72 - Inner product and norm give geometry

okay so we know what a norm is and we know what an inner product is okay so we have the inner product we have the inner product UV is an inner product and we have the norm of V given by the root of the inner product of V with itself and we know that this is a Norton and we know what these words mean it they they translate to a bunch of axioms that you have to verify and so on and so forth and we know concrete examples of inner products which lead obviously to concrete examples of norms okay and what I want to show is that in our two for example and maybe we'll work just in our two for simplicity just to be convinced in our two if we start with the inner product on our two that we defined the standard inner product on our to figure out what the norm is for that standard inner product and see that what we get is precisely how we measure lengths the vectors in r2 and then see how from these two notions we define angles that coincide with how we measure angles in r2 and see how from these two notions we can define distance that coincides with how we measure distance in r2 okay so we're gonna see how in r2 starting with the standard inner product we get the Euclidean space structure of our two in general starting from a inner product on a vector space any vector space with any inner product you get these notions but they give you geometry that maybe is something abstract right okay so let's look at what we get so in our to in our to the standard the standard inner product that was example number two I think for inner products is so suppose you have two vectors let's call them what did I call them here let's say U and V so u and V we're denoting it by squarish but by triangular brackets so U is now has two components so it's u 1 u 2 and V has two components V 1 V 2 and do you remember what the standard inner product was in R 2 you take you transpose it and multiply so it's just gonna be u 1 V 1 plus u 2 V 2 that's the standard inner product Finnerty do you agree right in our end in general you take this one transpose it and multiply that's how you get the number good ok so this is the standard inner product in R 2 what is the norm so therefore the norm of a vector in r2 v1 v2 by definition it's the root of the inner product of the vector with itself it's the root of the inner product of V V right that's the definition so maybe let's write it here v1 v2 inner product v1 v2 do you agree and this is nothing but V 1 times V 1 plus V 2 times V 2 so it's v1 squared plus v2 squared that's the length hmm the root do you agree okay so if you take a vector in r2 you get that it's norm its norm is in R to the standard inner product and norm R this is the standard inner product therefore this is the standard norm okay so let's see what we got what we just got if you think of r2 and think of it again as our good old arrows so here's an arrow in our to call it V right this is v1 its components are v1 and v2 okay so maybe let's write it let's write it like this if this is V then V 1 is here and V 2 is here therefore the length of this is V 1 and the length of this is V 2 do you agree and therefore the length of V the length of V is by the Pythagorean theorem is v1 squared plus v2 squared root by Pythagoras do you agree so we indeed reproduced the regular length of a vector that we know from Euclidean space that is the length of a vector it's the root of the squares of its components okay and that is precisely what we got from taking the standard inner product on r2 and the standard norm stemming from that standard inner product do you agree okay so that I think this rests the case for the length okay so maybe let's write that so the norm defined by the standard inner product on r2 or r3 you can do the exact same arguments on r3 right and r3 the length of a vector is the root of V 1 square plus V 2 squared plus V 3 squared okay and it's precisely what you would get okay the norm defined by the standard inner product on r2 or r3 give the regular familiar let's not write regular let's write the familiar length of a vector in the Euclidean structure okay I'm emphasizing this because you can impose different structures on r2 and r3 different norms and different lengths and different angles many different things this is the regular this is the familiar one this is the one you're used to okay and it's it precisely agrees with standard inner product in north okay okay what about angles so now now we know that length is norm okay in this example good what about angles what about angles we haven't seen angles yet in the definition of inner products and norms right those definitions didn't have a notion of an angle yet okay so let's see what do we do so what is an angle in in Euclidean space what is an angle in Euclidean space it's something we can measure right we know the angle between U and V is just take the angle between them right this is something we know okay so if we define if we define I'm gonna I'm gonna do something a bit differently now like it's oftentimes done in in in even in mathematics but usually in physics I'm gonna define the inner product for our two specifically starting from its geometric structure okay so suppose we take two vectors in r2 if we define in r2 I'm going to define the inner product of U and V the inner product of U and V to be this is tricky let's not call it the inner product let's call it something different let's call it what it's often called the dot product u dot V this is something completely different has nothing to do with inner product a priori nothing to do with inner product I'm going to define U dot V and that's what it's called the dot product or the scalar product to be the length of U I know what the length of U is right I can measure it using pythagoras and now I know that it agrees with the lens coming from the inner product but I don't care I know what it is times the length