70 - Inner product

we're ready to start our last topic of this course ready and excited to start our last topic of this course I should say that this is what we're gonna do is really just an introduction to a broader topic we're just gonna touch some aspects and and really get the ideas but without going into too many details and too many examples and too many theorems it's really more more of a survey of ideas that lead us toward where linear algebra continues so here's an introduction to this introduction so so far the the major mathematical structure which we which we considered in this course were vector spaces right vector spaces where the main object of of our studies and a vector space is something that has elements the elements are vectors vectors are sometimes matrices sometimes polynomials sometimes and tuples but they're vectors and there are two operations scalar multiplication and addition that was a vector space and and there of course some axioms which have to be satisfied but we know we know from from real life that even simple vector space is like let's say r2 or r3 have richer structure than just their vector space structure for example so maybe we'll write some stuff we know that even simple even simple vector spaces like r2 or r3 have richer structure than just addition and scalar multiplication right and what do I mean by richer structures so I'll do everything in r2 for simplicity it's easier to draw and easier to picture so if we think remember that we can identify point in r2 in r2 is just the two dimensional plane we can identify a point a B with a vector originating it with an arrow originating in the origin and terminating at the point a B right we took that point of view and we discussed complex numbers okay so if we if we think of vectors in r2 as arrows then we can discuss for example so we have angles what's an angle if you have two vectors and you think of them as as arrows so here's maybe one vector and here's another vector and everything is living in two-dimensional space so this is v1 and this is v2 there's an angle between them right this is a a geometric concept what's the angle between two vectors right and we can measure it this is something that that's there right but it's not encoded in the vector space structure of r2 it doesn't show up from the addition and the scalar multiplication do you agree there's another notion there's the notion of a length of a vector if we take the vector v1 which we identify with the point where it terminus we can measure its length right so there's a notion of so we have angles we can measure we can measure lengths right we can even discuss it elegant this is a geometric concept right this is something that has geometry in it we can even measure the distance between two vectors right what's the distance between v1 and v2 what it what is the distance between these two points and we know how to do this because we did it when we discussed addition of complex numbers for example so we can take the arrow connecting them this arrow and measure its length the length of this arrow is the distance between v1 and v2 and we know what this arrow is right if we call this arrow you what is what is U u is precisely if you look here v1 plus u equals v2 so U is v2 minus v1 do you agree so let's move this and write it here and U is just V 2 minus V 1 and the length of U is precisely the distance between the two vectors right so we can measure length and even distance an even distance let's call it D for distance the distance between v1 and v2 is d v1 v2 is the length of v2 minus v1 do you agree the length of V - the distance between these two points is the length of the vector connecting them do you agree okay so angles length distance these are all geometric concepts that exist in the structure of r2 and likewise they can easily be generalized to r3 and they're there and if you do courses in physics for example you use this stuff all the time and when you do multivariable calculus you're going to use this stuff all the time and these concepts these geometric concepts the geometric structure of these spaces is not encoded at all in the linear space structure okay it's not there it's additional structure okay and that's what we want to now discuss what we now want to build formally and abstractly such that this structure will appear not only in r2 and r3 and fir but for a general vector space okay and the key ingredient is going to be this mysterious thing called an inner product which we're gonna define soon so let me add that these these concepts concepts of angles length and distance turn r2 or r3 into not just a vector space but what we called it what we call an Euclidean space named after Euclid that's what that's what Euclidean space means okay it means these faces with their full rich structure in two Euclidean spaces okay and what we want we want to define these notions abstractly for any for any vector space okay and you may ask well why who why does it make sense to discuss an angle between two matrices right if that's where we're heading why does it make sense to define an angle between two matrices or why does it make sense to define the length of a polynomial right that's what we're gonna do if we're gonna define these notions for a general vector space a vector space can be a vector space of polynomials and we'll have the length of a polynomial why would that make sense why is it worthwhile doing right so many times in