71 - Norm

okay so people were laughing here this is indeed the complete title of this lesson norm but I want to first take you back a couple of boards to what we know so we know what an inner product space is an inner product space is a vector space over the real are the complex numbers and remember that most of what we're gonna do here is going to be over the real numbers for simplicity that has a notion of an inner product an inner product it's something that takes in two vectors and produces a scalar and satisfies three axioms symmetry or conjugates image over the complex numbers linearity in the first component which in translates into linearity in the second component except for alpha coming out with a bar if we're over the complex numbers from the second component and this positive definiteness property property number three okay that's an inner product we've seen examples for matrices for example the inner product of a and B is the trace of a B transpose that's an inner product on a vector space of square matrices for RN for example the inner product is just the sum of the products of the components the inner product of V and U is the sum you I VI okay what we haven't seen and that's what we want to discuss now is how does this notion of an inner product of which we've seen examples how does it produce geometry how from this do we get length distances and angles that's geometry okay and that's what we want to do now so the title is norm and that's the first and and maybe the key concept that translates inner products into geometry and I want to start with a definition definition let viii and remember we said this is not accurate because if I want to say that V is an inner product space I have to specify what the inner product is but I'm just going to call it V let V be an inner product space meaning that it has an inner product and the inner product is that triangular bracket thing okay define define for every V in V for every vector the what we're gonna call the norm of V denoted by a double bar on each side the Norma theme is gonna be by definition the root of the inner product of V with itself that's the definition and the notation okay V is called the norm the norm of V and big V is called a normed space it has an order okay clear so what is this norm thing what does it mean what does it do so first of all let's throw in here another word norm is kind of the formal name that we call it but in fact what it is is just length and we'll sometimes even use the word length which is a bit less formal so instead of norm we can just say length that's the length of okay so remarks remarks so remark number one maybe let's start here remarks there's a lot to say before this can really make sense so the first remark is that this is well-defined what do I mean well-defined I mean that this thing really compiles it makes sense and what does it need to be in order to make sense well I'm taking a vector good I'm defining its norm to be the rift of the inner product of the vector with itself inner product exists because I'm assuming that I'm in an inner product space and but I'm dating the root of the inner product and the inner product in general could be any number it could even be a complex number what's the root but according to axiom three the inner product of a vector with itself is always a non-negative real number that's axiom three and therefore this is well-defined even over C that's what we proved okay we proved it in the previous lesson it's not just in the axiom we proved it okay so if you forgot you have to no no no no even if you're over C the inner product of a vector with itself is always real and not just real non-negative okay so remarks by axiom three three we know by axiom three of the inner product of the inner product clear we know that V V is always non-negative and we mentioned and I'm saying it again whenever I write non negative for a complex number it implicitly says that this is a real number there is no notion of order on the complex numbers okay so therefore this thing is well-defined and therefore the norm of V being the root of something is well-defined clear the second remark that I want to meet is that this really gives the length of a vector it really gives the length of a vector in the regular sense of a length of a vector in r2 or r3 that I want to show you in a lot of detail but I want to postpone it until we say a few more remarks okay so this is our remark that's saying a promo for a bigger discussion that's gonna follow shortly okay the norm coincides coincides with the usual length the usual Euclidean Euclidean Lyn's okay why we need to show okay this will follow and in fact from this norm we're gonna be able to define angles okay which are gonna agree with the usual Euclidean angles when we're working in r2 or r3 okay with the usual Euclidean length in r2 or r3 okay but this definition is abstract we can define it for any inner product space okay but when we're working with the specific inner product space which is r2 or r3 with standard inner product we actually get back the Euclidean space structure okay that's this remark clear okay remark number three this is this is something that we have to discuss in a bit more lens this concept which I just defined is indeed a norm so what do I mean I just said that this is a norm what do I mean that this it is indeed in norm so there's a more general concept of what is a norm and when I'm saying what is a norm it means that a norm is an operation and here it's an operation defined on a single vector right it's not an operation taking in two vectors it's an operation defined on a single vector giving a number giving a scalar which satisfies certain axioms okay so let me write those down so V is indeed a norm ie satisfies the following three axioms being a norm is means that you need to satisfy three axioms not of the inner product but over the norm okay the following three so here are the axioms of