 The next idea we shall be introducing will be one of coordinates, but before that we will have to refine what we understand about basis and understand something called an ordered basis ok. So, just to motivate that discussion if I tell you that on the x y plane there is a point let us say 2 comma 1 it is an ordered pair. It means that along the x direction I go 2 units and along the y direction I go 1 unit and that is how I end up with 2 comma 1. If I had flipped this one it certainly would not be the same right because then you would go 1 unit along the x direction and 2 units along this and this is 1 comma 2. So, it is universally accepted by us that when we give the coordinates of points in the Euclidean space we go according to x y z not just any arbitrary order right that is universally understood so much. So, that these days we take it for granted if you want to push forward similar ideas therefore, we need to order our basis just think about it. Now forget about the special case of R 2 or a Euclidean space and let us say you have a vector space V of course, a finite dimensional vector space is what we have in mind right and you choose a V here. Suppose there is a basis for V given by V 1 V 2 till V n that means it is an n dimensional vector space. The moment I give you this V we have seen in the previous lecture that there is a unique way of representing this V in terms of the basis elements. We have seen that right if you choose 2 such ways that are not exactly the same then it eventually turns out that because of the linear independence of the elements in the basis they must be eventually the same. So, the representation with respect to a basis is unique except for one crucial detail there are permutations that are allowable which is to say that I can write this as. So, I can write this as alpha 1 V 1 plus alpha 2 V 2 plus alpha n V n right. Now think about it if you want to convey to your friend what a vector is and you and your friend are both privy or both are aware of a certain common basis that you have in mind. Then you do not even need to mention these V's separately you can just blurt out these numbers alpha 1 alpha 2 alpha 3 dot dot till alpha n and your friend will immediately know what vector you have in mind. But there is one crucial detail suppose your friend has jumbled of the order of the elements in the basis set then whereas you want your friend to multiply alpha 1 with V 1 alpha 2 with V 2 and so on your friend might erroneously multiply say alpha 1 with V 2 plus alpha 2 with V 5 and so on and say alpha 7 with V n and that will end up with a completely different vector is it not. So, the order matters. So, in order to assign some sort of a coordinate something analogous to a coordinate you understand coordinates already in Euclidean spaces right. So, what is coordinate more formally speaking given a vector space over a field we want to have a mapping of these elements in the vector space V to n tuples over the field where n is the dimension of the vector space. So, this is a so called structure preserving 1 to 1 correspondence that is how we assign coordinates. So, what do you need for that you need not just a basis, but an ordered basis. So, it is not just the objects they must also be accompanied by certain indices. So, that you know that this object is V 1 this object is V 2 this object is V 3 and so on just knowing the elements of that basis set is not enough to uniquely specify every object in V. In other words what I am saying is this fellow V look this is an arbitrary vector space this is not like anything that looks like n tuples of numbers, but what I can do is I can take this V to f n in the following manner. So, I have V which belongs to this V and this gets mapped to the n tuple given by alpha 1 alpha 2 till alpha n this is exactly the mapping this is how I assign coordinates and you see why is it structure preserving because look at the 2 main operations that you do in a vector space what are those scalar multiplication and vector addition. So, you take V 2 alpha V. So, V correspondingly maps to so this is from V 2 V right. So, this is scalar multiplication V gets mapped to another member in V which is just a scalar multiplication. Look at what happens now when you map this yeah this is V getting mapped to f n yeah that means this V has gotten mapped to alpha 1 alpha 2 alpha n. If you do the same scalar multiplication in the way that we understand over the field I am saying that this fellow gets mapped precisely to alpha alpha 1 alpha alpha 2 till alpha alpha n why because after all what is this fellow look at this fellow this fellow is nothing, but alpha alpha 1 V 1 plus dot dot dot till alpha alpha n V n. Now, if you assign this a coordinate yeah this is also a mapping from V to f n right. So, what are the coefficients of these V's it is exactly these. So, it preserves the structure in terms of the scalar multiplication whatever you do unto fellows in V the same is being done unto fellows in f n in the conventional sense of the operations that you understand in the sort of things that are analogous to Euclidean spaces the n tuples. So, this way any arbitrary vector space through