 So, inequality called as Cauchy Schwarz inequality. Now Cauchy Schwarz inequality states that if I am given an inner product space x and I take any two elements say x and y that belong to x then absolute value of inner product between x and y is always less than or equal to so we prove this fundamental inequality and I said that this was nothing but generalization of the result that mod of cos theta in three dimensions we have this we know this result this result which we know from three dimensions Cauchy Schwarz inequality was a generalization of this particular results from three dimensions in this particular case x and y are two vectors that belong to R3 are two vectors that belong to R3 and here so when any two vectors x and y that belongs to three dimensional vector space we know this result and this particular result is a generalization in any inner product space ok. Now the reason why we wanted to work on this particular inequality was two fold one was well we want to reach the concept of angle in a inner product space in a general space and at the same time we also want to prove triangle inequality in a inner product space why do I want to prove triangle inequality I want to define a norm using inner product ok I want to define a norm using inner product so that is why I want to prove the triangle inequality so how do I prove triangle inequality using this particular result so let us move towards that so this result is separate this result is just for your reference this three dimensional result is only for your reference so I am continuing again with our inner product space x I am going to take any two vectors x and y so and I want to find out inner product of x plus y with itself so this would be if I just expand this this would be x inner product x plus x inner product y plus y inner product x plus y inner product y now inner product a number it could be a positive number or negative number inner product need not be positive always norm is positive inner product can be positive or negative cos theta in this case cos theta can be positive or negative so I am going to just replace this I am going to replace this particular equality with an inequality so this is less than or equal to what is x inner product x always positive not a problem okay so x inner product x plus x inner product x plus two times absolute of x inner product y plus y inner product y you agree with me absolute value of this this could be a positive or negative number if this is a negative number absolute value is always greater greater than greater than this this value okay so I am just replacing this equality with this inequality I have fine even if even if these are complex numbers still this particular inequality will hold okay see now I am going to use Cauchy-Strauss inequality this absolute value is less than x inner product x raise to half into y inner product y raise to half okay I am going to use this inequality here so this will give me x plus y x plus y inner product is less than or equal to y inner product y raise to half okay is that fine just using Cauchy-Strauss inequality I get this so this is less than or equal to actually this quantity is less than or equal to I am continue this is a square now you can see this is a square okay but what is left hand side so I can write that can I say this all positive numbers all are positive numbers this is a positive number inner product of a vector with itself is a positive number I can take a square root okay I have express the right hand side as a square so I can take a square root what is this this is triangle inequality this is triangle inequality if I define see if I define now if I take if I define a norm if I define a norm okay which is like this then I have all three axioms satisfied what is the first axiom so we saw this the first axiom is norm of x 2 is greater than 0 if x is not equal to 0 vector and equal to 0 if and if and only if x is equal to 0 vector okay second we saw what trivially held was alpha times x 2 is equal to mod alpha norm x 2 all right and what is the third triangle inequality which now we have just now proved okay so my third result is norm x plus y 2 is less than or equal to norm x 2 plus norm y that is the result which I proved just now so inner product defines a norm very nice inner product this norm is defined using an inner product or induced by an inner product so can we draw now now that it is a norm it is a length measure okay can I extract something more out of Cauchy Schwarz inequality so what was my Cauchy Schwarz inequality let me see whether I can draw some more mileage out of Cauchy Schwarz inequality my Cauchy Schwarz inequality was x y less than or equal to which is nothing but norm x 2 into norm y 2 everyone with me on this this is a positive number this multiplication of these two positive numbers is greater than this positive number so only you can use one way so when you add a higher number on the right hand side you get an inequality if you wanted to use minus of something then probably it is different but there is only one way to use it when you derive triangle inequality from Cauchy Schwarz inequality I do not see any other way you could use it no we are not assuming I proved Cauchy Schwarz inequality I do not know whether you are present in yesterday's lecture so we proved it no assumptions Cauchy Schwarz inequality we have proved by logical arguments and now I am trying to see whether I can get some more insights through it so is this fine now we have defined a two norm okay now what is a norm ultimately it is a positive number these two are positive numbers right okay I can take positive numbers inside absolute value not an issue right so I would not be wrong if I say absolute of fine is this okay just compare I wanted to have x transpose y divided by two norm of x two norm of y take two unit vectors in three dimensions take in a product what do you get cos theta okay take two vectors in three dimensions and take two unit vectors in three dimensions and their inner product will give you cos theta exactly that is what I have arrived at in any inner product