of V times the cosine of alpha where alpha is the angle between U and V so here's you here's V u V alpha okay this is this alpha here okay so this is called the dot product okay then we can show and I'm going to show it in a minute I'm not going to just tell you to believe me I'm actually gonna show it that what we defined is none other than the inner product on r2 then we can show that u dot V is precisely is precisely u1 v1 plus u2 vitu and therefore it's precisely you v the inner product I'm gonna show it in a minute okay so it's not we can show we will show that this and this are actually the same thing okay so if we know the geometry we can define the inner product to be this if we know the inner product and we want to define the geometry so this is the reason therefore we define in general not just in r2 in general in general I mean in any V in any V which is an inner product space we define that the cosine of an angle between two vectors we define it to be we define it to be the inner product the inner product in fact the modulus of the inner product divided by the norm of U times the norm of V that's how we define so so if you want alpha you would take the arc cosine of whatever number you got here that's the angle okay why does this make sense well because in r2 or in r3 you would get that UV equals UV which is the same right UV equals UV equals UV cosine alpha okay so do you see the motivation for this definition No you you want to take two vectors so let's write it like this therefore we define for U and V in a general space V for you V and V in general what is the angle between them I'm not defining what cosine is here I'm taking two vectors let's say these vectors are two polynomials in a vector space of polynomials I'm defining what is the angle between two given polynomials the angle is you can you can if you want you can write it arc cosine of this thing okay this is not how it's usually written but that's what I mean take this number calculate its arc cosine that's the angle between the two given vectors whether they're polynomials matrices whatever functions okay that's how I define the angle between two given vectors in general okay so therefore we define for you V in general alpha to be the angle between U and V now you're gonna say that you don't know what the symbol is you and good okay so I yeah okay let lets let's restrict I'm getting my self into trouble here okay let's work over are for you V and V over R and then it will be better for you V and V over R in general okay and then I omitted this absolute value business okay and what I claim is so this is the same thing as saying that cosine of alpha equals UV divided by norm u times norm V okay this and this are the same thing do you agree okay now just a remark I still have I still want to prove this fact which is which is why this thing makes sense why it makes sense to define this right it makes sense because it agrees on our to right or r3 UV is you length V length times cosine of alpha okay that's why it makes sense this definition I want to show you one one remark note that when when you take something and say this is the cosine of an angle it better be between negative 1 and 1 do you agree cosines of angles are between negative 1 and 1 otherwise you can't plug it into the arc arc cosine functions it's not well-defined okay and this number a priori there's no statement that that the inner product is a number between negative 1 and 1 right but note that by cushy shvarts what did Koshi shvarts say it said precisely that in absolute value the the numerator here is less than or equal to the denominator that was Koshi shvarts that by Cauchy Schwarz u v is less than or equal you norm the norm that was precisely Cauchy Schwarz and therefore cosine of alpha an absolute value is indeed less than or equal to 1 the absolute value of this thing is indeed less than or equal to 1 and we can take our cosine it's well defined good ok so this is something that's worth mentioning ok so what I want to convince you is that these notions really coincide we now defined angles we defined angles on abstract vector spaces ok I want to show you that on our to this definition really coincides with how we start from angles and that boils down to showing the this showing that if we define this dot product we really get the inner product right do you agree is everybody following the line of thought here ok so let's show it let's show that if we define it like this we really get this so this is really just the inner product and therefore it makes sense to divide by this and define that to be the angle okay so here's our - we're gonna need a bit terminology so this is our - let's call it X Y ok so a vector of length 1 in the direction of X sorry a vector of length 1 in the direction of X is denoted by I hat it's a vector of length 1 in the direction of X ok we're on this board please vector of length one in the direction of Y is denoted by J hat these are vectors of length one in the directions of the axes okay this is geometric this is the good-old are two that we know okay do you do you agree that any vector a vector U is given by whatever it's u 1 component is times I plus whatever it's u 2 component is times J right if you take a general vector take a general vector may be let's take a general color like blue so take a general vector here here is a vector call it u its you one component it's you one component is this right and it's u 2 component is this and what is U its u 1 times this little guy which is precisely this vector plus u 2 times this guy which is this vector and edit so this would be let's take another general color like red this would be if this is I it's a unit vector in the direction of the x axis then this red vector is precisely u 1 times I right if u 1 is 5 then this is 5 I do agree and this vector which I can draw here or here doesn't matter it's the same vector same