20 – The space R^n

probably the most important structure that we’re going to discuss in this course is a structure called the vector space and we’re gonna start by considering one example of a vector space which is the space R N and I’m gonna describe it and tell you what it is and then we’re gonna kind of try to figure out what its properties are and then we’re gonna make that abstract that’s the idea so what is RN so by definition RN is the collection the set of all vectors all remember that a vector is a matrix with only one row or only one column that’s how we thought of them so I’m gonna write it as as a column vector but you could write it as a row vector it doesn’t really matter all the column vectors of the form a 1 a 2 dot dot all the way to a n so there are n entries here in this column vector such that all these elements are taken from the field are the field of real numbers so remember the notation the set of all vectors column vectors with n entries such that all the ai’s all these entries are just numbers from the field R so that’s the set RN as a set often you’ll you’ll you’ll see that that these column vectors are called n tuples ok sometimes they’re called that ok so these may be let’s write that so these are called vectors column vectors you could write them as rows or sometimes called n tuples ok so what is the structure on these on this space we’re gonna call it a space what is the structure on this well we know how to add things like this right so if we denote let’s maybe even start with equality when when is a so when we put an arrow like this this stands for the entire vector we use this notation previously when we discussed matrix form of systems of equations so how when are two vectors the same when is a equal to B if all the entries are are the same so AI equals bi for every I that’s when two vectors are the same right how do we add two vectors so a plus B a plus B how do we add them we just do it component wise right we add each AI with the appropriate bi so a plus B is just going to be the vector that has a1 plus b1 in the first entry a2 plus b2 in the second entry all the way up to a n plus BN right that’s how we add vectors we also know how to multiply a vector times a number okay so if alpha is if alpha is a number or how do we multiply alpha times a vector we’re just borrowing the ideas straight from matrices these are matrices so alpha times a vector is just alpha times the first coordinate alpha times the second chord in and so on all the way up to alpha times in okay so we have two operations here one is the addition of vectors and another is kind of a weird operation because it’s not an intrinsic operation two vectors it’s rather an operation between vectors and numbers from the field which are called scalars okay so this is what’s called scalar or scalar multiplication no bad terminology multiplication by a scalar I’m being careful about using the term scalar multiplication because further we’re not going to discuss it now but there’s a concept called a scalar product okay where you do multiply two vectors and you get a scalar but that’s irrelevant to what we’re doing right now so let’s call it multiplication by a scalar by the way this word scalar just means a number it just means an element

from the field so here our field is our field is our so these alphas come from the field are but the reason they’re called scalars is because what we’re doing is we’re scaling the the vector so if we had a vector multiplying it by 2 is scaling it by 2 and we’ll see the geometric interpretation of this in a minute and that gives more meaning to this idea of scaling and multiplying by a scalar ok so we have here what we have a space a space R and these are just it’s a set of elements and we have two operations on it one is this plus addition of vectors ok addition of vectors and there’s the other operation which we denote by maybe a dot but actually it’s we don’t write the dot often ok we just eliminate it so this triple of a space with these two operations addition of vectors and the multiplication by a scalar this structure now it’s a structure it’s not just a bunch of elements it has some structure on it this thing is called no it’s not a field the field is our okay this is called the vector space are in that’s what it’s called the vector space it’s a space of vectors okay so let’s try to see what properties these two operations have okay I’m just gonna make a big big list of them so for example so let’s write properties the vector space RN that’s what I wrote okay there are more vector spaces and that’s the the the broader concept that we’re going to discuss later this is just the first one an example okay so properties so for example maybe something trivial to say but we’ll see that it’s very important if we take two vectors a and B elements of RN this implies that if we add them which one do I want to do first yet if we add them the Sun is again a vector in RN okay we’re adding two things we’re getting something of the same nature right do you agree okay here’s another property if we take a vector in RN and the scalar in R then the scalar times the vector is what it’s it’s a vector it’s not in the field it’s in the vector space it’s in RN do you agree okay so these two properties share the name closure closure of the set under the operations this property means that it’s closed under addition we’re adding things were remaining in the same set in the same world this means that it’s closed under multiplication by a scalar okay so these are two properties of closure okay some more properties well-known properties so for example we know that if we add two vectors it’s just a special case of addition of matrices so we know this or you can see it that you’re adding component wise so it doesn’t matter if you do 1 plus 2 or 2 plus 3 for each component so we have this commutativity commutativity property for the addition right so this is the commutativity of addition a vector addition right property number 4 we have the distribute sorry the associativity a vector addition if we take 3 of them and add them it doesn’t depend on which order you add them in so you can first do a and B or frits to B and C you get the same thing so this is the associative

