21 - Vector spaces

in the previous video we discussed the vector space RN and we we saw that it has a ten properties which we which we focused our attention on properties of the structure of RN in terms of addition of vectors and in terms of multiplying a vector by a scalar a scalar is just an element from are a number okay and here is the list of these properties I left it here on the board and what we want to do now is define a totally abstract notion of a vector space which is going to be a structure a structure that is comprised of the following four things there's going to be a set of elements called the vectors but now they're abstract you can't think anymore of nth well you can think of those as examples but they're not necessarily going to be n tuples of numbers okay so those are the vectors the elements there's a field from which your scalars are coming from it can be any field q are the complex numbers C or any finite field any field we're going to denote it by F and they're going to be two operations how you add vectors and how you multiply a vector times the scalar how you scale a vector okay and they're going to be ten axioms okay we're going to require that this structure satisfies all these ten properties okay I'm not going to erase here I'm going to rewrite them because I want to UM well I want to rewrite them and the reason is that that there's a more compact way of doing it okay and you have to think for a minute look at properties one three four five and six look at properties one through six not including two do you agree that those five properties the first six without to only involve the addition do you see that okay the only involve the addition and these properties are are precisely the same properties that addition satisfies in a field do you see that right and in fact in fact they're precisely the five properties of what we called a group a group is a set of elements that satisfies precisely properties one through five not necessarily with three if it satisfies three then it's what we called an abelian group or a commutative group okay so if you happen to forget the notion of a group I'm just going to tell you what it is right now and you can look back at the I think it was in one of the very first videos of this course what is a group okay but a group is merely a set that satisfies properties one four five and six and a commutative group is one that satisfies three as well okay so there's one operation addition it's closed under addition addition is commutative that's a commutative group addition is associative that's property number four there's an additive identity that's property number five and there are additive inverses for every element that's property number six good so this is what I'm going to write now I'm going to write the definition of a vector space okay V is called a vector space / that's the term we usually use over a field F if is going to be our field if V is n a billion group and I'm going to remind you now an abelian group what this means so abelian group means that so this is in parentheses it's just a reminder a plus operation satisfying satisfying closure addition is closed so you add two things you get another element right closure this is a ello not a B no this is a CL not a D well satisfying closure associativity commutativity that's being abelian additive inverse well there's just additive so plus inverse and zero okay so so is it is it clear what I'm indicating by this okay it's really properties one three four five and six of that we saw in our n okay so it's an abelian group with an additional operation in operation of multiplication multiplication multiplication can you believe that in Hebrew the the word for multiplication is three letters multiplication by scalars by a scalar a a scalar is an element of F okay so there's an additional operation of multiplication by a scalar such that and now all I have to list is the five properties of multiplication by a scalar okay so the first property such that one and I'm going to list it abstractly remember it's no longer RN and it's no longer are so alpha times V belongs to big V big V is our vector space for every alpha in F and for every V in V make sure your little fees and big fees are little and big otherwise you'll get very confused now okay so what is this property number one exactly it's the closure right this is saying that multiplication of a vector by a scalar is closed you get again a vector okay property number two one times V equals V for every V in V good one is well-defined it's a field a field has a multiplicative identity that's this one and when you multiply a vector by one you get the vector itself back and remember that these vectors are no longer an tuples these could be matrices or functions or solutions to some system or whatnot okay three so this is these are abstract axioms in okay this is how we define a vector space three alpha times V plus W and note that being elements of the vector space is usually encoded in denoting them by little letters of the range uvw ok so I'm not going to add these arrows on them the arrows are more reserved for for tuples from from r2 and r3 that we're used to seeing okay so this could be a function okay so I don't want to put an arrow on it okay so alpha V plus W equals alpha v plus alpha W for alpha plus beta V equals alpha v plus beta v + v alpha beta V equals alpha beta V and these three are for every alpha beta in the field and for every V and W in the space good so this is the abstract definition of a vector space it has the structure has four ingredients the space itself the elements those are the little V's in the little w's the field the alphas and the beta's okay the plus operation satisfying five axioms which is here encoded in one word saying that it's an abelian group okay and the multiplication by a scalar operation which satisfies these five axioms okay and I'm just going to mention it again we mentioned it before there could be more structure there could be more axioms but that would yield a things that are more than just vector spaces so there's a notion of a normed space or a metric space or an inner product space and all these things involve further structure that is not relevant to this okay it goes beyond so we're only looking at these five properties plus