direction same length this vector is u 2 times J and their sum is you do you agree good okay likewise V is V 1 I plus v2 j right good okay now what is I dot I when I do this I just can't help myself from doing this sorry what is I dot I I dot I by definition what is the dot product is the length of I times the length of the other vector which is still I times cosine of the angle between them what's the angle between I in itself zero what's cosine of 0 1 so this is 1 this is 1 and this is 1 it's unit vectors it's vectors of length 1 therefore this is 1 do you agree ok what is J dot J also one do you agree for the same reason what is I dot J I thought J is going to be the length of I the length of J these are both ones but I and J what's the angle between them 90 degrees so we have cosine of PI over 2 and what's cosine of PI over 2 0 so this is going to produce 0 and likewise J dot I is 0 do you agree ok so what is u dot V u dot V I want to calculate this and show that what I get is U 1 V 1 plus u 2 V 2 that's the claim right that it's precisely the regular inner product so what is u dot V so I'm going to write it's u 1 I plus u 2 J dot V 1 I plus v2 j okay now formally I need to show that this that there's uh that I can distribute here okay that I can the products of each one separately and that's not a hard exercise okay you need to show it it's rather trivial from the definition of what the dot product is okay but what I get is what I get u1 v1 these are scalars times I dot I do you agree plus u1 v2 I dot J plus u2 v1 J dot I plus u2 v2 J dot J good and I dot I is 1 I dot J and J dot i are 0 and this is 1 again so what I get is precisely u1 v1 plus u2 v2 okay so what was originally defined what was originally defined in terms of lengths and angles translate to something that is defined purely algebraically in terms of the coordinates terms of the components okay and this is precisely the way we defined the standard inner product of U and V in r2 good okay so back to what this was trying to say what this was trying to say is that this dot product over here is really the same thing which is defined geometrically from knowing what angles are is really the same thing as the inner product which was defined algebraically okay they're the same thing on r2 therefore on r2 if we define angles like this we really get the angles we started with okay and therefore it makes sense to define angles like this for other spaces r4 r5 polynomials matrices fine okay that's the statement so now we know what angles are okay let's discuss what distances are ok so let's summarize this statement maybe add here that what about angles we define we define cosine of alpha to be the inner product of U and V divided by the product of the norms and this agrees just like here it gives the familiar lengths of a vector in Euclidean space and we get the familiar angles the regular notion of Engl in r2 but we can define this in general okay so what I want to discuss now is distances but since this is we have this fire alarm on this is a good place to stop and hope that they fix it and then we will discuss distances in a separate click so we have the notions of length and angles we need distances oh there it stopped good let's continue questions is it clear ok so we know what length is length is just the norm on r2 it agrees with the regular length that we know we know how to define angles once we have a norm and an inner product in general we know how to define angles on r2 it agrees with regular angle measurements that we know how do we define distance okay so let's continue so now I want to define distances III think that it was it was worth the time to discuss this dot product this thing that I'm erasing right now the dot product it was worth the time to discuss it because otherwise you know you may think wow what is that mysterious dot product that I've seen or will see in physics and how does it relate to the inner product it's the same thing it is the inner product it's all it's also called a scalar product you'll see that terminology as well it's the same thing it's the standard inner product on regular Euclidean r2 or r3 okay but there are other inner products in algebra you can impose different inner products giving rise to different norms and in fact you can have different norms that don't even come from inner products yeah you can have completely different norms that are not norms that come from inner products and each of these gives a completely different structure on your vector space and some of these structures are very interesting and they're not the regular Euclidean structures okay and you'll have to wait for future courses to see them okay but this is a fast topic fast topic and very very interesting very important you can you okay I'm getting carried away here okay what about distances what about distances distance we know length we know angles what about distance so let's define define the distance between two vectors in general completely general the distance between two matrices to be the norm of U minus V that's how I'm gonna define distance in a general inner product space I take the norm make it into a normed space and define the distance as the norm of the difference okay clean clean this is the regular distance between two vectors in r2 let's show that if we actually said it okay so this is the same as this is the same as the root of the inner product of U minus V and u minus V right that's that's what it is that's how we define the norm right so this coincides with good old distance with the familiar distance regular distance in r2 that we know okay in general its general but in r2 it coincides with the familiar distance let me just remind you what is the distance between U and V we set it not too long ago the last lesson so if this is you and this is V two vectors in r2 the distance between remember we we identify the vectors with points at