 So, what we shall next start with is the topic called inner products and inner product spaces, but I must warn you at the very outset that up until this point whatever we have dealt with barring a very few exceptions where we have pointed it out to you, everything was over general fields. So, anything that satisfies the definition of a field you can just fly with it and go ahead and plug those results and they will hold without any loss of generality. The moment we get into inner products however we have to restrict ourselves to just R and C. So, or F that is the first most important thing that you have to remember is either R or C and none of those other fields that you might have seen like those finite fields and so on. And there is a reason why this is important or rather we will not be invoking any property of the fields apart from those which come inherently with R and C ok, there is a reason why well it will be clear once we define what an inner product is most of you have a feel for what an inner product is because of your first brush with Euclidean spaces you might think of inner products as something that signify angles between vectors right. Now, here is the deal if you take even without getting into definition of inner products let us say you understand what inner product is think of them like dot products on Euclidean spaces. If you take the inner product of the dot product of a vector with itself and the vector happens to be a non-zero vector what do you expect to get and not just positive number, but also an idea about the length of the vector right and lengths obviously as you know have to be positive right. Now if you think of say for example the binary field and you take this now of course this is not a zero vector in the binary field by no stretch of imagination right, but look at this this is so this idea of the length and all this it just fails you cannot push your luck through with this right. So if we are considering z2 the binary field no longer does that idea carry forward. In fact if you think of angles like you know even without much sophisticated thought or understanding of inner products you can think of these dot products like some cosines of angles, but if you keep varying things constantly you expect the cosines to vary continuously as well right, but these are just some discrete points on the lattice I mean there are not even points in between when you think of a rotation you are rotating a vector continuously, but here there are just points on the lattice you cannot just go continuously and transit from one vector to another through a continuous path even right. So those are several other those and several other problems they preclude the use of any field other than r and c for inner products I think I had some hands raised there yes sorry oh sorry yeah yeah you are right it is zero yeah fundamental mistake thank you it is going to be zero right. So this is obviously an inner product or a dot product of a non-zero with itself leading to zero and it is exactly things like these that we want to rule out from any inner product ok. So now we will define an inner product and you will see that the one object that you are familiar with that dot product it just happens to be a special case of this inner product yeah. So what is the inner product you take this is how we denote an inner product this is the first argument this is the second argument all right and it takes objects in the vector space. So it is a binary and maps it to the field the field could be again either r or c ok satisfying the following properties one the inner product of v1 plus v2 with v3 is going to be v1 with v3 plus v2 with v3 ok. And alpha times v1 with v2 is equal to you can pull out the alpha outside v2 of course here v1 v2 v3 belong to the vector space v here alpha belongs to the field and v1 v2 belong to the vector space there is a second property together these two properties can be qualified as linear in first argument right that is what this means right it is just the classic definition of linearity if you just thinking of this as acting on the first fellow keeping the second fellow fixed and waiting the first fellow it is just linearity yeah. Third v1 v2 is given by v2 v1 the complex conjugate thereof so if you are dealing with the real field conjugation has no effect but if you are dealing with the complex field then of course you have to flip the sign of the imaginary part right excuse me and finally that is just the definition of the inner product nothing more than this nothing more or less everything else that comes is a consequence of these four points that appear in the definition of an inner product you can go ahead and cook up your own inner product name it after yourself as long as it satisfies these four properties. So this last property is what we will call positive definiteness so what about the second argument we have seen that the definition assures us that this is going to be linear in the first argument if it is also linear in the second argument we could have gone ahead and said that this is bilinear but is it indeed bilinear and if so then under what special cases or is it always bilinear these are the kind of questions that we should be asking ourselves because the definition seems to remain mum on the second argument but it leaves a hint as to how we could probe through the third property so if you want to find out what is the behavior of this inner product with respect to the second argument then probably we should go ahead and try to harness this right so let us probe this so let us consider v1 and v2 plus v3 as this now directly we do not know if the this is actually linear in the second argument however if we manage to bring it as the first argument then we might be able to invoke the properties that have been described in the definition so we can say this is nothing but the complex conjugate of v2 plus v3 and v1 the conjugation is just you know an action on a scalar does not really make much of a difference so we can just say that this is the conjugation of v2 v1 plus v3 v1 so you know that conjugation of a sum is equal to the sum of the conjugation does not matter it is just simple complex arithmetic