five properties okay only in addition and scaling and scalar multiplication good okay so let's start writing some examples okay so examples of vector spaces so examples so example number one is our n we know it's an example because we studied studied it separately and reveal the properties from RN right but if you want to claim that something is an example what do you have to do just like when you want to claim that something is a field what do you have to do you have to see that all the axioms are satisfied you have to check them one by one if they're satisfied good it's a vector space okay so this is the first example let's look at another example look at all matrices matrices of the same size of a fixed size so M by n matrices let's say Oh are so this is notation that we use so these are only matrices of size M by n M rows and columns with all the entries taken from the fields are okay I claim that this is a vector space okay so let's I'm not going to write everything because it's going to take too long and it's really easy to to see it in your head but if you need to play around with it a bit and get the feeling better do it don't don't be lazy to get your hands dirty okay but let's look at the properties the all ten axioms and see that it's indeed a vector space okay so first of all what are the elements the elements are M by n matrices those are the little V's okay so the little V's are no longer n tuples the little V's are now M by n matrices if it's if it's difficult for you to think M by n 3 think 2 by 3 okay what are the scalars well are if we're looking at matrices where the entries come from R then we're usually going to take the field to be R okay so those are the scalars what what's the operation I did what's the addition operation right addition of matrices right and what's the scalar multiplication multiplying a matrix by a scalar right so let's look at all da let's look at the definition there's a field F there's the vector space V these are M by n matrices this is RNA is it an abelian group so is there a plus operation yes there is a plus operation is it closed when you add an M by n matrix to another M by n matrix do you get an M by n matrix yes the operation is closed okay and it's not trivial than any op that any operation is closed if you would for example it's not relevant to this but if you would multiply two matrices you often alter the sizes write an M by n matrix you can only multiply by an N by L matrix and you get an M by L matrix right so not every operation is automatically closed do you agree okay but addition of me Juicy's of the same size as closed put a check what about associativity is addition of matrices associative it is right because it's done element wise is addition of matrices commutative is it true that a plus B and now think of the elements as capital A because it's a matrix and capital B a plus B is the same as B plus a right it's not true for matrix multiplication but that's not what we're talking about we're only talking about addition right okay is there an additive inverse to every matrix well first of all is there a zero matrix yeah there is all the entries are zero and is there an additive inverse to every matrix given a matrix can you find another matrix such you went when you add them you get the zero matrix yes you just flip the sign of all the entries do you agree okay so matrices of the same size a fixed size M by n are an abelian group okay is there a product of multiplication of a matrix by a scalar yes there is when you multiply a matrix by a scalar do you again get a matrix of the same size yes you do so it's closed when you multiply the scalar one this is not the scalar matrix it's just the number one times a matrix do you get back the same matrix one time's the matrix is the same matrix good and do these three properties of all kinds of associativity n' and distributivity laws do they hold and the answer is yes we know that we mentioned that as properties of addition of matrices and scalar multiples of matrices previously right if you do five times the matrix a plus the matrix B it's five times eight plus five times B and you can verify that again element twice to be convinced okay and and likewise four four and five okay so M by n matrices are a vector space good okay before we give more examples let's list some more properties okay so more examples more examples soon more properties so so what do I mean by properties well if I look at our end I can just say hey it satisfies also that dot dot okay but here now I'm holding in my hand something completely abstract all I know is that it satisfies these five axioms plus being an abelian group which is another five axioms can I say that it satisfies more things and the answer is yes but I have to prove it okay I can't check it on examples I have to prove it okay so it's going to be called a theorem it's not going to be a difficult theorem but it requires proof so let V be a vector space let's abbreviate vector space like this over and I have to mention what the field is so the field is going to be F then so the first property is going to be that zero times V is zero for every V in V now there's something tricky here what is this zero right this is the scalar zero because it's multiplying a vector it can only be the scalar zero there's no other notion of multiplication right what is this zero this is the vector zero right we're not putting arrows but we should see that these two zeros are completely different this could be the zero five by six matrix whereas this is just the number zero okay you could it would make life a lot well not harder but slower because I'm going to have tons of arrows over everything and it's not necessary okay sometimes things just like you don't put a dot here and you know that this is multiplication sometimes you get things that are not implicit explicitly written but their implicit and they're easy to to know them to understand into and you do just need to to specifically emphasize them okay one hour do this use of the arrow just to help understand like it was unnecessary earlier hey what do waitresses it was not yeah you could get along without arrows at all some people do that okay in the world of vectors in terms of vectors of RN it's it's very common to use arrows of different types some some