assignment of coordinates can be made to look like vectors and matrices in the conventional way that you understand it to be right. So, by the same token let us say you have V 1 plus V 2 I mean you have say this is V cross V and this gets mapped to V. So, you have V 1 plus V 2 which is nothing, but let us say summation alpha I V I is V 1 plus summation beta I V I is V 2 right. If you want to map this if you want to map this. So, you want to map this to f n cross ok let me just say you want to just map this through addition operation. So, you want to just map it to f n right. So, you have f n here and this is f n cross f n what is this? This is basically getting mapped to fellows such as alpha 1 dot dot dot till alpha n plus beta 1 dot dot dot till beta n and what is this? This is exactly the conventional addition right. What you end up with here is just alpha 1 plus beta 1 to alpha n plus beta n sorry this one this no this is basically I am picking out objects. So, I am picking out a pair of objects a binary operation. No no it is basically I am picking out objects from this. So, I am picking out two objects a binary operation. You can just say I mean this also you can say that this of course, came from V cross f that way. So, it is basically I am picking out a binary operation. So, that is why I just you can also say V it does not matter. It is taking two objects from V and mapping it to a single object. It is just a little detail, but the idea is that look at this addition operation here. This addition operation is defined as per the vector addition rules in the vector space V, but this addition operation here is defined as per the addition operation rules in n tuples of f. So, there is a structure preservation here by the same token here look at this multiplication that is going on here. This multiplication has happened in terms of the rules of V, but this multiplication here has happened as you know the scalar multiplication to exist in n tuples of numbers. So, it is a structure preserving one to one correspondence from V to n dimensional if V is n dimensional then to n dimensional fn n tuples of field members just stacked up right. So, this is a very important idea because once we understand this idea we will see that it makes sense to understand finite dimensional vector spaces through studies of matrices because ultimately whatever you are doing to objects in arbitrary vector spaces if you are performing so called linear transformations on objects in vector spaces what you are doing is to n tuples of numbers you are applying some matrix multiplication some matrices acting on them. We shall see more of that later, but this is a very important a very crucial idea right the idea of assignment of coordinates in this manner ok. So, we shall see another important question now yeah this. So, it is two operations two elements picked and this is the operation performed between them you are picking out two elements from fn. So, you are picking out this and this yeah and you are performing this operation which is defined in fn here you are performing this operation which is defined in V, but this addition and this addition they conform that is the idea. They are happening like this is there because they are preserving the structure. So, by knowing what to do with elements in n tuples in terms of addition you are basically doing the same thing here. So, this makes this our knowledge of this operation kind of irrelevant. If you know how to assign coordinates then all that you need to do is all that you need to know is basically how to operate additions of n tuples over the field by the rules of the field that is the preservation of the structure because they preserve the operations characteristics and structure. Exactly. So, that is why V is over that field f. So, this cannot be over any arbitrary f this is over the same f over which V exists I had written that I have erased it now. So, this is only n tuples of members from where V draws its scalars V is obviously defined over some field of scalars and it is those field of scalars that you stack up as n tuples and then you think of the analogous operations right. So, because the operations are inherited from the properties of the field over which V is defined that is why this carries forward that is why the knowing knowledge of this addition of over f n's suffices to do everything that you wanted to do over the original vector space V in terms of its own addition yeah of course. So, only then can you as only then can you call it a legitimate coordinate assignment. Otherwise it is not a coordinate assignment if you are changing up these rules if you are saying oh this is defined over real numbers or I am going to here take complex numbers or the other way round then this is no longer a coordinate assignment. So, basically I would say V to V V to f n V to f n. So, that is why two objects the same. Exactly. Then you cannot call it a coordinate assignment then that is not a coordinate assignment. No it is a different field then right if you are if you are taking a different field here. So, if you are taking suppose some other field and you are multiplying it with members of original f