that you have done in your plus two levels yeah so this is what v2 v1 plus v3 v1 that turns out to be again if you flip double conjugation is back to the same value again so this is v1 v2 plus v1 v3 remember this we did not mention in the definition but it is a consequence of whatever is there in the definition we have just used those properties in the definition so it seems we are still on track towards pushing for bilinearity rv so there is yet another property that needs to be verified so let us see what happens when we take v1 and alpha v2 the alpha belongs to the field again by field I am only referring to r and c nothing else right so this is what this is equal to conjugation taken over alpha v2 v1 now of course what is this this is nothing but alpha v2 v1 the conjugation of the whole thing conjugation of products is product of conjugates so again this is nothing but alpha bar or alpha star if I call it that and that is the problem you see okay let me retain the bar as the notation for conjugation all right so this is v2 v1 the whole conjugate which is nothing but alpha bar times v1 v2 so this is what spoils the party it is not really bilinear unless unless the field is real so if your field happens to be r then it is indeed bilinear if it is not then it is sesquilinear almost linear but not quite I mean it meets half the property of linearity which is this summation property but when it comes to this homogeneity property or this scaling property it fails not by much but exactly by tweaking the number to its conjugate but that is not linear right so this is sesquilinear if you are talking about the complex field but it is bilinear if you are talking about the real field right so this is note that this is all a consequence of whatever is there in the definition we do not have to include this separately in the definition because these are all derivatives of the definition itself right so now let us look at some examples of inner products apart from of course the most natural dot product that you might have come across so okay let us start with the most natural example say CN okay over C right and what we have is let us say V1 V2 like so given by V1 transpose V2 conjugate okay you can go ahead and check that this meets those four properties right of course V1 and V2 are objects that are n tuples of complex numbers right go ahead and check that though there are those four properties alone that define an inner product and this satisfies those four properties if you hadn't taken this conjugation you would be in trouble where exactly would you have been in trouble if you hadn't taken that conjugation of the second argument where would the definition fall through sorry I would say what about the positive definiteness so for example again if you take one and I one and I right again you would be in trouble this is one squared plus I squared is equal to zero whereas obviously this is not a zero vector right so you have to take the conjugation now if you take the conjugation no longer is this an issue because now you would have a minus sign here and therefore that changes matters completely and it does give you an idea over the length of the vector as you understand it right so you need this conjugation of course if it's real then whether you take the conjugation or not it matters not but in general over n tuples of complex numbers this is what the dot product or the inner product reduces to but is this the only kind of inner product well of course not so let's do a bit of an exercise and see how far we can push our luck right so suppose you take C2 shall we take C2 okay let's take art to the simplest case let's take art to cross art to to our how in the following manner so you have V1 V2 and you have V1 hat V2 hat right this gets mapped to alpha times V1 V1 hat plus beta times V1 V2 hat plus gamma times V1 hat V2 plus delta times V2 V2 hat let us pose the problem in the following manner we are interested in answering the question as to whether there do exist some suitable choices of alpha beta gamma and delta that allows us to call this a legitimate inner product and through this process we will understand two things one of course that the kind of inner products that you are used to seeing are not the only ones that exist and second it will also strengthen our understanding of the definition because we will be invoking all those checks that we have defined in the description for the inner product in the first place right so that's the goal of this exercise this is an exercise okay but we will do it alright so first things first we have to ensure that what does this property what property does this meet as a first check if you take two different objects as the first argument then the resultant somebody is that not obvious from here I would say that that first check in fact even the second check the first two checks of linearity are actually quite obvious right linearity in the first argument is actually quite obvious what we need to actually check is the third property first non-trivial property is a third property because the third property in this case it's an inner product over the real field so you don't need to take that conjugation so if you flip the positions of the first and the second argument you should still end up getting the same result same resultant scalar which means that please ask if this is not clear which means that the first condition that I should invoke here is that alpha v1 v1 hat plus delta v2 v2 hat plus beta v1 v2 hat plus gamma v1 hat v2 is going to be equal to alpha v1 v1 hat plus delta v2 v2 hat plus beta look the cross terms will change if you flip the order or the arguments then the cross terms will change so this will become v1 hat v2 plus gamma v2 hat v1 this must be true if the third point in the definition has to hold then this must be true isn't it but of course I can already get rid of certain terms so what does this mean go ahead and check out what is common with beta and what is common with gamma it is the same thing right so what you have is beta minus gamma times v1 v2 hat minus v1 hat v2 must be 0 identically is it not this must be true for any arbitrary choice of the first vector and the