mathematics there's a lot of a lot of ability to do abstract stuff okay once you have anything you can generalize it abstractly but if it's gonna be useful if people are gonna be interested in it it better have applications it better indeed be useful other than just for the fun of it okay and it turns out that these notions that we're going to define of angles lands and distances and the formal structures that that we're gonna that we're going to construct metric spaces and normed spaces where we're going to see this stuff gradually is in fact very very useful and gives rise to many many many many examples and theorems and theory and applications which we're not going to be able to touch too much in the scope of this course that's why I said this is really just an introduction okay we'll see a bit of each but that's all okay so the key concept the key ingredient is the inner product which is the first thing that I'm going to define okay so is the introduction to this introduction clear is it clear where we're heading in what we want to do yeah what we call Euclidean space what we call Euclidean space is r2 or r3 and in fact RN with the additional structure of angles lengths and distances but we haven't defined it formally yet okay we just said what it is because we know what it is for r2 and r3 from prior knowledge or from from just understanding what I just explained okay but we didn't define formally what is an angle what is the length of a vector okay in in in r4 for example we can't touch it anymore in r4 right we can't draw r4 okay so there's still definitions needed here and that's where we're building - okay okay so here's our first definition now now we're into the details and now we're back to formality it's no longer just an intuitive introduction so let V let V be a vector space a linear space and everything we're going to do from now on all our vector spaces just like they were throughout the course in fact we didn't discuss although we had F in general for our field usually that F was R or C right so now it's going to be only R or C okay so let V be a vector space over either the real numbers or the complex numbers okay these are the only vector spaces that we're going to consider an inner product that's what I'm defining an inner product on V is is another operation is another operation is an operation defined on pairs of elements just like addition addition we define for two vectors right pairs of vectors we can call them vectors we know what they are but while addition for example when we add two vectors we get a vector right the sum of two vectors is a vector the inner product of two vectors is going to be a scalar okay the inner product of two vectors is going to be a scalar a number okay so is an operation defined on pairs of vectors which gives a number this operation is often called scalar product okay scalar product especially in the Euclidean context and are in okay which gives a scalar it is denoted by these triangular brackets u comma V that's how we denote the inner product of U and V this is not the only notation out there this is the one I'm gonna use okay you'll see various notations for example in physics or in other places sometimes you'll see a vertical bar instead of this comma here between them okay and sometimes these notations have slightly different meanings so don't be afraid of notation but this is the notation I'm gonna use if you see different notation and various sources like a book or a different course you have to make sure that you understand if it's precisely this or it's something slightly different it is denoted by UV and satisfies three axioms so just saying there's an operation usually doesn't say too much but we're assuming that the operation satisfies some properties just like for addition or scalar multiplication in the definition of a vector space there are properties that they need to satisfy in order to be called addition or scalar multiplication okay so what are the properties what are the axioms so there are three so here's axiom number one the inner product of U and V is equal to the inner product of V and U but not exactly rather with a complex conjugate okay so remember that the result of the inner product is a scalar okay the result is a scalar u comma V is a number okay if we're working over C then this number and this number are complex conjugates okay so if this gave us a plus bi this would give us a minus bi if we're working over R if we're working over R then they're actually equal okay so maybe we'll emphasize here over R u v equals u right because real numbers the complex conjugate is themselves right so over R this becomes just equality that's the first property property it's called conjugates symmetry this is called symmetry obviously this is called symmetry and in general it's called conjugate symmetry okay not symmetry but conjugate symmetry okay I'll write that so number one is called conjugate symmetry good so that's the first property property number two is linearity linearity of this operation in the first in the first factor meaning if you take alpha u times V remember that this is called a product so I can call it x alpha u times V this is the same as alpha u V okay so this is linearity you can pull the scalar out right and if you take addition u plus W comma V that equals u comma V plus W comma V ok so this is linearity in the first factor do you agree okay so let's maybe write that linear linearity in the first first