there is no guarantee that the things you end up with is also part of that original field. So, in that case can be over the field f. Which f do you mean by that? The initial ones. No, no they are not they are not because these fellows themselves are not a part of that field. So, you are acting on them using some other. So, if this is a complex for example and this is defined over real. So, you multiply a real with a complex number you end up with a complex number at the end of the day. So, obviously it is not a real n tuple anymore, but you would have wanted real n tuples because the vector space is over real field right. So, the field must agree. So, now let us revisit that important question we had where we saw something like say w 1 t is equal to e to the I do not exactly remember the function plus 2 e to the minus 2 t plus 5 e to the 2 t w 2 t is equal to e to the 5 sorry e to the 2 t minus e to the minus t and w 3 t is equal to e to the minus t plus 7 e to the 2 t minus 4 e to the minus 2 t minus t sorry yeah minus 2 t is it yeah alright. And we ask the question as to whether does w 3 t belong to span of w 1 t and w 2 t ok. Now, on the face of it this might seem like a challenging question because you know these are time functions and all right. But now think about it in terms of coordinates yeah. So, suppose you consider a set B to be given by consider an ordered basis. So, I leave it to you to check that what I am going to write is indeed a basis ordering is of course, trivial the way I choose the order is the ordering. So, it is e to the t e to the minus 2 t e to the 2 t ok ordered basis B for W 1 t and W 2 t the span there off. So, I leave it to you as an exercise to check that this is indeed a basis for the span of W 1 and W 2. So, what you have to check you have to check that any vector that you write as a linear combination of W 1 and W 2 can be written indeed as a linear combination of these 3 fellows. So, it is a spanning set yeah and the fact that it is linearly independent right. So, that is what you need to show ok. Now, given that we understand what we do about coordinates if this is my ordered basis how can I represent this. So, now can I not write W 1 t and this is the notation for it in terms of this basis this is the notation ok when I assign it I put it in a box bracket with a subscript for the basis the ordered basis this turns out to be look at the order oh is it minus t minus t sorry ok. So, it is 1 2 and 5 1 2 and 5 similarly W 2 t in terms of this ordered basis now turns out to be minus 1 0 and 1 and W 3 t in terms of the same basis turns out to be 1 7 and 1 minus 4 and 7 1 minus 4 and 7. So, all those time dependences and things are gone you have ripped it up apart and torn it down to its bare minimum which is an in tuple a 3 tuple in this case and all that you are asking now is do there exist scalars C 1 and C 2 such that C 1 and C 2 are the same basis. So, all those time dependences and things are gone you have ripped it up apart and torn it down to its bare minimum which is an in tuple a 3 tuple in this case and all that you are asking now is do there exist scalars C 1 and C 2 such that C 1 times this vector plus C 2 times this vector is equal to this vector. So, that is analogous to asking whether the following 1 2 5 minus 1 0 1 times x 1 x 2 is equal to 1 minus 4 7 whether this matrix whether this linear system of equations has a solution or not and you can go ahead and apply your test right the row reduced echelon form and whatever find out the rank of this a matrix augmented with this and check if the rank increases if there is a solution then of course it belongs to the span if there is not a solution then it does not belong to the span right. So, you boil it down to the same question that we started on the very first day of this course which is solvability or lack thereof of this equation a x is equal to b right. So, you see after doing all of this once we have an ordered basis and we are able to transform things to coordinates it makes sense then again to study this fundamental question that we started with which is solution of a x is equal to b right basis for a vector yeah basis should be contained within yeah yeah yeah yeah you are right. So, maybe I should cook up another function yeah yeah you are right yeah yeah we need another another you are you are right yes exactly so that is a very good point very nice thank you yeah did you get his question what he asked is a very legitimate question. So, maybe I should cook up another maybe call this w 4 and expand this using another x 1 x 2 x 3 okay. So, maybe I should cook up another w 3 t which is some 5 e to the minus t plus 6 e to the minus 2 t plus 4 e to the 2 t yeah. So, the point is making is the following thanks for that question yeah my bad so the point is making is that the basis elements of course must belong to this span right. So, you should be able to combine fellows here in to cook up each of these elements, but because I had fewer fellows there because I had only 2 of them. So, I could not have cooked up all 3 of them independently yeah so that basis is actually the basis for a bigger vector space. It is basically a basis