second vector so in general this is not 0 in general this term is not going to vanish so the only way that you can ensure that this is 0 is by choosing beta is equal to gamma so you must choose beta is equal to gamma right so this leads us to conclude that beta must be equal to gamma right that is the first condition that we must choose so we cannot choose any arbitrary beta and gamma and things like this already you are seeing the effect of the definition the restrictions that are imposed by the definition you could not have chosen any arbitrary alpha beta delta gamma and call data inner product at least beta has to be equal to gamma coefficients of the cross terms must match right what about the second next condition yes why we are swapping because we have the definition that v1 okay maybe do not confuse with this v1 let us say we are taking some v1 and v2 inner product that must be equal to v2 v1 but because this is a real vector space I mean over a real field therefore this conjugation matters not so if this conjugation matters not I can get rid of this if I get rid of this then basically I would say that switching the locations of the first and second arguments does not matter which is what I have done so if this is what it does to the first and second argument then this is what it does to the second and first argument when they are flipped and they must still be equal because of the third point in the definition of inner products clear okay so that is the first observation what about the positive definiteness right let us take a look at this we have alpha v1 v1 hat plus delta v2 v2 hat now that we know that beta is equal to gamma let me just call it the same right no point in carrying forward and calling them beta and gamma separately right so we can just say this is plus beta v1 v2 hat plus beta v1 hat v2 and also from this second term here alpha v1 v1 hat plus delta v2 v2 hat okay let me not invoke the second term here for positive definiteness what do we need to do we need to pass on the same vector as the first and second argument so in that case v1 is equal to v1 hat and v2 is equal to v2 hat right so if v1 is equal to v1 hat and v2 is equal to v2 hat then we need what exactly alpha because it is real so I can just go ahead and square it alpha v1 squared plus delta v2 squared plus this is now the same right v1 v2 twice beta v1 v2 and what was this be greater than or equal to 0 is it not because of the positive definiteness right now suppose v1 is not equal to 0 then I can divide throughout by v1 squared right then what do I get alpha plus delta times v2 upon v1 squared plus twice beta v1 upon v2 is greater than or equal to 0 sorry v2 upon v1 okay thank you so if this is the case then what am I to infer if I plug in v2 is equal to 0 then what must I have for and these are legitimate choices right because this must be true for any arbitrary vector so I have chosen one such vector where v1 is not equal to 0 and v2 is equal to 0 right further let v1 sorry v2 is equal to 0 then I must have what alpha should be greater than or equal to 0 similarly by exactly the same reasoning and assuming v2 is not equal to 0 rather v1 is we should also have delta is greater than or equal to 0 what is the other interesting thing that can happen yeah we can but I mean are we going to get some new results from there so the first let's say this is the second condition and this is the third condition so is this enough that's the question right thank you that's exactly what we are looking for this quadratic thing here in v2 by v1 or if I like I can flip it over by v1 by v2 and you will check that the condition is the same it should not have any roots if it does then it will violate this condition no it has to change sign then so I need the discriminant of this quadratic term to be negative so what is the discriminant of this quadratic term right this is ax squared plus bx plus c so I need the discriminant which is b squared minus 4ac which is 4 beta squared minus 4 alpha delta should be negative which means beta squared should be less than alpha delta or alpha delta is greater than or alpha delta minus beta squared is greater than 0 I need strict inequality because if I have just equality I will have multiple roots I don't want that to be the case sorry sign won't change but I don't want 0 for something that is non-zero I cannot tolerate 0 unless the vector itself is 0 so if I have a solution that is non-zero that is also not acceptable to me right now just think about this what is this the way I have represented it it may not appear very obvious but what is this representation at the end of the day is this not the representation of the following form sorry yeah it is the representation in this form so you have v1 v2 and you have alpha beta beta alpha times v1 v2 so what you are asking for is that this matrix that encapsulates this operation had better be symmetric its determinant had better be positive sorry delta sorry yes thank you so this determinant had better be positive yeah and these terms individually should be non-negative yeah that would ensure that what you end up with is a legit inner product later on we will see that such matrices the generalizations there are at least not necessarily only 2 by 2 the generalizations of these matrices are what we call positive definite matrices right so taken in by this idea we will now give you another example of inner product that is cn cross cn to c which is given by 1 v2 that gets mapped to or let us just take it real for the time being yeah let us not mess things up with complex all the time yeah so we will not take the Hermitian ok so this is what this inner product does with ok go ahead and check that this is in general an inner product of course the special cases when this is identity then it is exactly what you are familiar with right when this A transpose A is identity right there are special classes of matrices called orthogonal matrices where u transpose u is identity right but you need not have that you can have any generalization like so and that is also going to be an inner product we will see a few more examples of inner products or spaces that are a little more exotic in the next module