meaning there are left component okay well we'll see that's the axiom it has to satisfy this will see what so you're asking a natural question would be well is it linear in the second component and the answer will be not an axiom it will follow from the axiom so we don't have to impose it as an axiom okay and three and by the way the answer is going to be not exactly a bit surprising but not exactly and property number three is that the comma V if you take the inner product of a vector with itself you always get something non-negative saying do you remember what it means to say that something is non-negative when we're discussing complex numbers when we have remember a priori we're working over C this may be a complex number when we're saying that this complex number is non-negative we're implicitly saying it's real okay there's no order relation on the complex numbers we mentioned that okay so by this statement is implicitly saying that this number is in particular a real number okay and a non-negative real number okay and furthermore it could be equal to zero it could be zero right because we're saying it's non-negative it could be zero but that's if and only if V is zero okay so the only situation where the product is zero is when we're taking zero times zero okay okay this this property number three is called positive definiteness these are just names and we're probably not going to use them too much but nevertheless I'm mentioning them okay so these are the three properties of being an inner product okay now it may look very mysterious very abstract and probably not too related to to what we just discussed about Euclidean space so we have to carefully work out stuff and see why it is okay what's the relation and what other properties follow from this and see some examples ok so there's stuff to come so let's start with a few remarks before we do examples the most important thing is examples because when we see examples things become clearer we understand what we're facing but before that I want to make some quick remarks a few quick remarks so let's write them here remarks V comma V V comma V so this is remark number let's call it 1 V comma V equals by property number 1 V u equals the conjugate of U V right so V V equals the conjugate of V V by 1 do you agree ok but when is a number its own conjugate its own complex conjugate when it's real so from here it follows so V V is always a real number even if we're working over C ok and that makes the statement in in part 3 that it's non-negative real number make makes it reasonable makes it make sense ok good okay so this is for everything so is this remark clear okay remark number two addresses the issue of what happens in the second component is it linear in the second component so the answer is not quite almost so let's look at the product of U and alpha V so I I inserted a scalar in the second component now and I want to see if I can pull that scalar out okay so what I know by the axioms is how to pull out a scalar from the first component right and what I also know is how to reverse the order of the components that's axiom number one so by axiom number one I can force the scalar to move to the first component I get alpha v u but with this complex conjugate right this is axiom one do you agree and now I know how to pull a scalar out from the first component it just pulls out that's axiom number two so by axiom number two this equals alpha v you conjugate right now the product of two complex conjugates or sorry that the conjugate of a product is the product of the conjugates so this equals alpha bar v u bar right and then this in turn using property two again property one again sorry this equals alpha bar UV do you agree okay so it's really playing with the definitions we get if you look at both ends that alpha in fact can be pulled out of the inner product but when it's pulled out from the second factor it comes out with with a complex conjugate clear so it's not linear in the second component it's conjugate linear in the second component clear what about addition what about addition in the second component it turns out that for addition it's it is linear just like for the first component so if we take you and then V plus W right what we get we can reverse the order by adding a complex conjugate so it's V plus W u bar and this in turn we can write as v u+ w u bar right so this is by property two and this is by property one everybody with me okay and now the bar of a sum is the sum of the bars so this equals V u bar Plus W u bar and then using property one again this equals U V Plus u W okay so UV plus W is UV plus u W so it the the linearity with respect to addition holds for the second component as well okay and it follows from these three axioms so I don't need to write it as an axiom as another axiom clear okay these are fun right it's actually a kind of fun playing with these things abstractly do you agree let's do one more just for the fun but it really is it really is just fun I mean we need to see examples in order for this to to really be something meaningful okay so let's do just one more what is - V V for example - V V so I can pull out the scalar - one from the inner product so it's minus 1 V V right do you agree and I can throw it in well minus one is real so I can put a bar on it for free it's the same thing so it's minus one bar do you agree - one bar V V and once I have an alpha bar out here I can throw it into the second