textbooks just do the vectors in bold there are many many different ways of indicating that but really we'll see well at the beginning it's going to take getting used to like any new thing but you'll see that when you see something like this it would be obvious that this is the scalar 0 this is the vector 0 okay and by the way if it still confuses you it's fine put a little arrow here to remind you that's fine okay okay now it looks obvious right you multiply 0 times a vector of course you get 0 what what else are you gonna get but it's not one of the axioms ok so the fact that it's obvious here and it's obvious here doesn't make it true you need to prove it and what can you use in order to prove it only the axioms right that's tada those are the rules of the game we're playing in in abstract is sizing stuff okay here's another property alpha times 0 equals 0 for every alpha in F if you take the zero vector multiplied by Y yet whatever you're going to get the zero vector back okay looks very convincing but that's not a proof you need to prove it property number three alpha v equals zero implies implies that alpha is 0 or V is 0 so if you take the product of a scalar times a vector and you get the 0 vector either alpha was zero this is the zero scalar this is zero in F or V is zero this is the zero vector and this is the letter the letter O in English good ok let's prove I'm not going to prove all of them maybe I'll pro leave you leave one for you to prove on your own and I'll do two and three just so you get the idea so I think by now we can erase these properties of RN because we rewrote them abstractly and now we have an extra board okay so let's do let's do some proofs so proof of two for example so we want to show that alpha times zero is zero and our hands are tied we can only use the axioms we've been playing this get these games when proving things about fields right we're already not completely puzzled by doing things like this and still in the beginning when you see stuff like this it looks kind of what's going on here what why are we doing this isn't it obvious that that multiplying something by zero gives zero no it's not okay you need to prove it good everybody happy smile thank you okay so alpha times zero do you agree that this equals alpha times zero plus zero why you can't just agree because it looks convincing you have to show why this follows from the axioms because zero this is the vector zero has this property that whatever you add it to you get back the thing you added it to right so if you take zero and add a zero to it you get zero back do you agree okay that's the property of the zero in an abelian group okay the property of the additive identity okay so this holds now I'm going to use one of those distributivity properties number three in the definition this equals alpha times zero plus alpha times zero do you agree okay so this is again I'm concretely saying in each step what I'm using what I'm relying on in order for this to hold only by the definition okay good so now that I know this this implies that if I take alpha times zero and I add to it it's additive inverse and this additive inverse exists any any element as an additive inverse so maybe we need some more parentheses here it's plus minus alpha zero right every element has an additive inverse that's a property of an abelian group on the one hand what is an element plus its inverse zero do you agree on the other hand this equals and now I'm going to replace alpha times zero by two of them okay and I can do that because of what I showed here so I get alpha times zero plus alpha times 0 plus minus alpha times zero do you agree ok now I'm going to use the associativity law for addition so instead of writing like this I'm going to write alpha times zero plus alpha times 0 plus minus alpha times zero here I used associativity of addition which is part of being an abelian group part of being a group ok and what is this now zero it's an element plus it's additive inverse so this just equals alpha times zero plus zero and adding zero is like doing nothing that's again a property of being zero so you get alpha times zero so if you look at the two ends alpha times zero is zero ok we proved it now we know that it's true once we proved it we know that this is true for any vector space in the world once you think of and ones you haven't thought of yet clear it's a property ok and it's as valid as any of the axioms because we now proved it we can use it ok let's prove number three do you agree to this statement that this is not hard it's not difficult but it's a bit confusing at first it's not it's not something you're used to do in fact right you're not used to saying hey zero times something is zero I need to prove that okay because it's something new okay you're in your head you're always running examples of things you're used to but it's abstract it's gonna cover many many things that you're not used to yet okay so you have to prove it okay let's prove number three that this we did number three was look here that if we multiply multiply a scalar times a vector and we get zero then either the scalar or the vector where zero this we proved something very similar when we talked about fields saying that there are no zero divisors and feel that if you in a field if you multiply two elements and get zero one of the elements has has to be zero and the proof is going to be very similar if you look back at that proof almost the same okay just different because now this is a vector and this is a scalar but the idea is very very similar okay let's do it so let alpha times V equals zero does anybody remember how we did this okay so I'll remind you we first assumed that alpha is zero okay so if what we need to prove that either alpha is 0 or V is 0 if alpha is zero we're done da right well if alpha is zero then alpha is zero that's what we want to prove so we need to prove so saying that one of them is zero is saying okay suppose alpha is not zero let's prove that V is zero do you agree that that would prove it so if alpha is not zero then then and the reason I'm doing this the reason I'm separating this case is also also useful for what for what I'm going to right now remember that alpha is an element in the field okay if it's not zero it has a multiplicative inverse in