space same inequality no difference okay so I am going to say now well let me define an angle so now let me define an angle in any inner product space okay how do I define an angle so let me define two unit vectors x divided by two norm okay let me define another unit vector y cap which is y divided by two norm of y and then then cos theta okay or theta equal to cos inverse of inner product of x is it okay is this fine so inner product generalization of concept of dot product to inner product allowed me to prove a very very important result from three dimensions into a any general inner product space so we could define angle between two vectors now inner product space could be any set of objects like we had set of continuous functions over an interval okay and when you study your undergraduate you come across many such functions they are called orthogonal functions they are called orthogonal polynomials why are they called orthogonal what is the basis okay it is basically you are talking because there is an underlying inner product space the inner product defined on it and that inner product okay allows us to define concept of angle between two vectors as in elements of the vector space they could be continuous functions that is why you know we have all those results when you look at Fourier series see if you take this inner product space set of continuous functions over minus pi to pi okay then we are told when you study Fourier series that and let me define a inner product here inner product is defined between any f and g two functions as integral minus pi to pi ft gt dt let me take this inner product space okay and let me define this particular inner product well what we are told is that inner product of sin t cos t is 0 because they are orthogonal okay when you hit upon this concept first time that there are two functions why are they orthogonal in what sense because when you think of orthogonality you are trained to think in terms of three dimensions i, j, k and so on okay but what you should realize is that is in the same sense i, j, k three unit vectors in three dimensions are orthogonal these two vectors are orthogonal in that inner product space set of continuous functions over minus pi to pi or the change to 0 to pi 2 to pi so this is interesting so this allows us to talk about orthogonality of functions orthogonality of general vectors in any vector space any inner product space so far so good so we have defined angle obvious thing that comes next is orthogonality okay we say that two vectors are orthogonal inner product is 0 simple the inner product is 0 then these two vectors x and y any arbitrary vectors x and y for which inner product is 0 in a inner product space they are orthogonal vectors okay what are the other concepts that you need when you start orthogonality well one thing we talk about is that a vector is perpendicular to a plane right we have to use a notion of a vector being perpendicular to a plane or a set let us say so this when I if I have this plane okay I am I can say that this particular vector is perpendicular to every vector in this plane right every vector in this set I could talk about an entire plane I could talk about this limited set and this particular vector will be perpendicular to all the vectors in this set okay so if you have a subset s which is subset of inner product space s and a vector x that belongs to inner product space is such that x is perpendicular to y for any y that belongs to s then we say that s is perpendicular we say that x vector is perpendicular to s if s is some subset of inner product space okay and I take any arbitrary vector x in the original space this is a subset this is a this is a vector if this vector is perpendicular to every vector that belong to the set then the vector is perpendicular to the entire set all these concepts will require more and more once we once we progress okay so what result that I wanted to generalize what was the best result in geometry that you keep using all the time Pythagoras theorem okay can we prove Pythagoras theorem? What is Pythagoras theorem? What do you what is the statement of Pythagoras theorem in three dimensions. Let us let us look at three dimensions what is the statement how will you state Pythagoras theorem in three dimensions if you are given any two vectors x and y okay I am given any 2 vectors x and y in three dimensions and let x be perpendicular to y. how will you state Pythagoras theorem norm of minus x y plus x, x and y are 2 vectors what will form a triangle x plus y will form the triangle y minus x also can form a triangle both will hold then so you want to say that can I say this is equal to x transpose x plus y transpose y as he is rightly pointing out this could be said even for x minus y okay so x minus y transpose x minus y will also give you x transpose x plus so this is my Pythagoras theorem this is my Pythagoras theorem in 3 dimensions do you agree with me is anyone has a doubt here what is x plus y if I take 2 vectors in 3 dimensions what is x plus y just try to visualize unless you visualize you will not get it let us say this is x vector and this is y vector what is x plus y this is my x vector this is y vector what is x plus y parallelogram law in fact now I would expect the parallelogram law to hold in what is do you remember parallelogram law everything will hold I mean you are just generalizing this is my x plus y vector right this is my x plus y vector and this is x minus y vector okay so now if x and y are perpendicular x and y are perpendicular okay then what we are saying is square of so we are looking at a scenario where y is like this sorry x is like this x and y we are looking at a scenario where x and y are like this and x plus y is actually this right so square of length here is this square plus this square that is all I am stating here okay is this fine so now all that I need to generalize this in inner product space is to use the concept that if 2 vectors in the product space are orthogonal then that inner product is 0 I just start with the same thing so in an inner product space x let x and y belonging