 Now, comes the second part which is algebra and well you are more or less right about algebra when you say that you know you have these variables that come up symbols that come up and that is how you describe an algebra, right. But you have done more sophisticated algebra than that, maybe not in a formal setup which is why you do not recognize it. How many of you have heard of something called sigma algebra, sigma algebra? I am assuming you have done a course on advanced probability or something stochastic processes or something of that sort where you encountered this. Do you recall what a sigma algebra is, right, right, right. Thank you. So, basically what he said there there are two important components it is a collection of objects. So, you have to first have some object on which you define an algebra and then you have a set of operations which are legitimate which are well defined which you can carry out and then you set a rule base, right you impose that, that gives it a structure. So, sigma algebra for instance a sigma algebra you have a set A and you have a set U which is a collection of subsets of A, okay, such that so these are symbols by the way get used to these looking at these symbols such that if A i, A i is a member of this collection of subsets, okay. If A i belongs to U then the complement of A i must also belong to U that is the first rule, okay. The second rule is this is one, the second rule is if countable unions, okay, countable unions of subsets of A, sorry if I should say if A i belongs to U then the countable unions of the subsets must also belong to U and so should the countable intersections. So, if A i belongs to U you do not need to copy these just writing them for the sake of completeness. It is not something we shall be dealing with in this course I mean just giving you an example of an algebra that you are familiar with. So, this is an intersection of A i must also belong to U, okay. This is the rule. So, there are a few terms which we have already introduced and that is the reason we learn about definitions in mathematics what is countable does anyone know what is countable set of integers is a set of rational numbers countable it is not, okay yeah we will actually be dealing with such things about mappings and showing that if there are two sets contain an equal number of elements or not yeah if there is some structural similarity between sets and stuff and the like we will get into those concepts of course pertaining to linear algebra not in a very generic sense of analysis or topology or things like that. But anyway, so thank you for that answer so you can have countable infinities and you can have uncountable infinities as it turns out right there are infinities and bigger infinities and so on and so forth. So, the infinity of the set of integers is denoted by aleph not so a symbol that you might have encountered somewhere aleph not okay that is the set of infinity that is the infinity of the countable set anyway this is a bit of a digression the point is that there is a set of rules. So, if you have a set and you cook up the collection of all subsets okay collection of all subsets of a and then if they follow this rule such that if the set a i comes from this union the compliment must also come from there by the way this is not a very trivial operation complimentation is it is anyone of you aware of certain paradoxes and certain contradictions in set theory yeah would you know the name of that paradox okay not very important yeah it is Russell's paradox can you explain does anybody remember what it was is all about right okay. So, when you talk about these operations they have to be well defined and as one of your classmates has pointed out or rather clarified Russell's paradox for us you see if you have a set yeah and if you consider the set of all those sets which do not contain themselves that that is like almost every other set right a set of oranges yeah is it an orange it is not right a set of oranges is an object is a construction that we have created it is not an orange right so a set of when you when you cook up this sort of an object so is it a is a set a member of itself so there are things you call a set normal if it is not a member of itself and you say call a set abnormal if it is a member of itself okay so as as your friend pointed out just a while back that you take a look at all these sets that are not contained in themselves right so are they normal or abnormal right so think about all those collections of sets which are not contained in themselves right so you have for example the original set let us say 1 2 3 yeah and you cook up the set of all subsets what are those it is 1 2 3 itself it is 1 2 3 and then you can have like take 2 out of these and then of course you have to end up with the empty set 2 right so this set looks quite different from any of the subsets individually right it is a different object altogether you agree right because that is where the first understanding of this comes as I said the orange and the set of oranges they are completely different objects right so now if you define something like in a contradictory manner in a very layman's language suppose as a village where you have a you have set a rule base so all this contradiction comes from how you set the rule base so in a village if you have the following rule that there is exactly one barber and every man in the village no one has a beard okay so the condition is that every man either shaves himself or he gets shaved by the barber the question is who shaves the barber look at the rule the rule itself contradicts it right it is either or you cannot have both so you either shave yourself or you get shaved by the barber and nobody keeps a