the field it has a 1 over alpha or an alpha to the minus 1 okay then there exists alpha to the minus 1 in F there exist a multiplicative inverse of alpha do you agree ok such that and now we can write what is V on the one hand V is 1 times V that is one of the of the axioms that's axiom number 2 so we can use that instead of 1 we can replace 1 1 is the scalar 1 we can replace 1 by alpha minus 1 times alpha for any element in the field if you multiply it by its multiplicative inverse you get 1 do you agree we're just here using a property of the field not of the vector space now we're using the property of the vector space property number 5 that we could that this product here it's a product in the field and here it's a product in the vector space is associative so we can write this as alpha inverse times alpha v do you agree but what is alpha v alpha V is 0 that's the whole part that that's the assumption of what we're trying to prove so this is alpha inverse times 0 the 0 vector ok and what is alpha inverse x 0 y which axiom does it follow from very good it's not an axiom it's it's part two of the theorem which we just proved that any alpha times zero is zero so this equals zero by part two of the theorem that we just proved so if you again look at the end if alpha is not 0 then V has to be 0 nope I need to prove that either alpha is 0 or V is 0 if alpha is 0 we're done the neither alpha is 0 or V zero right if alpha is not zero then well we showed that V has to be zero do you agree it's logic it's not it's not basing on any you need to prove that either alpha is 0 or V is zero okay if alpha is zero you proved it well you can't say if alpha is zero we're done and if V is zero we're done because what if but none of them are then you didn't prove anything but that's not what I did I said if alpha is zero we're done if it's not zero and that's the complimenting case there are only these two options either alpha is zero or it's not if it's not then V is zero so in we exhausted all the possibilities and we showed that in each of the possibilities we get either alpha 0 or V 0 good ok so try to do try to do number 1 on your own and also try to do so here's a here's homework try to show for example that alpha times minus V minus V is the inverse of the vector V in the group in the in the vector space equals minus alpha times V minus alpha is the inverse of of the scalar alpha ok and this equals minus alpha V ok try to show it using only the axioms the definition plus what else can use right you can use the new three properties that we listed and proved ok try to do it it's really feels at the beginning like playing the game but it's a game you need to play by the rules ok good ok let's write more examples of vector spaces so more examples I promised you so let's do them and once again I'm not gonna for each example explicitly write all the ten axioms ok I'm gonna kind of wave my hands and and try to convince you that well they hold but if if you feel that it's too fast if you feel that you need to think about it some more I think that's probably what you should feel it's it's the natural thing at this stage you need to think about it some more you need to take each example and run it through the the definition and verify that indeed everything holds so here's another example we had two of them so let's continue the enumeration so example number three often denoted F to the N okay so these are just n tuples over F okay so we had R to the N RN but you could have for example a C to the N and topples okay column vectors where all the entries are complex numbers a plus IB okay so that's also going to be a vector space over the complex numbers for exactly the same reasons okay there's addition of complex numbers the addition is sorry there's addition of of column vectors where the entries are complex numbers just like addition of matrices where the entries are complex okay everything works you just have to verify it to think about it carefully one by one and be convinced okay yeah for any field F over F over the field F so in this course I said it several times in this course the the world of fields that we're going to make Yusef is rather dull we're usually going to only consider R and C the real numbers are the complex numbers those are going to be the F's that we're usually going to use okay but there are many many more fields and you may encounter them in further studies and when I say something and put an F it means that it would work just the same for any field even if you don't verify it right now okay okay number four so three and four are just expansions of 1 and to take matrices M by n matrices over a field F not necessarily over R again it's going to be a vector space and I'm not going to run through all the properties because it's the same statement ok but here are some things that maybe are new so for example what we denote by F with X in brackets ok this is notation for so let's write here M by n matrices over F where all the entries are from F and and it's it's I'm not writing it but it's implicit and it's well understood what the addition is what the scalar multiplication is right everything is clear we add matrices just like we add matrices okay so this is polynomials over F over F means that the world of scalars that we're using all the coefficients all the roots everything are from the field F ok so I'm claiming that this is a vector space why well let's think is there a notion of addition of polynomials yeah we know how to add polynomials when you add two polynomials do you get a polynomial yeah yeah good when you add two polynomials does it matter in which order you add them no it's commutative is there a zero polynomial yeah zero is the zero polynomial you add it to any other polynomial you get you get the polynomial right is there an inverse to a polynomial is there given a polynomial can you find another one such that when you add them you get zero yeah right you just put minuses on all the coefficients do you agree okay is there now we need scalar multiplication if I multiply a polynomial by a number do I get a polynomial again yeah I do if I um is it true do the distributivity law hold so for if I take three plus five times a polynomial is it the same as three times the polynomial plus five times the polynomial Yeah right