for the span of these 3 rather than that of w 1 and w 2 okay. So, now if I have 3 hopefully I have not chosen very silly numbers maybe you can check the calculations hopefully now that I have all 3 of them can be you know it is now a 3 dimensional you can check that these are now linearly independent so that they span and of course now the answer to this question would be hidden within 5 6 and 4 yeah yeah thank you that is right. So, you will have 5 6 and 4 and w 4 t will then be of course 1 minus 4 and 7 yeah. So, that is now the question you are trying to ask for this of course in this case you can see that it will be a solution. So, now it will belong to the span right in the other case also you can take that up as a subsidiary question that if you want to cook up a basis for the span of these 2 right then what are the elements that you want in your basis that you can take up as a question okay maybe I will put it up in a problem somewhere it gives me ideas alright. So, this is an important question that we have seen that any arbitrary vector space can now be treated as in the same manner as you treat n tuples of numbers, but another important question now arises suppose 2 people were to choose 2 different ordered basis and suppose you have knowledge about both of those ordered basis is there a way to get from one to the other like what is the relation. So, let us erase this and show it pictorially. So, suppose you have this vector space v and one person by choosing a particular ordered basis let us say b 1 has given it a particular structure let us say with respect to b 1 alright and another person by choosing a second basis has given it second ordered basis has given it this structure the link or the connection that we are now trying to establish is between these suppose you know b 1 and b 2 can you figure out how these 2 are connected alright. So, this is an important question because quite often you are required to go from one choice of a basis to another for several reasons one certain basis may be more handy than the other in the context of the problem that you have to deal with right. So, you might often want to go from choosing one set of basis to another alright. So, several types of functions for example. So, I mean later on we shall see this even in more detail if any of you have done things like Fourier series alright when you do Fourier series you just write it down as sums of those signs and cosines right, but those are actually basis alright a very convenient basis because they turn out to be what we shall see as orthogonal basis. We have not gone into those ideas yet I am just giving you dropping some hints at the moment. So, you might have also depicted those periodic waves or periodic functions by choice of some other functional basis right it is totally up to you by the end of today's class we shall actually see one example of this basis transformation before that before we are done with today's lecture okay, but first we want to answer this in the general sense given that you have vector space v one basis and a second basis you know how to get from this to representing in terms of ordered basis 1 and in terms of ordered basis 2, but suppose now I delete this original vector space. So, that all that you are left with is and this how do you go from one to the other what sort of knowledge do you have to inherit from here that allows you to get from one to the other it is if I may put it like and you will see the analogy when I have come up with the expression finally, so suppose you have two languages one of which you are very familiar with right and you want to say something in the other language which you are not familiar with what do you do you write the particular sentence in the language that you understand ask Google translate to do the translation for you and Google I mean sometimes it messes up really bad because of the structure and the grammar, but it back in the day when you did not have Google translate what would people do, so you say English to French you are familiar with English you are not so good at French, so what do you do you take an English to French dictionary okay, you look up the English words that you want in your sentence translate them to French and then you string them together and hope that the structure and everything is right, so that a French person would understand your sentence right, so it is exactly something similar that happens when we get from one to the other these two bases are like two different languages you can think of them like that you are familiar with one of them if you want to go from the familiar basis to the unfamiliar basis it turns out you need to know what each element in the familiar basis is called in the unfamiliar basis what each word in the familiar language is called in the unfamiliar language okay that is what we shall see next and it is a beautiful result, so how do we go about this let us say you have vector space V of course finite dimensional vector space and you have the basis V1 which is V1 till Vn and a basis B2 which is let us say U1 till Un okay, so we will stop here for this module and we will carry on with this proof in the next.