component without the bar that's this statement alpha bar goes into the second component without a bar right so now I can write it V - V good so this is by this property here so minus V V is the same as V minus V and this is true for every good so these are just a bunch of remarks and these kind of show you how to prove or or find or explore more properties and work with these inner products abstractly okay let's see examples of inner products on concrete vector spaces that we know okay and what do I mean by see examples I'm gonna give you a formula that somehow takes two vectors and produces a scalar for example takes two matrices and produces a scalar or takes two polynomials and produces a scalar okay and I'm gonna claim hey this formula is an inner product in order to to be an inner product we're gonna have to verify that all the axioms in this definition here's the definition we're going to need to verify for each of the statements for each of the examples that these three properties look on this board please that these three axioms of being an inner product indeed hold please look on this board thank you so we're gonna give examples of concrete V's with concrete formulas that take two vectors two matrices two polynomials two n-tuples and produce a scaler and we're going to show that they satisfy these three properties and therefore they are inner products okay and note note that the word product is not necessarily what we know as product right so for example when we had products of numbers the result was a number okay when we had products of matrices the result was a matrix a product of polynomials in the regular sense is a polynomial here this is something really new okay keep in mind the product of two vectors is a scaler okay it's really something new okay so let's do some examples some examples oh now I'm already there examples so here's our first example sample number one let's take our vector space to just be the real numbers that's a vector space most in fact all the examples that are gonna give that I'm gonna give now are gonna be real examples okay let's take a peek back at the definition all our examples are going to be over R and when we're working over R things become simpler why simpler because there's the complex conjugate business is not there anymore so we have real symmetry for axiom number one and this implies that for the the second axiom when we're talking about the second component again we have real linearity in the second component as well right because for the second component we had here in this remark that a scaler pulls out with a complex conjugate but if we're working over R it's just alpha do you agree okay so things are simpler when we're working over R and since we're really not getting into too much details all the examples that I'm gonna give are gonna be over R where things are simpler because this complex conjugate business is not there okay so now let's take the simplest example possible where the spaces are itself okay so the vectors are really just real numbers okay and in this case I can define an inner product by just taking a product okay the product of two numbers is a number okay but we still have to verify that this satisfies all the axioms so is it true for example that I'm not gonna write it because it's really trivial but let's just do it without raining is it true that XY and YX are equal for numbers yes it's true because multiplication in R is commutative do you agree is it true that if we put here a scalar but a scalar is just another number right when these are the scalars and the vectors are the same they're all just numbers so if you put here alpha x times y we get alpha XY is it the same as putting the Alpha out here yes it is and even writing it is a bit ridiculous it's just writing it right is it true that if we take x times or X plus Z times y we get the same as X y plus zy the answer is yes and why is the answer yes no but why is it true so why is it true that X that X plus Z X plus Z y equals x y plus Z Y this is part of property number two of axiom number two of being an inner product okay so if you're gonna say that this is an inner product you have to argue that this is the dis Holtz okay and the reason it's true is that here what we have is X plus Z Y right that's how we defined it and here we have X y plus Z Y do you agree and why is this true why is this true why are these two equal because R is a field this is a distributivity property that holds in our clear good so sometimes showing that the simplest thing hole the simplest things hold can be tricky in the sense that you have to understand what you have to clean why is it not it's trivial okay it's trivial but there's something in here it's because of the fact that multiplication in our which is how we define this inner product is distributive good okay and the last property axiom number three was that x times X itself is always greater than or equal to zero and that's true because X X is just x squared and x squared is always greater than or equal to zero and it equals zero only when X is zero okay good okay okay so this is an inner product this defines an inner product on our good okay let's extend that a bit and let's take V now to be our n so now our elements are n tuples okay so for X X is now going to be an N tuple so it's going to be X 1 X 2 dot X N and Y is gonna be again an n-tuple so it's gonna be y 1 y 2 dot dot dot y n for two vectors I'm gonna define define