their tips right the point u1 v1 is this point in r2 which is the the tip of that arrow right and the distance these two points is precisely the length of this vector right and we already agreed that this measures length right this measures length the norm and this is well if you want to be consistent with what we had there then we should call this one V and this one U and here in blue is U minus V that's this blue arrow and the length of U minus V is precisely the distance between U and V right okay so this coincides with the regular distance in r2 everybody with me okay by the way it coincides with the regular distance in R as well how do we what is the distance between two points on the real line the absolute value of the difference also in R in art itself right the distance between x and y is just the absolute value of x minus y which is precisely what you would get there right because the inner product on R is just the product right and therefore the the the norm is just X minus y times X minus y square root it's absolute value of X minus y do you agree okay so this fits in with what we know and therefore it makes sense to generalize in that direction okay so this thing this distance formula that we defined I want to remark just like we did for the norm we said hey here's a norm and then we said wait this is not just a norm this is indeed a norm by the definition of what a norm is I want to remark that for this as well this is indeed a distance function and the formal terminology for a distance function is called the metric okay this thing is indeed a metric space V I want to prove it I want to say what it means they're gonna be four axioms and I want to show that they holds not very difficult okay so let's do that let's erase this so ah let's see let's say it like this D U V defined by the norm of U minus V is a metric on V so what does it mean to be a metric by definition a metric satisfies four axioms and you should think okay suppose something is a function that measures distances in operation this is again an operation on two guys right it takes in two vectors U and V and gives out a number and that number is the distance between these two guys what would I expect from the distance between two vectors so or the distance between two points okay what is reasonable requirements reasonable reasonable things to impose on something if you want to call it distance okay so for example or not for example its list all of them first of all the distance between any two elements is going to be non-negative right you don't non-negative real you don't want to say that the distance between two numbers is negative five doesn't make sense right but I'm saying doesn't make sense I'm saying naively doesn't make sense everything can be generalized to two realms of roof okay but where is simple-minded where naive okay distance should satisfy this do you agree okay another thing you would expect is that if the distance between two points is zero they better be the same point if in it only if u and V are just the same okay another thing you would expect is that the distance between U and V probably the same be the same as the distance between V and u right would be weird to think that the distance between Moscow and New York is different than the distance between New York and Moscow depends which way you go that's true and depends on the trade winds as well on your speed but not the distance but but if you think of the trade winds just a couple of days ago of a British Airways flight broke the the non concorde record for flying between new york and london okay that because of very very high trade winds that reached 300 kilometers an hour or something like that that the flight was at an average speed of I don't know almost 1,200 kilometers per hour which is just on the brick of the speed of sound okay and that I think they made it and I know just a bit over five hours from New York to London but on the other hand those flying to the states had kind of a long flight because they had to fight those winds anyway sorry sorry distance between two points would be the same doesn't matter in which direction and the fourth axiom is again the triangle inequality the distance between U and W is less than or equal if you want to go from one point to another the shortest distance is just go and it would be longer if you go via a third point right so it's less than or equal to D u V Plus D V W this is the triangle inequality again okay these are the four axiom they make sense for something called a distance do you agree let's see that this thing which is defined via norm which is defined via an inner product indeed satisfies these four axioms okay so let's do it it's it's simple it's really simple in this case yeah almost you just have to be careful to be careful in what you do okay so so the norm the norm satisfies so this would follow directly for from the triangle let's show it okay so for example let's show this one what is duw so let's prove prove some stuff in in and we don't have the properties of the norm on the board anymore so let's let's just do it what is the distance between U and W the distance between U and W by definition is the norm of U minus W do you agree okay now the norm of U minus W I can write as the norm of U minus V plus V minus W I just added and subtracted V that's like doing nothing okay and now I can use the triangle inequality the triangle inequality for norms that I already know is satisfied I checked it right that the triangle in I checked it I showed it from cauchy-schwarz which we didn't prove but we discussed it let's see so this is in turn less than or equal to this is the the norm of a sum u minus V plus V minus W is less than or equal to the sum of the norms so this is the triangle inequality for Laurence and this in turn equals D u w + d VW sorry what did I write d u V Plus D V W so that's precisely the triangle inequality right