beard it is not allowed so what does the barber do if he shaves himself then he is the barber he cannot shave himself and get shaved by the barber at the same time see the contradiction there right so this kind of a contradiction is very obvious in set theory so you have to be alive to these nuances when you are dealing with certain kinds of objects thankfully we shall not be dealing with all these things so we shall not go crazy thinking about all of these ideas but it is once in a while important to understand why things need to be well defined and why we must understand the limitations of what our definitions are right if something is embedded within the theory itself there is nothing you can do that contradiction is kind of there you cannot evade it it is part of naive set theory right whenever you define something over it this contradiction will always exist all right so moving ahead with all of these contradictions we will nonetheless try to still keep our focus eyes on the ball as they say right keep our focus on solving systems of linear equations and just because I said that we will not talk about specific applications excuse me because you know this is after all a course with a doubly tag to it so I will just show you one application which I am sure every one of you has encountered maybe in your plus two courses where you have solved linear equations yeah which is nothing but circuit containing just resistances straightforward application so let us call this R1 R2 R3 R4 right and we want to solve for currents in this circuit of course you might argue just use series parallel you know simplifications and just solve for the current why going to this business well I just want to illustrate to you that this lends itself to the formulation of what we call system of linear equations so what we will call is this current as I1 and this current as I2 and this as a voltage source V so the first equation of course then becomes what is it I1 with a coefficient R1 plus R2 minus I2 with a coefficient R2 and the second equation there is no voltage source in this loop so of course this side you have 0 maybe I should put it there other side where you get the idea so this what will it have minus R2 I1 plus R2 plus R3 plus R4 I2 right what are the unknowns the currents are the the voltage is known so I might jolly well write this equation in the following manner whose entries I leave it to you to fill up of course there is nothing really to it right so this is an equation of the form ax is equal to b right that is what we shall be trying to solve that is problem one not the circuit but this is the first problem we will be solving in this course much of the course in fact probably up to 60 percent to three quarters of this course will consider problems like this and how to get an answer to solving this problem there is another problem which we shall try to solve in this course whose motivations I shall not introduce right away I will introduce it only as and when we shall be dealing with it because there is a concern that you might not remember it by the time I actually deal with that problem but that is the so called eigenvalue eigenvector problem which is of the form ax maybe I should use the same x from next class I will use just little x here ax is equal to lambda x where again x is an unknown but apart from that lambda is also an unknown okay so these are the two problems we shall solve in this course all right we already seen one example just to keep things light and not very heavy I will give you another example we will not get into any technical details or any mathematical rigor today I will just give you one more example and let us see how much time we have depending on that we maybe delve into another and again excuse me because this is also come somewhat coming from my experience with electrical systems and electrical engineering but I am sure again anyone who's done a course on basic physics which is everyone presumably in this class you would also find it familiar so you know this equation that we come out come across in field theory which says that what the gradient of the potential yeah is equal to what the field right the field intensity isn't it and what else what is the divergence of this not this is electrical the electrostatic it's not magnetic the electrostatic it's row it's row right it's a charge density yeah it's just row so if you combine these two together what do you end up with divergence of the grad it's what we call the Laplace's equation right so what we have is this this is nabla the symbol is nabla so you take this Laplacian of the potential and you end up with the charge density now this is what you're looking to solve what does this generally look like if you have three dimensions this nabla squared kind of thing is defined in the following manner right rings a bell looks a little familiar I'm sure right so suppose we ignore the third part of this and we only consider a planar sheet on which we want to calculate the potential right then we have to solve for this two-dimensional Laplace equation so we have to solve we have to solve something like we have to solve for this and let's say the region over which we want to solve for this is given like so right so of course it's not very straightforward to solve for this computationally we think of solving this as kind of creating grades or tessellations to this entire region like this and then if we have a lot of computational resources on our hands we create finer and finer tessellations because we don't mind having lots and lots of variables to calculate if on the other hand computational power is at a premium then we take coarser tessellations coarser grids and we say okay there's a lot of margin