 We need to think of this also as an inner product because we are you see if you look at it what is the discrete time Fourier transform really. Let us look back at the inverse discrete time Fourier transform to fix our ideas. So the inverse discrete time Fourier transform says that you can reconstruct xn from the discrete time Fourier transform in the following way and we have already given this an interpretation. These are like the coordinates with different directions you know different omegas are different directions so to speak or different omegas give you different axis and you see these are different basis vectors. I mean they are different basis vectors for different values of omega. Now let us get this idea clear what we are saying is that for different values of omega over any contiguous interval of 2 pi each omega gives you a different so called I mean you can call it a perpendicular direction I mean I do not know whether it is a correct thing to do but you can call it a perpendicular direction it is a different axis so you have as many like for example in the time domain every point every value of n gives you a gives you one degree of freedom. In fact if you look back at the way you constructed a sequence from its samples there was the idea of a basis there you said that xn is summation k going from minus to plus infinity xk delta n minus k and here xk were like the coordinates if you recall and these were like the orthogonal basis vectors. Now here of course this is exact different delta n minus k for different values of k are truly perpendicular in fact if you were to take a dot product of delta n minus k1 and delta n minus k2 I mean you have two sequences it would be 0 for k1 not equal to k2. So if the impulses are located if the impulse sequences have the non-zero one located at non-zero sample located at different points their dot product is automatically 0 that is obvious because the sample the non-zero sample do not overlap. So these are all perpendicular directions so to speak so any sequence can be constructed from all these infinite countably infinite perpendicular directions. Now you know if we go to finite dimensions we can recreate this situation to some extent but as I want you in the previous lecture one must not take literally the conclusions of finite dimension to infinite dimension. However we can get a good clue about what to expect in infinite dimensions when we look at finite dimensions. So let us look at the very simple finite dimensional case of two dimensions you see let us consider this two dimensional space in which the paper lies in which the sheet lies I mean you can visualize the sheet extending to infinity constituting a two dimensional space and let us put the origin here let us draw two pairs of perpendicular vectors. So one pair is like this u1 cap and u2 cap I draw another pair of perpendicular vectors u3 cap and u4 cap it is very clear of course that u1 cap u2 cap as also u3 cap u4 cap form an orthonormal basis. What is an orthonormal basis? An orthonormal basis is a collection of vectors from that space which are mutually perpendicular take any two of them they are perpendicular and together these vectors span that space the word basis means they span that space span means you can construct any vector in that space as a linear combination of these. So of course it is very obvious that you can construct any vector in two dimensional space as either a linear combination of u1 cap and u2 cap or a linear combination of u3 cap and u4 cap that is very obvious. In fact let us to emphasize that point draw a vector and illustrate what I am saying in fact we will not draw one but two such vectors. So let us redraw this you have a u1 cap there you have a u2 cap here you have a u3 cap here and you have a u4 cap there and you have this vector I will draw a long one v1 and you have this vector v2 and of course it is always possible to write v1 cap as dot product of v1 cap with u1 times u1 plus dot product of v1 cap of v1 sorry not v1 cap with u2 times u2 you see this is the beauty of an orthonormal basis in orthonormal basis you can find the component of a vector along one of the orthonormal basis elements by taking the dot product of v1 with that basis element and this can be done for each of the basis elements. So when you have an orthonormal basis this is the bone the coordinates are easy to find when the basis is not orthonormal you know you can of course have a basis that is not orthonormal what I mean is you can have a collection of vectors for example in this collection u1 u2 u3 u4 you can take the pair u1 u3 u1 u3 also form a basis because you can express you can express v1 in terms of just u1 and u3 or you can express v1 in terms of u2 and u4 and you can do it by using the parallelogram law you can construct a parallelogram with sides parallel to u1 and u3 and they will give you a linear combination of u1 and u3 which gives you v1 so using the parallelogram law you can always express v1 in terms of u1 and u3 or u2 and u4 so u1 and u3 together form a basis but not an orthonormal basis similarly do u2 and u4 together. So the beauty of an orthonormal basis is that finding the coordinates is very easy you see if you take u1 and u3 it is a basis but not an orthonormal basis you can of course find the coordinates by using the parallelogram law but finding the coordinates is not a decoupled process that means I cannot find the coordinates of along u1 and along u3 independently I need to solve two equations for two coordinates as opposed to that when I have an orthonormal basis the job is very easy I simply take the dot product of the vector along each of these orthonormal basis elements and there I am the coordinate comes. Is that clear to everybody any doubts on this? So now we have done this for v1 we of course can do the same thing for v1 with the basis u3 and u4 so you know you can write v1 is also dot product of v1 with u3 times u3 cap plus dot product of v1 with u4 times u4 cap and of course if you like I can complete this by putting 1, 2 here so I can do this both for v1 and v2 please read this as v1 so there I have 4 I mean coordinates for v1 the coordinate with respect to u1, u2, u3 and u4 and similarly 4 coordinates for v2. Now let us write these 4 equations down what we are saying is v1 is of the form v11 u1 cap plus v12 u2 cap which is also v13 u3 cap plus v14 u4 cap where the dot product of v1 with uk is v1 k and similarly for v2 so v2 is v21 u1 cap plus v22 u2 cap which is also v23 u3 cap plus v24 u4 cap similarly for v2 k. Now the dot product is very easy to calculate because this is a perpendicular basis so the dot product of v1 with v2 in the simple 2 dimensional space is v11 v21 plus v12 v22 and it is also v13 v23 plus v14 v24 so that is what I am saying the dot product has nothing to do with which basis you use it is independent of the basis the dot product or the inner product is independent of basis and that is not very difficult to see you can use if you use orthonormal basis then calculation of the dot product is easy you just take products of corresponding coordinates and add but the inner product or the dot product itself does not depend on which basis you choose and that is not difficult to see at all I mean you know if you go back to the drawing a couple of slides ago if you were to take the dot product of v1 and v2 inherently it has nothing to do with what basis you have used to represent v1 and v2 the dot product is the property of the 2 vectors not a property of its representation. However given a representation one can of course calculate the dot product with convenience and ease now this is exactly what that relationship is saying and now let me put it back before you with this renewed understanding so what we are saying in this relationship look at this relationship once again look at this relationship once again we are saying the dot product of the sequences x1 and x2 is independent of the representation of those vectors you could think of the vectors represented in their natural domain or you could think of the vectors represented in the frequency domain the dot product is unchanged and of course in the frequency domain the dot product is defined in this way you need to multiply corresponding points on the frequency axis and integrate over all these points instead of add because the frequency variable is continuous so it is a very elegant and simple interpretation once you think about it.