being the norm and what I want you to keep in mind what I want you to keep in mind is that the three axioms that I'm gonna write should be the most reasonable things you would demand from something that gives you length okay if it's gonna be length it's gonna satisfy these three things and those are the axioms of a norm okay so the first is that if you take a scalar multiple okay if you take suppose let's think of vectors and think of the norm as giving the length if you take three times the vector what do you expect to get is the norm of three times the vector three times the norm right three times the length so that's the statement equals alpha V but what if you take almost I'm gonna fix this in a minute what happens if you take negative three times the vector what is the length if you take negative three times the vector you're just reversing its direction right what do you expect the length of negative three times the vector to be three times the norm not negative 3 times the North right so therefore we have here this the absolute value of alpha alpha is a scalar okay so this is the first axiom the second axiom I should have probably written this down is the first axiom the norm is always a non-negative number you don't expect to have a vector of length minus four okay for every V and the norm equals zero if and only if V is the zero vector right make sense in the third axiom of a norm is what's called the triangle inequality okay if you go if you measure the length of U plus V it's always less than or equal to the length of U plus the length of e so this is the triangle inequality okay and you should think of a triangle if you measure the DA if you think of a triangle and you measure an edge it should be the length of the of a single edge should be shorter than taking the lengths of the other two in addition you could think of addition is just like we did addition with arrows in the complex numbers okay so does this compile do these three axioms compile so a function that takes a vector and produces a number is called a norm if it satisfies these three properties claim clean the function that takes a vector and produces a number a scalar by taking the root of the inner product of the vector with itself is indeed a norm satisfies this three properties these three properties let's prove this okay so proof so remember this maybe I should add V which is the root of V comma V this specific function is indeed a norm okay that's the claim so I need to prove three things so let's start with number two number two is really straightforward really trivial right because what we're claiming is that the norm is always non-negative but this is a non-negative number we're taking its root it's always non-negative right we're claiming that the norm is 0 if and only if we're looking at the zero vector and again we know that this number is zero if and only if we're looking at the zero vector that's X that's part of axiom three of the inner product let's take a peek here it is it's zero if and only if we're looking at the zero vector right so do you agree that axiom number two for the norm is trivially satisfied good okay so two follows immediately from axiom three of the inner product do you agree okay what about number one what about number one so I want to prove it so let's take alpha V and calculate its norm its norm that arises from the inner product this norm okay so what is it by definition it's the root of alpha V comma alpha V that's how we define the norm of alpha v right now I have an alpha here in the first component and alpha in the second component I know how to pull them out right from the first component alpha comes out just as an alpha that's the linearity of the first component in the inner product from the second component alpha comes out as an alpha bar right so what we get here is the root of alpha alpha bar v comma V do you agree what is alpha alpha bar these are now complex scalars numbers what is alpha alpha bar it's the modulus squared of alpha remember that times V V right the root of a product this can come out as just the modulus of alpha so I get the modulus of alpha times the root of V V which is just the modulus of alpha times the norm that proves one right using the axioms we know for the inner product we can prove that this specific norm is indeed an order to satisfies these axioms okay now three is tricky three is usually for any norm when you want to show that something is a norm this is where you usually meet complications okay it's not a trivial fact okay so three is tricky what do I mean by tricky if you just try too straightforward to provide a straightforward proof you're gonna see that doesn't work okay you need some some more subtle observations more and more so you need something more okay and I'm in fact not gonna give a complete proof of three I'm gonna put down remark number four which is a certain inequality that is satisfied for this okay for a norm coming from an inner product and then I'm gonna prove three using that remark okay but I'm not going to prove that remark okay so there's there it's not a complete proof of this statement okay so three is tricky we will see how it follows from and I'm gonna write here CS c s stands for cauchy-schwarz which are names of two people co she you know from calculus he's a famous calculus guy but he's a famous mathematician and Schwarz is another one I'd remember if we did we okay I don't remember but we didn't there's the famous Schwarz lemma and there's anyway so CS is Cauchy Schwarz which is going to be our remark number four okay so we're in remarks concerning the norm so let's write what we mark number four is the mark number four is called the Cauchy Schwarz I think you spell it without a team like this and sometimes especially in the former Soviet Union it was attributed to another mathematician called Booya coughs key so I'll add him as well the Cauchy Schwarz Boone yakov ski inequality says the following