to x well I am writing all everything in this cryptic language because is faster to write otherwise and you will get used to it after some time x is perpendicular to y right I pick up 2 elements x and y in the inner product space which are perpendicular all that I have to do to prove Pythagoras theorem is to start with x plus y x plus y this is equal to norm x this is equal to x inner product x plus x inner product y plus y inner product x plus this is 0 this is 0 x and y are perpendicular inner product with 0 okay what follows is the classic Pythagoras theorem generalized to any inner product space a grand generalization of ideas same ideas what you should not forget is the ideas of geometry which we are using from your school three dimensional vector space which you are used to in your college same thing is being generalized in different spaces okay so if your geometrical ideas in three dimensions are clear you will understand what is happening here you cannot visualize what exactly this means in a function space visualization is not so easy or I do not know whether it is possible or in n dimensions or but geometrically it is the same thing what is happening here when you have two perpendicular vectors and writing Pythagoras theorem geometrically it is not at all different that is important to understand okay same geometrical concepts are we are not able to visualize this okay so what next how did orthogonal vectors help you in three dimensions what were they used for can somebody throw a light standard basis orthogonal basis very very useful right we use orthogonal basis all the time so you had three unit vectors which are orthogonal in fact you chose them orthonormal what is orthonormal unit vectors orthonormal their magnitude was one okay so orthonormal vectors define help us to define any arbitrary vector in terms of its components along certain directions right so we write a vector x component along i direction and y component along j direction this is the first time when you start looking at coordinate geometry this is how you start representing a vector right so we need now exactly the same thing we need to generalize a set of orthonormal basis vectors in any inner product space and then we should be able to express a given vector in terms of a orthogonal basis right because orthogonal basis has many many advantages in computations as compared to non orthogonal basis okay okay so in three dimensions how many ways you can construct a basis what is the basis in three dimensions so for example yeah three independent vectors okay so so in three dimensions just like this just like this you know this let us say k this is unit vector here say k i and j just like these three unit vectors form a basis okay I can take some three other vectors say even e2 e3 as long as these three vectors are linearly independent they can form a basis there are infinite ways of defining a basis in three dimensions given that there are infinite ways of defining so I can if I give you some arbitrary vector in three dimensions say this vector this vector v okay I can write vector v as v is equal to say x1 e1 plus x2 e2 plus x3 e3 where e1 e2 e3 are three basis vectors I am perfectly allowed to do this okay yet we prefer to work with so same vector v we find it convenient to write in terms of you know some component xi plus yj plus zk and so on so we prefer this basis over this basis okay so likewise are there some special basis which are more useful when you do computations it turns out that there are okay now for example I have a function continuous function which is a polynomial okay can I define a orthogonal basis for a set of polynomials then I can express okay I can express a polynomial something like this using orthogonal components I can express a function using orthogonal components I can express a function using orthogonal components along certain orthogonal polynomial directions so I am going to generalize this idea I do not like this when I work in three dimensions I prefer this so I need orthogonal basis when I go to an inner product space so now I am using a concept of orthogonal set in this case i, j and k are is a set of orthogonal vectors in fact they are set of orthonormal vectors right set of orthonormal vectors so in inner product space if I give you a set okay and if I pick any two elements in that set okay and if the inner product of any two elements is 0 then that set is orthogonal set when we will call it as orthonormal set if each one of them is a unit vector then it is a orthonormal set so if each vector as unit magnitude then we called as set as a orthonormal set okay now moving on to okay so do you remember how do you construct if I give you in three dimensions if I give you three vectors which are not orthogonal I want to construct an orthogonal set starting from a non orthogonal set how do you do it does if I say Gram-Schmidt process does it ring a bell no you do not know what is Gram-Schmidt process okay we will study what is Gram-Schmidt process so I like orthogonality because it helps me to represent vectors in very nice way and so if I am given a set which is not orthogonal I would like to construct a set which is orthogonal okay if I am given a set which is not orthogonal then I can construct an orthogonal basis which helps me to represent vectors okay so first I am going to start looking at three dimensions we generalize to set of polynomials we will then go to functions space and so on so okay let us go back to our three dimensional vector space we have this vector v we have this vector v here and so let us call this x y and z directions and this our i ijk are three unit vectors okay if I wanted to compute component of v along x how do I do it dot product with unit vector in that direction right so right I use the unit vector v transpose i so this vector this will give me in a product dot product actually we should not say v transpose v dot i that is the right v dot i not v transpose i v dot i so dot product will give me x component okay dot product of v with j will give me y component and dot product of okay so suppose I find