X Y I'm gonna define something that I'm gonna claim as an inner product so I'm gonna define it as X transpose that's just making X from a column vector into a row vector once it's a row vector I can multiply it as matrices with Y okay a row vector times a column vector gives a number right and what is that number I can even say more explicitly what it is so it's the number 1 X 2 dot dot X n x y 1 y 2 dot dot dot y n this is X transpose times y and what do I get I get XY x 1 y 1 plus x 2 y 2 plus X 3 y 3 plus dot x and y n right so I get x 1 y 1 plus X 2 y 2 plus dot dot X n y n do you agree or I can write it in in in summation notation I equals 1 to n X I Y that's the same thing do you agree so the claim is that this is an inner product on are in okay and this is not just any inner product so this is called let's write it here this is an inner product inner product on RN called the standard inner product okay clear so we take two vectors we just take the sum of the products of their components okay first of all do you see that when are when n is 1 when n is 1 so V is just R then these vectors only have one component and this sum just becomes x1 times y1 so it reduces to the first example do you see that so this the first example is a special case a particular case of the second example ok now why is this an inner product right we have to verify the axioms there's no shortcuts right so let's think of why the axiom holds why the axioms hold and and remember that ok let's look again at that board it's hard to hop between the boards let's recall what the axioms are and then verify them at least the idea so we're working over R so we have to have symmetry UV has to equal to VU we have to have linearity in the first component and we need to have that the product of two guys is always non-negative and 0 if and only if V is 0 ok so these are the three properties that we have to verify let's just look at this and see if it indeed holds ok so the first thing is X Y the same as Y X Y X is going to be y transpose X it's going to be writing the components of Y here and the components of X here but the product is going to be the same right the product is going to be y 1 X 1 plus y 2 X 2 and so on but these are real numbers they commute do you agree ok so the first axiom holds what about the second axiom suppose we have a little alpha here can we pull it out so if there's a little alpha here it means it's sitting in each of these components right it's alpha X 1 alpha X 2 alpha X and it means it's gonna appear here as alpha X 1 y 1 plus alpha X 2 y 2 plus alpha etc here we can pull it out by distributivity right so we're gonna get alpha times the entire sum alpha times XY clear so that's how you check properties okay I'm not gonna do the other two they're as trivial okay but you need to verify them you need to spell it out and see that it holds and usually it's gonna fall not usually in these simple examples it's gonna follow from simple properties of the underlying structures like distributivity good okay let's do a couple more examples maybe slightly more interesting let's look at a space of matrices for example so now in example 3 our V is gonna be n by n matrices and we want to define something some operation where we take two matrices okay in order for it to be an inner product before we even think of the axioms so this is example number 3 V is now matrices n by n matrices over R an inner product has to be something that takes two matrices a and B and produces a number okay so if we were for example to say okay let this be a times V they're square matrices they're n by n a times B is defined but a times B is not a number it's a matrix so it's not an inner product because an inner product has to produce a number okay so some somehow we need to use some concept that we know from the world of matrices which takes a matrices a matrix or two matrices matrices and produces a number how can we for example produce a number from a matrix the determinant that a good idea right the determinant of a matrix is a number so it would make sense to write here some determinant that involves a B may be the determinant of a B for example okay you could try you could try so it would it would compile but you would still need to check the three axioms and see if it is or it isn't okay what I'm gonna write is again called the standard inner product on matrices and it's not going to involve the determinant it's gonna involve the trace remember what the trace is the sum of the diagonal elements that is again a number you throw in a matrix it spits out a number right so what I'm gonna write is the trace and now it has to involve a B right so it's gonna be the trace of a times B transpose okay may look a bit mysterious may look a bit mysterious but it turns out that this is an inner product okay whereas maybe more naive attempts would not produce necessarily an inner product and you and it's very good practice to try very good exercise okay so let's show that this is an inner product it's a times B transpose no if I wanted to transpose everything I would have to put the parentheses on a B right it's the transpose just to be okay no no no no it's possible would be what would be your if they're not square matrices okay ah so you're saying the trip yeah okay okay yeah things could extend I don't want I don't want to to go beyond the very simple examples that I'm giving here okay one needs to check okay so you're asking for example