now how would you prove three for example this is the proof of a four how would you prove three you have to say that the norm of U minus V and the norm of V minus U is the same thing that was not one of the axioms for the norm right the norm only addressed what is the norm of a single element so you need to prove it okay you need to prove it in and the way we're gonna prove it is we're gonna reduce back to two how the norm was defined via an inner product where is it on this board let's look here a second here this is what the norm is right the norm of U minus V is the inner product of U minus V times u minus V root okay so let's prove it so proof of of three let's prove three proof ready so what is d UV d u v by the definition using the inner product is the norm of U minus V which is the root of the inner product of U minus V times u minus V do you agree this equals now I can start using linearity of the inner product ok so this equals u u inner product minus u v inner product I can pull the - out - V u inner product plus minus times minus V V that's the linearity properties and all of this has to live inside a square root let's write to the power 1/2 ok I have space limitations here so I have to be careful clear these are linearity properties of the inner product and now I can rearrange the orders the order these are numbers so I can write this as V V that's this one I wrote it first - v u - u v + u u root to the power 1/2 and this is precisely the root of V minus u times V minus u right v v- v u- u v + u u and this is the distance between V and u good you prolly can do it even without resorting back to the inner product I'm probably just saying too much but never mind this is good good good practice okay good so it's indeed a metric it's not just an abstract thing we defined for R 2 or 4 we define something a general concept called a metric that gives a structure of distances between two points in a vector space okay there's an entire course in mathematics called metric spaces an entire course ok so this is really just introduction to some touchy yes touching vast topics which give rise to a lot of mathematics and beyond we're really just touching it ok good so let's summarize let's summarize everything we we just said let's do it maybe here so conclusion of all this discussion any any inner product space which is a vector space that comes equipped with an inner product that has some inner product function is a normed space and the metric space where the norm the inner product is this thing some UV the norm is defined by the root of the inner product of the vector with itself and the metric is defined by the UV equals the norm of U minus V ok so any inner product space this is the inner product is a normed space where this is the norm and a metric space where this is how we define the metric and these notions these notions give rise to geometry ok and these notions these structures provide geometry on any inner product space V okay provide distances provide lengths and provide angles by the way we define it the inner product divided by the product of the norms okay and maybe a remark remark again things that I've mentioned but I want to say and right there are many there are many norms and metrics some very interesting ones even if you opened the Wikipedia entry for what is a metric you would find different metrics on on spaces some are very interesting like there's one named after the the train system okay you can okay I'll leave it to you to read some stuff and get interested and they don't necessary don't necessarily not every metric corresponds to a norm not every norm corresponds to an inner product okay we we built a specific ordering we took an inner product constructed from Anna Norman from data metric that agrees on r2 and r3 with Euclidean space structure but there are many other norms and metrics that are not related to inner products okay and this is a vast topic there are many norms and metrics not let's let's even say that even on r2 even on r2 there are norms that don't that aren't defined via inner products that in fact don't they can't be defined that can't be defined via inner products okay so this is a very rich very rich area and and and we're really just touching it really just touching it duh there is still we still have one lesson left and there's still one more thing that I want to do and that's relating all of this stuff to the things that we did so here we kind of took a detour and and everything we said no no longer had anything to do with linear maps with Basie's with matrix representations with eigenvalues and eigen vectors and we discussed a lot of things in this course and this is as if completely unrelated right this is just taking a vector space and providing richer structure on it using this notion of the inner product ok what I want to discuss the last topic that we're going to discuss that is how this inner product and norm how do they relate to things we already know so for example if you think for a minute think of r3 r3 has a basis a basis of three elements the standard basis right those three elements 1 0 0 0 1 0 and 0 0 1 now that we have all this geometric structure they're all if you think of them geometrically they look like this right there the their unit vectors in the directions of the axis right there all of length one there all of length one and they're all perpendicular to each other okay so that's a special basis that has this property that all its elements are of length 1 and perpendicular that's called an orthonormal basis now that we have these notions we can define it and we can see what it's good for so that's what we're gonna do next time and that's gonna end our our course this semester

Channel: Technion

Views: 19,197

Rating: 4.8947368 out of 5

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

Id: 1Vj67d6qIfI

Length: 53min 46sec (3226 seconds)

Published: Mon Nov 30 2015