for error I mean it's like if you if you dish out more money I'll give you clearer and better results but if you don't have that much computational capability you're only only going to get coarser results that's basically what it is so then it turns out that we have to discretize this equation in space right so let me not just draw this like a parallelogram let me just draw this like a grid as if you're looking at it from a top the top view of this and take a small toyish example again the poor artist in me is showing its true colors right there's a reason why I've drawn them in two different colors it'll be clear in a while the point is how do we now formulate this problem we want to solve for the potentials at different different points on this grid right we can measure the charge density let's say we are given the charge densities how do we solve this problem turns out it'll be a linear system of equations let's just give it a name let's call this point as p q r and s all right and let's say the potential at this point is v p v q v r and v s right and let's say this is 1 2 3 4 5 6 7 8 9 10 11 12 let's just take the sample point p and pass it through the discretized version of this equation shall we so what is the discretized version of this what do you think it is what is a gradient after all it is just a substitute for a differentiation which is when in the discretized form a difference right so if I do take the gradient of this it's just v q minus v p but if I take the divergence of that gradient what do you think it is that going to be is the difference between v q minus v p and v p minus v 5 along the x direction plus I'll also have to account for the y direction so what should I write the first sample equation for the point v p is going to be as follows v q minus v p minus v p minus v 5 that takes care of the x direction plus v r minus v p minus or let's say up over we go is the higher direction so it's v 2 minus v p minus v p minus v r is equal to rho at p what is known and what is unknown here and how many equations should I write it turns out that's where the domain knowledge of the problem comes in which is why I cannot teach you very specific applications because my own knowledge is limited by the kind of problems that I have handle in my life right it turns out in this case you will know the potentials at these points why what's the argument those are the boundary points so it's a boundary value problem you would know what the values of the potentials at these points are there are essentially four unknowns here yeah there are four unknowns and you will cook up equations for all four of those right you will know the charge densities at each and every point so the terms that sort of pretend to be potentials and therefore unknown and not really unknown term such as v 5 term such as v 2 you have to know how to massage this equation in the proper fashion so that you end up with a system of linear equations that contain four unknowns and therefore in number so you set up this Laplace's equation for all four of these points considering that you know the values at the boundaries right and if that sounds like oh that's too much knowledge I know the value at 12 points I am just figuring out the value at you know four points it's like an overkill then I urge you to consider a finer grid that's 200 cross 200 and consider the number of unknowns and the number of knowns at the boundary point yeah probably that would make it more meaningful yeah so that is clearly another example of where ax is equal to b kind of equations are useful which is why again we are justified in engaging or devoting or energies behind this course okay any questions as of now from anyone this one so what is this if you take for example the function let's say this function what is the first derivative of this function first derivative of this function is the slope at a point what is the second derivative of this function it is the rate of change of the slope so for instance you can have a function that is growing like this this function is an increasing function because it's slow at all of these points is positive on the other hand you have another function that's also like this that's also an increasing function where do you see the difference between these two if you take the second derivative in this case the function is increasing so is its slope in this case the function is increasing its slope is positive but its slope is decreasing so the second derivative is negative right I urge you to look at this as something like the second derivative so what is the second derivative doing the first derivative is telling me about its tendency how this thing is varying how this potential is varying across the space that will be told by the difference between vq and vp what is the rate of increase or decrease of this slope that can be said by how much has it differed between point p and q and how much did it differ between point 5 and p was this increase more or less than this right so that's what this second difference is in fact if I write this in a compact fashion I would have sorry that was vq plus v 5 minus 2 vp and this also I can write minus 2 vp that's generally the pattern right so if you are interested in not formulating this as four equations in four unknowns and if you want to write all those potentials pretending as if these are all unknowns then you will end up with a system of equations where you have 16 variables so you have a 4 cross 16 matrix right of course there will be constraints because many of those variables will be known right but you can formulate this as 4 cross 4 you can formulate this as 4 cross 16 it's a matter of taste but you are solving essentially the same problem right.