suppose we take matrices of size M by n okay if we take matrices of size M by n we can't multiply them but we can multiply a times B transpose right because a is M by n B is n by M okay and then we can take the trace is it an inner product great question verify ok so there's a reason that this transpose is here and okay good very good let's let's verify here it's gonna take here it's gonna be less than just more than just an observation to say that this is an inner product we're gonna have to write a bit of stuff let's do some of it at least okay so this is an inner product this is an inner product let's see what goes into showing that the axioms indeed hold okay so for example I'm gonna verify let's choose some of them I'm gonna verify some and leave you the others as exercises okay so for example is the inner product of a be the same as the inner product of B and a okay so what's the inner product of a and B by definition it's the trace of a B transpose right that's how we defined it now recall two properties that we know for matrices before we ever discussed inner products we know that in general if we take two matrices a and B multiply them and transpose what is that equal right it's the product of the transpose matrices but in reversed order remember this property so this is a general property and another statement is that you take a matrix transpose it and then transpose again what do you get you just get the matrix back right these are very simple facts that we know from our very very very first discussion of matrices remember we had a long dictionary of matrix matrix operations and we proved some of them I think we may have even proved this formally showed that they agree for not necessarily for square matrices even okay for matrices that can be multiplied we show that the sizes agree that the dimensions agree and we showed that element wise they agree remember how we prove things like this okay okay so we know these two properties so without taking the trace yet a B transpose a B transpose is the same as B a transpose transpose tell me if you agree because be a transpose transpose is the transpose of a product is the second one transpose so it's a transpose transpose which is a time's the first one transpose B transpose so do you agree that these are the same okay and now and now we know in general the trace of a square matrix is the same as the trace of the transposed matrix right so trace of a is the same as the trace of a transpose right do you agree so BA transpose transpose once it's in the trace I can throw out that outer transpose it's this good and this by definition this equals BA that's how we do you agree good so this proves by properties of the trace and properties of transpose and properties of matrices this proves the first axiom if we define the inner product like this then taking the inner product of a and B and of B of a is the same this proves property number one which we called symmetry good ok let's look at another property let's show for example and sometimes what's more complicated to show is sometimes even though what looks is a simpler axiom so Y is a a for example this has to be non-negative right why is it non-negative let's prove it so what is a a it's by definition I'm gonna erase this or write it smaller so by definition it's the trace of a a transpose right I want to claim that this is non-negative that's part of axiom number three that this is a non-negative number right so why is it non-negative let's calculate what it is okay so what is the trace of a matrix it's the sum of its diagonal elements so it's the sum I equals 1 to N of a a transpose i I do you agree the sum of the diagonal elements of this matrix which is itself a product of two matrices do you agree ok good this equals this equals what is the I element of a product the iife' element of a product I know the formula in general for the I J's element of a if we have in general the IJ element of a B it's the sum K equals 1 to n of a IJ b b j IJ b JK right remember that's so let's write that but now it's I I and now it's special matrices it's not just a B it's a a transpose so this is the outer sum which I'm not touching I equals 1 to N and now I'm summing over a new index K equals 1 to N and I have to take the eye case element of a let's call it a by K times a transpose K J but I'm not looking for the IJ element I'm looking for the I ice element so it's K I do you agree everybody with me sure ok what is the K ice element of a transpose right it's the case element of a so this is the sum I equals 1 to N of the sum K equals 1 to N of a I K times 8 I K AI K squared do you agree and that's it this is a non-negative number these are all these may be negative the entries of a may be negative but they're squares are already non-negative and therefore I'm summing N squared non-negative elements so this is necessarily greater than or equal to 0 right and moreover it could be 0 if and only if all of these entries are 0 right nothing can cancel out here so in fact a equals 0 if and only if a is the zero matrix do you agree okay so this proves axiom number three what remains to prove is axiom number two right the linearity and I'm leaving it for you to try as an exercise okay you're you're you could guess that properties of the transpose are gonna show up there right we're gonna have a plus B or a plus C B right and then we're gonna throw it into this trace things are gonna require some properties of