out if I am given vector v okay I find out the component along x okay let us call this vector as vx what is vx vx is you know component along x I am going to call it as vx okay so what will be v-vx what will be this vector v-vx yeah it will be two components that are remaining along y and so everything that was along x has been removed okay now what remains is only okay so in fact you would expect that component to lie in which plane yz plane right okay now this idea I am going to use to come up with this concept of Gram-Schmidt process okay is this clear what I talked about just now that you find a component along a particular direction remove it from the original vector what remains is along the remaining orthogonal components okay so this is a very important concept well Gram-Schmidt orthogonalization can be done only in an inner product space not in any vector space because inner product defines angle orthogonality and the things that you really need to construct an orthogonal basis okay idea of orthogonal basis cannot be thought of in some other arbitrary vector space where inner product is not defined okay so definition of inner product is crucial when it comes to okay now let us start with R3 okay so I am taking three vectors v1 v2 v3 which are linearly independent in R3 but not orthogonal okay they are not orthogonal they are just okay I am given three vectors in R3 and then I want to construct a set which is orthogonal basis right I could have constructed a basis from this which is a non-orthogonal basis this basis would be you know one way of constructing a non-orthogonal basis will be v1 upon right and v2 upon norm v2 and v3 upon I can construct a unit vector I can construct three unit vectors but they are not orthogonal okay so I would like to go to orthogonal set from this okay so let us define let us define vector e1 okay this vector e1 I am going to define as v1 divided by two norm of v1 is this fine I want to construct this is unit vector so I got one unit vector I want three unit vectors which are orthogonal in fact I would like them to be orthonormal then what I am going to do is I am going to remove the component I am going to define a new vector z2 I am going to define a new vector z2 okay which is v2 minus component of v2 along even how do I find component of v2 along even dot product times this is a scalar right this is the component this is the scalar component along even so this vector minus minus this will be everything now that is left which is not along so z2 will have everything that is not along even is even perpendicular to z2 you can just check that z2 dot product even what is this this is v2 dot product even minus v2 even even dot product even or even in a product even what is in a even even one so this is this is one so what do you get here 0 okay so I have constructed a vector z2 which is orthogonal to okay so z2 is perpendicular to even but z2 is a vector which is not a unit vector I like unit vectors ijk okay so how do I get a unit vector I will take e2 which is z2 divided by norm z2 okay so I have two vectors see this z2 and e2 are aligned along the same direction magnitudes are different right so e2 and even are also perpendicular so even is perpendicular to okay is that fine now what next I want to now construct a third vector so v3 so I will construct a vector z3 which is v3 minus component along even because very easily check that z3 is perpendicular to even z3 is perpendicular to e2 not difficult to check okay just take inner products you will see that z3 is perpendicular to even z3 is perpendicular to e2 okay so even e2 e3 are mutually orthogonal okay so how do I create e3 now take a unit vector along z3 okay so take a unit vector okay so we started with a non orthogonal set and we got an orthogonal set we got an orthogonal set I can do this why just in three dimensions you have some doubt see this e3 is a third vector which I am going to define just by taking unit direction along z3 okay see I started with what did I start with I start with v1 v2 v3 these are not orthogonal okay so from v1 I constructed this even vector okay then I removed component along even from v2 okay whatever was left was perpendicular to even okay next then I defined this z2 I defined a unit vector along z2 okay then I removed component of v3 along even v3 along e2 right whatever was left was perpendicular to both even and e2 you can just check this see because even and e2 are orthogonal if you take inner product of this inner product of even with e2 will be 0 and inner product of e2 with e2 will be 1 okay so it will just nicely follow so you started with three non orthogonal vectors finally I got this z3 which is not a unit magnitude vector so I am just making this unit magnitude vector here okay so I can generalize this process in n dimensions if you are given if you are given n vectors in n dimensions okay how could I construct an orthogonal set in n dimensional space how can I go on doing this Gram-Schmidt orthogonalization so I could systematically go from 1, 2, 3, 4 you know so this is even is x1 then z2 equal to x2 minus inner product x2 even even and e2 equal to and so on I just go on methodically doing the same thing okay then e3 then e4 then e5 then e6 I can go up to en so starting from a non orthogonal set okay so what we will see in the next class is we will take an example in three dimensions construct an orthogonal set we will take a set of polynomials which are not orthogonal construct set of orthogonal polynomials if you just follow Gram-Schmidt process what will pop out is Legendre polynomials okay I think you have heard this name Legendre polynomials and then you must have heard shifted Legendre polynomials and then you must have heard Bessel polynomials all these things will fall into line if you understand Gram-Schmidt process okay it is some orthogonal set constructed on some inner product space of interest okay and those sets can be constructed simply following the simple idea from three dimensions Gram-Schmidt orthogonalization okay that is the message okay so next class we will look at examples of this