product being 0 means they are orthogonal which means that what is this point x 1 minus x 2 and y 1 minus y 2 is it not the vector that you have drawn like. So, let me use a different color. So, one of those vectors is this x 1 minus x 2 y 1 minus y 2 if I am saying that this A B vector is orthogonal to this vector what must be its direction perpendicular to that line right. So, this is the direction A B maybe not I should write it like this in other words the coefficients are always giving me the normals and what is that right hand side giving me this alpha it is a bias I mean if I play around with this alpha I will be moving this line parallel to this existing line. So, in other words every equation can be viewed as a bunch of numbers which constitute the normal and a bias. So, these are all so called hyper planes right every equation corresponds to a hyper plane and every hyper plane let us give it this name is described by described by a normal and a bias that is it. So, essentially every constraint being imposed on your system of linear equations if you view it in the Euclidean space it is basically every time I am giving you a new set of normal and bias and saying that you have to adhere to this you have to meet this constraint that whatever points you are giving me they must satisfy this normal bias constraint right every time I am giving you a new h i. So, essentially the solution must be in the intersection of all these h i's yeah that is the solution for you that is a geometric interpretation but there is a risk of going too far with this you see in 2D you might think oh hang on I know exactly when the solution does not exist when you have the same normal but different biases in two equations the train shall never meet and therefore, you have no solution right. So, you might think ok in 3D it is the same thing this does not look like a plane we call it a hyper plane but in 3D it actually looks like a plane. So, let us say you have a plane like so and you cook up another plane which is parallel to it and you might think yeah I know I know exactly that is the so that must be it that is the reason why I do not have a solution when I have three equations and three variables must be those two planes they have messed up they have different biases but hang on let us look at a situation like this ok let me not draw the thing let me just try to draw the figure again suppose you have something like this ok again it is like putting three cards together you see three planes no two of them are parallel to each other do you think this is going to have a solution pair wise there is always a line over which you can have a solution but when you put them like that together there is no one single point in this whole space which satisfies all three of them together. So, do not get too carried away with intuitions that you build in lower dimensional spaces I know this goes counter to what I said when we talked about that and problem right but you should also know that that is the reason why we try to do a bit of algebra we do not always want to rely on things you see already in 3d this is messing up and 4d we cannot even visualize right. So, that is why we want an algebraic structure. So, although it is good to have a geometric inside every once in a while let us not be over reliant on that geometry but rather trust what we do algebraically right which is why we need to understand things like dimensions and other things other notions which we use so freely and so loosely speaking in a much more formal structure right. So, therefore, what we are now going to start what we have done up until now is rather you know free flowing maybe you figured that this is just viewing the usual things in new light which may be interesting but it is ok it is kind of like everything that someone understood I mean something that everyone understands. Now, what we are going to try and see is a little more formalism introduced we have already seen how to go about the business of finding solutions using this row reduced echelon form we would like to know the limitations of that method any half decent scientific method should tell you not just its merits but also the limits of its usage we do not do that do not trust it it is proselytizing it is not science. So, when we are doing this row reduced echelon form we should also know where we cannot push it beyond a certain point what are the operations that we carried out when getting to this row reduced echelon form of course, you will say matrix multiplications what do those matrix multiplications imply see we had to divide numbers we had to add numbers we had to multiply numbers but these operations do not necessarily have to be the way we have viewed them since our childhood for instance and this is where algebra comes in when I say root 3 what do you think 1.732 and some nasty decimals thereafter which are non-recurring right. But now let us say I define the operations in a different way suppose I have a set of numbers instead of giving you all the numbers to play around with I give you the following 1 2 so 0 1 2 3 4 5 6 suppose this is the set of numbers that is all you have you have actually done something similar when you dealt with binary arithmetic you just had 0s and 1s but I am giving you something more I am giving you 7 of these objects and I am defining the addition operation in a special manner I am saying that for a b belonging to S I define a plus b this is my addition now as a plus b modulo 7 how many of you have heard of modulo operations ok most of you but still I will then repeat it for those who are not familiar modulo essentially means you add them divide them by the number 7 here in this case take the remainder so of course if you take any two numbers here there is a chance that the number might be more than 7 if you take 1 and 2 it is just 3 you divide 3 by 7 it is 0 times 7 plus 3 so 3 is the remainder hmm but if you are if you take 5 and 6 it is 5 plus 6 that is 11 divided by 7 that is 7 times 1 plus 4 so 4 is the answer because 4 is the remainder right that is how it is defined and by the same token we define the multiplication also in this manner a multiplied by b modulo 7 ok now what do you think that is root 3 mean in this case is it 1.73 does not make sense 1.73 to whatever real number you have in mind is not even part of this number system does it mean root 3 does not exist maybe I do not know I have not checked but if it does exist what is it what should I be looking for if I want to solve for root 3 I should be looking for a number which when multiplied by itself in the sense of this operation that I have now defined not the conventional get everything you have learnt so the way to learn about this is unlearn everything you have learnt up until this point just focus on the operations that I have defined do not have any baggage carried from the past just go for these operations and figure out if any of these numbers multiplied by itself leads to 3 then that is a square root of 3 right. So, let us check 1 squared is 1 2 squared is 4 yeah 3 squared is 9 9 modulo 7 is 2 4 squared is 16 16 modulo 7 is again 2 5 squared is 25 25 modulo 7 is 4 6 squared is 36 that is 1. So, in this number system unfortunately however you might have already noticed if I had set instead to find out square root of 2 did I have anything there what was it right. So, when you see weird things like