property good ok property number 5 there exists remember this symbol there exists a 0 vector a 0 and I’m going to add a little arrow over the 0 to indicate that it’s the 0 vector namely it’s just the vector that has 0 in every entry so there exists a 0 vector in RN and what does this 0 do what is its special its special feature that when you add something with a 0 vector you get back to something so such that a plus 0 equals a for every a and are in do you agree okay good next property note that all the properties so far well except maybe number two all the properties are first properties of the addition okay we’re gonna then reveal some properties of the scalar multiplication so number six every every or for every we even have a symbol for that for every a in RN there’s an additive inverse there exists remember for all or for every bear exists a minus a which is again an element in the same vector space what is minus a you just throw in minus signs on all the entries right such that what is the property of the additive inverse what does it do write a plus minus a gives you the additive identity gives you zero right okay for more properties we’re gonna need another board property number seven one times a equals a for every a in RN what is this one right it’s the scalar one it’s the number one in the field do you agree we don’t know we haven’t defined any notion of multiplication of vectors okay so putting an arrow here would not make sense okay but if you take the scalar one it satisfies that if you multiply it by any vector you’re just multiplying each of each entry by one so you’re not changing anything okay good and now three more properties which I really really want to fit on this board and I’m gonna do it so I’m going to erase here and erase here a bit and make room here for three more properties which are all all involved the scalar multiplication end and the addition so property number eight alpha times a plus B equals alpha A plus alpha B what would you call this property right distributivity and nine is gonna be that alpha plus beta times the vector a now we’re adding scalars and multiplying them by a vector equals alpha A plus beta e what would you call this property it’s also distributivity okay its distributive distributivity is a proper property that relates addition to some form of product to some form of multiplication okay these two pluses are very different pluses this is a plus of vectors this is a plus it looks the same I mean even from here it looks precisely the same but it’s a totally different operation right this is a plus of numbers of scalars and here we’re distributing the plus of of addition of vectors with respect to the multiplication by a scalar here we’re distributing the plus of addition of scalars with respect to the multiplication by a scalar okay so both eight and nine are forms of distributivity so let’s add here for one and two these are the closure properties one and two and let’s add here that’s that here not sure I’m gonna be

able to do this but I’ll try this three beyou TV T and this U is a V good yeah it’s the same names that we’ve discussed before this is the the additive identity we mentioned it this is the additive inverse okay it’s the same names the same sort of properties that we’ve encountered when we discussed fields okay okay one more property property number 10 says that a B times a sorry L alpha beta times a where alpha beta are scalars equals alpha times beta a okay so so these it’s sort of an an associative any property but with respect to two different operations right here it’s a product of two scalars of two numbers whereas here it’s the product of a scalar times a vector do you see that there are two different notions of multiplication here both of which do not have a symbol at all but they’re different do you see that okay good okay so for all of these maybe you should add for every alpha beta for all of these three for every alpha beta in R and for every a and B in R in this relates to all these three properties okay probably equals yes you can add that you can multiply alpha by a and then multiply by beta it follows you don’t need to write it explicitly because it follows from the fact that you can commute the multiplication of the scalars here right and then just use this property again so I’m not gonna list it separately but it’s true okay okay so these are some properties we just collected things that are easy to observe that the vector space RN satisfies okay do you agree okay and where are we heading so what we wanna do is to say there are many many many things in mathematics that satisfy these ten properties okay some of them you know very well maybe you haven’t thought of it okay and we’re gonna talk about it shortly but many different mathematical structures like functions like matrices like solutions to homogeneous systems of equations many many things satisfy these ten properties so we want to strip this strip these ten properties from their concrete vector space RN and say hey let’s define an abstract notion a general notion called a vector space where RN is just going to be one example of such a thing and we’re going to call it V capital V for vector space and what is a vector space a vector space is a bunch of elements that satisfy a bunch of elements plus a field where you take the scalars from plus an addition operation plus a product of elements from the field times elements from V that satisfy these ten abstract properties okay it’s going look very abstract very abstract okay but then we’re gonna see that in fact we’re just rewriting these ten properties that we know that n-tuples satisfy okay and then we’re gonna see in examples that in fact many many other things satisfy them and why is that good the reason it’s good is if we prove a theorem about a vector space well one reason it’s good if we prove a theorem about a general vector space it automatically holds for all the examples for our end for matrices first solutions of non-homogeneous systems for whatever okay so making something abstract has a very powerful idea behind it you prove something in the abstract setting you get it for free for all your examples okay