is that is that is one that the scalar one times a polynomial does it give back the polynomial yeah so do you see that everything holds okay good let's put a 5/5 prime here and this is going to be or let's call it a remark remark before we continue this list mark there's also um notation so maybe let's leave the notation for a minute but polynomials one could ask suppose I take only polynomials of degree seven not all this is all the polynomials all of them suppose I take only ponit polynomials of a fixed degree of degree n only those is it a vector space nope so why not so you're thinking what's going to change what we just mentioned exactly all the properties and we saw that they hold so my claim is one of them is going to fail there's one property that's going to fail very good closure under addition you can take two polynomials of degree three suppose one starts with three X to the power three let's not use three all the time four X to the power three and the other one is negative four X to the power three there's some is no longer a polynomial of degree three do you agree that's another thing that fails zero is not a polynomial of degree n no you can't degrees of polynomials are well-defined zero is not a polynomial of degree n if n is is is let's say three or four or seventeen okay so some of the property is a V of being a vector space a ving a vector theta of being a vector space fail okay zero is not there and you can add two of them and get a polynomial of a different degree okay so closure for addition fields okay so this is not are not a vector space good clear okay ah-6 look at all the functions you know functions right you're you're studying them in calculus and you've met them before all real valued functions functions from R to R for example the set of all functions from R to R any function that you want it's a vector space there's addition of functions right over R there's addition of functions you add two functions you get a function you can add them in whatever way you want it's associative right good you can multiply a function by three you get a new function all the distributive laws all if you multiply a function by one you get back the function do you agree there's the zero function x equals zero right everybody good these are just polynomials here I'm taking all the functions e to the X sine X arc tan of X all the functions and many functions you've never thought of in your life and many functions that don't even have a formula okay any function you want okay all the functions only real valued that go from R to R do you agree that it's a vector space okay let's take it a step a step further um well I don't know if you've already encountered these notions in your calculus studies but I'm going to mention them uh and just intuitively explain them if you haven't continuous functions okay continuous functions intuitively you're going to formalize this in calculus intuitively continuous functions are functions that you can draw without lifting the pen off the board or the pencil off the paper or the marker off the air no that doesn't compile so that's a continuous function continuous functions are also a vector space you add two continuous functions you get a continuous functions that's something that intuitively makes sense it of course requires proof and and those proofs are in calculus okay there's a notion of a differentiable function a differentiable function is a function that has a derivative in every point okay that and intuitively you can think of a function whose graph doesn't have any cusps as a smooth graph nope okay differentiable functions are a vector space if you add two differentiable functions you get a differentiable function okay so let's maybe call this a six prime are continuous functions and um let's call it six prime prime differentiable functions if these words are don't mean enough to you at the moment don't worry about it so but when you get to calculus and and encounter these notions then you can uh flash back on this and say hey I know why continuous functions are a vector space because one of the theorems are going to prove is that if F is continuous and G is continuous then F plus G is continuous okay good so this is a very rich list of many many things that you've encountered in mathematics and the list goes on and on we're going to stop here it's just a list of examples but we're gonna I'll mention many more examples many more examples as we as we proceed as we go along okay so questions about about all of this okay so once again to summarize it's very abstract it takes time to get used to seeing something that's completely abstract but keep in mind the examples okay whenever you want to suppose somebody asks you is it true that in any vector space the following property holds what do you do how do you know if it's true it's false if you want to prove it if you want to give a counter example well you think examples okay so if somebody tells you alpha times zero is always zero then you start thinking an example and in examples if you take alpha times the zero polynomial is it the zero polynomial yeah if you take alpha times a matrix is it the zero matrix yeah if you take alpha times a function is it the zero function yeah so it seems to work in all the examples well probably it's true let's try proving it right and then you prove it using the axioms okay and if you find one of the examples where it doesn't work so suppose somebody tells you that I don't know six times a function gives you the zero function you say sorry six times V gives you a V then you're going to say no that's not true six times the matrix is not the same matrix it's a different matrix right there's a counter example okay good that's how you approach these abstract things you always think examples examples are where you feel more secure where you know what you're doing and then go one step up to the abstract okay and and and don't be intimidated by the abstract because you get used to it okay we're going to do many many things in the abstract and gradually you'll get used to them okay so what we're going to discuss next is the notion of subspaces coming up next

Channel: Technion

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

Id: OK3Wn_47QPY

Length: 46min 18sec (2778 seconds)

Published: Mon Nov 23 2015