square root of 2 is equal to 3 be not alarmed because it might be that the operations have been cooked up in a wicked fashion to trick you. So, you have to keep your minds open to these possibilities that is why we do algebra because the trick that you have learnt incidentally of taking inverses and all are not just applicable for the conventional number system where you know real numbers, but also for other different types of number systems which together constitute what we define as fields. So, we need to understand what are fields in order to understand where we can apply those tricks and techniques that we have learnt about row reduced echelon forms and where we cannot. So, let us yes. So, this was what we call we will see later what we call a finite field. So, you can keep checking over this. So, yeah it is not necessarily an easy problem yeah it is not easy I never said it is easy I just told you what the principle should be what you should be actually looking for. So, you should not by default assume the point I made was you should not by default assume that square root of 3 is 1.732 square root of 2 is 1.414 and so on. So, we will not get into that here those will be like you know getting into efficient algorithms to do that those are not those are not part of this sort of a course yeah I am just trying to get you introduced to the idea of doing abstract abstract kind of operations here yeah and what those symbols means. So, the square root symbol what it means is that you have a number multiplied a certain number of times twice precisely with itself to yield another number subject to the operation that has been defined. So, for that first we have to understand maybe it is better if you understand what is a group. So, again I am going to keep it semi formal if there is something like that. So, we have a set S yeah with some operation how should I denote it let us call it just a plus it is not just an addition it is just any operation. So, this is a set and this is a binary operation binary operation means I take two elements from the set at a time and I get them to operate under this ok. So, a binary operation ok. So, there is a set this is a binary operation this is said to be a group if it satisfies the following what is the first thing it is got a name it is called closure which means for all a b belonging to S a plus b also belongs to S. If I pluck out any two members from that set and pass them through that operation what results yeah should also belong to that set yeah. Second is a property again you are familiar with is associativity which is when we say that fall a b c coming from the set S it does not matter in which order you do the operation it is all the same. So, that actually you can get rid of the bracket and be unambiguous about it ok. So, you have the property of property of closure you have the property of associativity third you have the property of existence of unique identity ok what it means is there exists a unique this is the symbol there exists a unique. So, remember I am just using the symbol 0 as a placeholder this is not the number 0 yeah unfortunately we ran out of better symbol. So, we call it 0, but it is the identity element because I have denoted the symbol as addition. So, I have used the additive identity as we understand in case of numbers, but this is just the identity element such that for all a in this set S we have a plus 0 is equal to a. So, when you take this identity element get it to act or perform the binary operation with any number of that set what you get back is the original number ok operation with the identity has no effect no change it remains identical with what it was the fourth existence of an inverse what it means is for all a belonging to S there exists a unique minus a again this minus should not be read as the negative of the number essentially it is just a placeholder a symbol such that a plus minus a gives you back the additive identity or in this case of course, we have not said addition yet the identity ok. If these four properties are held together we call such a set with this operation to form a group. In fact, there are things called Cayley tables if you are interested you can look them up maybe we will have the occasion to drop one or two in this course. Cayley tables which basically describe for finite number of elements in a set with an operation they actually tell you what happens when you take them two at a time and what the resultant is I we have not talked about commutativity yet sorry you will have to impose that condition you will have to impose that condition. So, yeah same for inverse you will have to impose it a priority because you have not yet talked about commutativity I will come to commutativity and then I will use it with a different color and talk about this. So, that is the reason why I held it back yes we will revisit it with the white chalk again but before that we will just write something else with a yellow chalk and it gets a different name yes yes this is the same yeah to but this inverse is not the same for every element for given element the inverse is unique yeah. So, do not go about thinking every element will have the same inverse no of course not, but given that you have one element then there is a unique inverse you cannot have multiple elements which serve as the inverse of a particular element that is what this uniqueness means. So, there is uniqueness in both the cases but this is one size that fits all this is the identity that works for everyone but the inverse does not work for everyone for everyone there is one inverse this uniqueness just means that in addition if you have a fifth property that for all a b belonging to this set a s sorry a plus b is equal to b plus a which is the property of commutativity alright then it is not just a group but what we call an abelian group and then as your friend has just pointed out because this commutativity is something that has to be imposed. So, in order to have it in the definition itself here we will have to impose this here and if you do not impose it in the definition it is not implied automatically unless it is an abelian group in which case it does not matter in which order you are carrying out the operation. So, this is an abelian group with respect to one operation in the next lecture we shall build on this and we shall see that instead of considering a set with just one operation we will consider a set with two operations and we will see what sort of structures we impose based on these two operations and in combination with these two operations we will have other properties such as distributivity pop up when we consider both of those operations together yeah and we will have structures that we call as rings with identity then subsequently integral domains and fields and so on and so forth and then we shall see that fields are exactly the objects we want to concern ourselves with because it is exactly when you are dealing with fields that everything we have carried out in total for solving the system of equations a x is equal to b can be replicated. We will also see some interesting features and properties of fields maybe particularly with finite fields, but we shall not go in a full detail of an algebra course which if you are interested you can go through a preliminary book by I mean preliminary course based on a book by Michael Artin on algebra. Thank you.