good clear okay but before we do that I want to discuss the vector space or in just a bit more so so what I want to say is that there are there there’s more to RN than just these ten properties okay for RN and in particular for r2 and r3 we have a geometric visualization of them we know how to think of them in terms of geometry okay so for r2 which is just pairs two tuples pairs of real numbers and r3 which is triples of real numbers we have a geometric point of view a geometric realization okay so we don’t just think of them as as algebraic mysterious things we actually think of these vectors and I’m quite sure you’ve seen this we even mentioned it when we discuss complex numbers you think of them as arrows right so for example for example r2 are too we think of of the plane which we often indicate like this and the elements we think of as little arrows okay so so if you have what is an element in r2 it’s just a point a comma B we think of a is the x-coordinate and if B is the y coordinate and here’s a point in r2 right this is a comma B we can write it as a row vector or a column vector it doesn’t matter right so this is a point in r2 and furthermore we often identify this point with an arrow that starts at the origin and ends at that point do you agree okay maybe I shouldn’t call it a B but call it a1 a2 that would be more more reasonable for what we’re going to say in a minute do you agree and then this would be the vector a that’s how we think of elements in the plane right and then we can have another one let’s draw it in red let’s say we have somebody here and this corresponds to the point b1 b2 so this is B and we know what this edition of so here the scaling becomes very very obvious right what is multiplying this by 3 it would precisely be multiplying the x coordinate by 3 buck-buck-buck multiplying the y coordinate by 3 and getting a vector in arrow three times the length so scaling by three multiplying by the scalar three would be precisely scaling it by three do you see that you see it right how do we add these guys right we adding vectors in RN in general is adding component wise right so we do a 1 plus B 1 that’s the x coordinate and ay 2 plus B 2 is the y coordinate and that corresponds to just taking the X component of this adding the X component of this which would throw us to there some in terms of the axe and there’s some in terms of the Y and geometrically we know what we get we get the diagonal of the parallelogram right we would get this vector here in black this would be a plus B right so even the addition has has an interpretation a geometric interpretation that we can visualize we can see and by the way this this can be done in r3 there’s just a missing axis here and think of another axis going this way and everything works exactly the same okay that the drawings become a bit more challenging but it’s exactly the same idea right so there’s a geometric realization for r2 and r3 as what’s called two dimensional space and three dimensional space where we all live okay and there’s even more to it in r2 and r3 we we often define more operations more products okay so for example there’s a product of two vectors called the scalar product and there’s a product of two vectors called the vector product the scalar product can be extended to any RN I’m not telling you what it is right now the vector product is something that is really intrinsic to

r3 it’s it’s a special thing about r3 and cannot be reduced to r2 or extend it to two higher ends okay but we’re not gonna discuss them here I’m not even going to tell you what they are there you go beyond the idea beyond those ten core properties of being a vector space they give it even more structure okay and the rest of the things that we’re going to discuss the rest of the examples of vector spaces that we’re going to discuss are not necessarily going to have these properties and therefore we’re not including them in the properties of in the properties we’re trying to isolate and identify and restrict it okay so I’ll remark that our remark further structure further structure such as the dot product or the scalar product also called scalar product or the cross product also called the vector product the reason it’s called the dot product is it’s a product of vectors which we denote by a dot and this is a product of vectors which we denote by a little cross that’s the reason for the names are not are not part of being a vector space so I’m not saying anything of the form these are not important they’re crucial they’re very important in applications and so on but spaces with these further structures or other further structures are not just vector spaces they have further structure and it is generalized okay there are things called for example metric spaces okay metric spaces are spaces that are not just vector spaces but they have additional structure and this for example yields the metric structure which means that you can measure distances okay and measure angles and things like that okay so the the idea of being a vector space is really just the algebraic part of this of these structures namely the addition and the scaling that’s the that’s what we’re we’re considering in looking at the vector space structure are in okay clear okay good yeah so so what we want to do now what we want to do next it’s going to be in a separate video what we want to do is take these properties erase the are ends everywhere and erase the R’s everywhere replace R by capital F standing for a field any field it could be C it could be other fields which we usually we don’t discuss in this course but you mentioned that they exist like finite fields Zn and so on and replacing our n by something general as well V standing for a vector space okay and that’s what we’re going to call a vector space then we’re gonna see many many examples of those and then we’re going to start building the theory of vector spaces okay so that’s coming up next