if you take the inner product of two vectors this is a number a scaler take its absolute value or in general it's its modulus it can be a complex number this is always less than or equal to the norm of U times the norm of V okay this innocent-looking inequality which is not really hard to prove but I don't want to do it now it takes there's it takes some time in an effort not too much okay but this is arguably one of the most important inequalities in mathematics believe it or not okay it shows up in many many forms and disguises in many many contexts okay and this is one form of writing it the Cauchy Schwarz inequality and it holds for a norm stemming from an inner product okay I'm not going to prove it what I do want to do as an exercise in using Cauchy Schwarz is to prove the triangle inequality axiom for the norm from this ok so let's do that so let's show let's show that the triangle inequality follows from Cauchy Schwarz ok so remember so don't get confused now let's remember what we're trying to do and what we need to show so we have an inner product space so we know what the inner product is we define the norm to be the root of the inner product of V times V ok now we want to show that the norm that we defined satisfies the triangle inequality namely the norm of U plus V is less than or equal to the norm of U plus the norm of V ok so remember that when we work once we expand this notion of the norm there are going to be roots there and the roots are kind of in the way in it's easier to work without them so you'll often see that when you want to proof prove statements about the norm you prove the statements on the square of the norm okay so what I'm going to do is I'm going to take suppose we have two vectors call them X and y gonna take the norm of X plus y and square it calculate calculate calculate calculate and show that it's less than or equal to the norm of X plus the norm of Y squared and therefore by taking roots I get the triangle inequality okay so this is just for not having these roots all along the way so what is X plus y squared it's the inner product of X plus y with X plus y right the root of it but I'm squaring so the root is gone do you agree okay now by the linearity properties of the inner product I know that I can break this into four products where each one meets each one okay so I can write this as X X plus XY plus y X plus y Y and I can't combine these into a single one unless I know I'm working over the real numbers but this is general this is true over the complex numbers and therefore these two are not the same right okay so this is from linearity and with respect to addition which I know holds for the first term as well as for the second term right with respect to addition now I'm gonna replace so what is X X by the way X X is the norm of x squared right because the root of X X is the norm of X and here I'm gonna leave this as X Y and replace this one by X Y bar this I can do right plus the norm of Y squared do you agree good now what this is a number this is a complex number this is its conjugate what do I know about a number plus its conjugate not just the real twice the real trice the real so it's plus two times the real part of X Y do you agree good okay now remember that the real part of a complex number as well as the imaginary part of a complex number are always less than or equal to the modulus of the complex number remember so this is less than or equal to the norm of x squared plus two x instead of the real part I'm just going to take the length of this month number the modulus plus the norm of Y squared good and now I'm gonna use cauchy-schwarz so this is less than or equal to and here I'm using koshi Schwarz this is less than or equal to x squared plus two times X y plus y squared right and these are numbers a squared plus 2 AP plus B squared which is the same as X plus y squared and now if you look at the booth ends this is less than or equal to this therefore the roots of them are also equal right and that's precisely what I wanted to show that's the triangle inequality good so the triangle inequality follows nothing tricky but still not completely just you know a trivial observation from this property the Cauchy Schwarz inequality which requires proofing we didn't prove it ok so there's there's it's not trivial stuff the root of what can be negative know the norm the norm is always a real number that we proved right that was axiom number 2 the norm is a non-negative real number right ok because what was it follows from axiom 3 that it's the the root of a real number ok good ok so that was remarked number 4 and that's that that completes my my remarks about the norm ok so now we know that this norm is indeed a norm okay it's satisfies these properties what we still don't know and that's that's why we're here is why is this thing precisely what we want for for geometry why is this why does this coincide at least in R 2 and R 3 with the regular Euclidean structure that we that we know that we started with right so let's see that so I'm debating maybe will will yeah maybe maybe let's break this year instead of having a very very long lesson and post thrown into the next lesson so that this is really abstract as we haven't seen any examples right but the examples are gonna be precisely in R 2 and R 3 where we're gonna see how this gives rise to the Euclidean structure but I want to do it in a separate clip so are there any more questions about this ok we know the notion of a norm the notion of a norm when we have an inner product space it gives rise to a norm ok we want to see this remark number 2 here why this norm coincides with the usual Euclidean length and how it gives rise to angles and distances as well ok questions ok so let's stop this one here and continue in a separate one

Channel: Technion

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

Id: HgqIWKL7L58

Length: 30min 24sec (1824 seconds)

Published: Mon Nov 30 2015