 Welcome to the 10th session in the 2nd module in the course on signals and systems. We continue discussing the relation between signals and vectors that we had embarked upon in the previous session. Now we need to know a little more about vectors. We said something about the dot product. We need to say a little more. How do we calculate the dot product between two two-dimensional vectors where you know the components. So let me take a graphical view of this. So I have two vectors. Let me call them v1 and v2. And I express these in terms of their components. So I associate with them their perpendicular components. I am assuming that these are the two perpendicular directions here. I will just show them in blue with the perpendicular unit vectors. Let me call them u1 and u2. It is customary to use a cap over the unit vector. And therefore I have these components here. The red one for v2 and let me use black for v1. So I call this v12 because this is v12 u2 cap. And this is v11 u1 cap here. And this is v22 u2 cap and v21 u1 cap. And there I have a very simple relationship. v1 is v11 u1 cap plus v12 u2 cap. And v2 is v21 u1 cap plus v22 u2 cap. And let me now use the symbol for dot product which is a dot. So v1 dot v2. We all know it to be essentially the product of corresponding components and then sum. So sum of products of corresponding components which I can expand as v11 v21 plus v12 u22. So this is an important concept that we often encounter. If I have perpendicular components for a vector and if I do the same thing for all the vectors in a space, I can take the dot product of 2 vectors in that space by simply multiplying the corresponding perpendicular component lengths and add it these over all the component directions. So of course in 3 dimensions I would have 3 such perpendicular components. And you know although we cannot visualize more than 3 dimensions, we can certainly conceive of them conceptually at least for a finite number. And then after all what is a two-dimensional vector? A two-dimensional vector is just an ordered pair of two well for the moment real numbers. But later we will also allow complex numbers and you know why we so fond of complex numbers. Now you have some idea why we like to use complex numbers too. Well, let us get a better idea of what I am trying to say. Let us now relate vectors and first sequences. So let us do that. Vectors and sequence. What is a sequence? A sequence is a discrete sigma so to speak. You know that. So suppose you have a finite length sequence with only two points. Let us say again without any loss of generality that the two points are at 0 and 1. So you could for example think of V11 V12 as a two-point sequence and so also V21 V22. So what I am trying to point out is that you could think of the two vectors. If we drew a few minutes ago V1 and V2 as two two-point sequences and you could think of three-point sequences, four-point sequences, ten-point sequences. Essentially if a sequence has ten non-zero values, contiguously locate ten possibly non-zero values meaning all other values are guaranteed to be 0. Then you could think of it as a ten-dimensional vector and as long as the sequence has finite number of non-zero points, you are quite alright in calling it a vector of finite dimension. There is an exact correspondence and now you also get an exact correspondence between the dot products of two sequences and the dot product of two vectors. In fact it is almost trivial. If you think of these two sequences which we call V1 and V2, you could take their dot products, go back here. So I have a product of the corresponding components, multiply V11 and V21 and V12 and V22, multiply the corresponding components and add. Now you could of course take this two three dimensions, you could take it to ten dimensions. So defining a dot product between two finite dimension vectors or finite length sequences is not a problem at all now. Now you could take one step further. You could take this to infinite length sequences. The only change is now you have a series instead of just a finite sum. So you can visualize two discrete signals or two sequences where potentially all the points are non-zero. Let us take an example. Let us consider the two sequences x1n which is equal to half the power of n un or it is essentially half to the power of n for n greater than equal to 0 and 0 for n less than 0. And let us assume x2n is one-third to the power of n un which implies that it is one-third to the power of n for n greater than equal to 0 and 0 for n less than 0. Can we conceive of a dot product between x1n and x2n? Remember it is not see when I say x1n one must not misunderstand it to meet the mean that specific point you know x1 x2 are entities their sequences. So the dot product is essentially summed over all n the corresponding products of component. In this case this would become summation n going from 0 to infinity half to the power of n one-third to the power of n which is essentially summation n going from 0 to infinity 1 by 6 to the power of n which is a very simple sum to calculate it is the first term which is for n equal to 0 equal to 1 divided by the common ratio 1 minus 1 by 6. So the denominator is 5 by 6 and therefore you have 6 by 5. This is the dot product between x1 and x2. Now you see here of course you had an infinite number of dimensions associated with each of those so-called vectors. So now you know how to think of discrete sequences as vectors that is not too difficult to generalize starting from the finite dimensional vectors that we are familiar with. Of course here I took what is called a one-sided sequence that means I had a non-zero point all on one side of a specific chosen point. I could choose what I call two-sided sequences that means I could have potentially all the points being non-zero all over the set of integers. A minor variation the summation from n equal to minus infinity to n equal to plus infinity needs to be maintained as it is you cannot contract that summation that is all. Anyway I therefore leave it to you as an exercise to consider one such case. Let me write down that exercise. Calculate the dot product of the sequences x1 n is half raised to the power mod n for all n. Now here this is what I call a two-sided sequence or you know a sequence which is non-zero all over the set of integers and x2 n which is one-third raised to the power of mod n for all n which is again a two-sided sequence. Now that is a simple exercise and I leave it to you to do it alright. So we are quite convinced that we know how to calculate dot products. Now you know the first thing I am going to do is now to give a more formal name to dot product like we did you know we called perpendicular we said is a term we use more in high school geometry dot product again a term we use more in high school. Now in college we would like to use slightly more formal terms which also mean the same notion in a more generic context and therefore we will use the term inner product. We will use the term inner product for dot product henceforth. Formal term or more general term for dot product and we will use this to denote the inner product you know you put a comma and you have two triangular brackets and you put the arguments of the inner product on two sides of the comma. For example in the exercise that I gave you and the exercises I have been working out the arguments are x1 and x2. We will see more about the inner product in the next session. Thank you.