either the trace or the or the transpose okay good it's the idea clear okay so this thing is an inner product on matrices you take two matrices you produce a number that's fat satisfies all these axioms good I want to give you one more example one more example this last example I'm gonna write it here this last example is something that you may know a bit or if you haven't seen integrals in calculus yet you may have forgotten it since you've seen integrals in high school but nevertheless I want to write this so example number four example number four I'm gonna take V to be the space the vector space vector space functions function not even necessarily polynomials functions defined on some interval a B okay functions on some interval a B and I'm gonna assume I'm gonna assume that these functions are what's called integrable okay and if you don't know what integrable yet what integrable is yet then you're just gonna take it as a some property of all it means is that you can integrate them yeah you can stick them into the snake okay so if you haven't seen this a more formal this word this word I'm saying all it is and like all it means is that you can stick it into the snake there there's like four hours of discussion to explain really what it means for a function to be integrable okay so it's it's it's I'm really cheating you a bit here okay so and by the way this is no longer a finite dimensional vector space okay so this this this example is really just to show you the things can get very interesting and and very interesting and and this thing gives rise to very important stuff so I'm just touching touching stuff that's really beyond this course okay so I can define an inner product now these are two functions two integral functions on a B and I can integrable and I'm going to assume they're real valued okay over are integrable real valued integrable real valued functions on a b and i can define an inner product to be the integral from A to B you don't see many snakes in algebra right this is really a calculus calculus ish sort of thing but calculus and algebra are not two that the intersection is far from empty okay so these merge into many many topics like functional analysis and in in operator algebra and so on and so forth and I can go on forever here so I'll stop of the product f of X times G of X DX this is an inner product okay and if you just think for a minute and you know a bit of stuff about integrals then you can see that the linearity holds okay so if you take because an integral is a linear operation and by the way notice that we're taking two functions and producing a number right the integral from A to B of a function is at least intuitively the area under the graph it's a number okay so we're taking two functions and producing a number okay okay so this is an inner product and there's really no way for us to appreciate the importance of this and things that stem from it at this point okay it's really just you can call it a teaser I don't know it's it's really hinting to vast rich mathematics that that's beyond our scope right now okay okay so this this wraps up the list of examples that I wanted to give I I just want to say one more thing one more definition maybe I'll add it here on this board so what what I want to say is that a space that has an inner product is called an inner product space how's that for a trivial definition okay so let's write it here up here definition if V admits an inner product v is called no longer just a vector space but an inner product space good clear okay so we know what an inner product is the next step is to see how from this notion of an inner product which at least thus far still we don't see how from this we get angles how from an inner product we get angles how from an inner product we get length how from an inner product we get distance okay those are the those are the concepts that we wanted to pin down right in general but this does give give rise to them but we're gonna see that in a in a separate lesson in the next place okay so we know what an inner product is we saw examples now we want to see how it gives rise to all these geometric concepts that we had for for Euclidean spaces okay that's next questions so again this is a big topic which we're not getting into it in fact we can admit more than one inner product okay you can define different inner products on the same space okay so in fact this is a bit of a Vinnie actors a bit of inaccuracy here because you have to write V comma what's your inner product okay and V together in it with with a specific inner product is called an inner product space okay so that's that's an good question right are there spaces that have no inner products on the spaces that we've seen can we define different inner products later on do these different inner products give rise to different geometric structures on the same spaces answer yes very interesting very exciting beyond our scope unfortunately semester ends and three days okay okay but I'd be happy to discuss it later if you want I just can't we can't afford to go into all this but but I hope I'm intriguing you too and tempting you to take further courses along these lines and these ideas okay so that's the end of this one

Channel: Technion

Views: 37,091

Rating: 4.9169812 out of 5

Keywords: Technion, Algebra 1M, Dr. Aviv Censor, International school of engineering

Id: raW8dUc1Ts0

Length: 65min 8sec (3908 seconds)

Published: Mon Nov 30 2015