 So let us continue our review of linear algebra and we're going to do it in the broader category of our modules So let V be an R module, which means that V by itself is an abelian group and that a ring R is acting upon V Of course if R is itself a field that makes V into a vector space and let W be a subset of V We say that W is a sub module of V We which we would denote that as W is less than or equal to V If it is itself an R module under the restriction of operations of addition and skill and multiplication So first of all W needs to be a subgroup of V with respect to addition, but then we also have to have the restriction with skill and multiplication gives us a Module as well. So remember when it comes to skill and multiplication This is a function from R cross V into V So when we restrict it to be R cross W So the vectors no longer have to be from V, but they have to be from W We need that the product is always in W as well and that gives us a sub module This is analogous to the notion of a subgroup sub ring that we've talked about before now in this special case That R is a field then V would be a vector space and we would call W a subspace So given any any module V and any subset W W will be a sub module if and only if the following three conditions are satisfied first W contains the zero vector And particularly you'll need that it's not empty, but it'll necessarily contain the zero vector. It's closed under addition So you take any vectors V and W inside of W There's some V plus W is also in there and that also needs to be closed under skill and multiplication So you take any scalar in the ring of scalars And you take any vector inside of the subset W our times V is inside there as well And so I'll leave it as an exercise For the viewer to prove that these three conditions are Equivalence to being a sub module. It's the exact same proof one would use in linear algebra to show that a when a subset Is a subspace or not? Let me give you a few examples of such things here So consider the polynomial ring f a join x here We're assuming f is a field therefore f a join x is a vector space and consider the subset of Polynomials where we adjoin only x squared that is to say that we only take even powers of x and so the coefficient of any Odd degree monomial is zero in that situation This forms a subspace of the polynomial vector space f a join x clearly f a join x squared contains the zero polynomial because the to be to belong to f a join x squared your Coefficient of odd powers has to be zero. Well, this one's coefficient zero in front of everything in Discriminate so yeah, the zero polynomials in there Why is the sum of two polynomials that come from f a join x squared also in f a join x squared? Well, look at the look at the coefficients of the odd degree monomials if you take something like x well the coefficient of one Polynomial zero the coefficient of the other zero and zero plus zero zero. What about x cube? Well, the one has coefficient zero the other has coefficient zero, so their sum is zero Voila, and this will happen for all of the odd monomials. Why is it closed under scalar multiplication? Well, if you take a polynomial and f a joined x squared and you times it by r We're looking at the coefficient of odd monomials. You're gonna get r times zero, which is still zero So the coefficient of odd monomials is still zero And so we're gonna get that f a join x squared is closed under skill and multiplication This shows us that f a join x squared is a sub module of Or I should say a subspace since these are vector spaces here of f a join x It's also an infinite dimensional vector space though and this this this strategy works for any natural number whatsoever If we take f a join x to the end It will be a sub module of f a join x and if you are curious zero does belong to the natural numbers here If you take f a join x zero right here This is just f which of course is a sub module of f a join x like so It's a subspace because this is closed under scalar multiplication because it's just a bunch of scalars anyways So that's a that's a nice example to provide here So some other vocabulary that's relevant to when we talk about sub space Some spaces and some modules is the idea of linear combinations and generation suppose that we have a our module called v and take some subset of v We're gonna call it s prime right here and it contains the vectors v1 v2 v3 up to vn and For this for this definition here assume that this set s prime is a finite subset Then we can define linear combinations of the set s i to be sums of the following form We're gonna take the sum of things that look like a i times v i where a i is a scalar from the ring V i of course is a vector from the set s prime right there and we take arbitrary sums of this So if you expand it looks something like this Something times v1 plus something times v2 all the way up to something times vn add it all together And so sums that have this form are called linear combinations of s prime This is how we define it when s prime is a finite set if s is instead an infinite set Then we say a linear combination Of something in s is it comes about from some finite subset s prime and a linear combination in that situation as well So another way of saying that is if you take a finite So if you have this set s which is which is infinite if you take a finite number of Vectors inside of s that makes a finite set like this And if it's a if it's a linear combination over s prime then it's a linear combination over s So you just if it's an infinite set you just push it on the finite subsets and you you get your linear combinations there So take any subset of a r module and that could be infinite or finite doesn't matter We can define the span of s which is commonly denoted as span of s Are some people use angle brackets to suggest that it's some type of Submodule generated by the generators s analogous to what we see With ideals and with subgroups in group theory and such and it's gonna be a set of all linear combinations of things from s So the span of s is then defined to be the collection of all things that look like these type of sums AV times V we take a sum of all these different V's where V ranges over the elements of s prime and s prime is some finite Subset of s here. The idea is we don't want to have an infinite sum. We don't we won't we don't want to do calculus We're not doing infinite series or anything like that. We're just trying to take finite sums So when it comes together when it comes to adding things from s all but finitely many of the coefficients are gonna be zero Some finite amount could be non zero and that's when we put all those together We get the span of that set s and this is going to generate the smallest Submodule of V that contains the set s itself So if you take the collection of all of all Submodules that contain V contain s inside of V and you look at their intersections That's gonna be the span of s So it is the smallest submodule generated by those but maybe I'm getting ahead of myself our spans even modules Submodules, right? Well, they contain the zero They contain the zero vector because you could just put zero in front of everything zero times anything is zero You add a bunch of zeros together. You're gonna get zero. So the zero vector is in there. What about an arbitrary What about sums of arbitrary linear combinations? If alpha and beta are two linear combinations and s then there's some set we'll call it s prime for which the elements of The sum of alpha have non-zero coefficient. We call this the support of alpha Likewise beta has some finite subset of s Which is its support that is this is set of all the vectors which have non-zero coefficients Now s prime and s double prime are finite sets if we take their union It'll still be a finite set and so without the loss of generality We can think of alpha and beta as linear combinations on the set s prime union s double prime Because those elements That were belong to s double prime that don't belong to s prime We can assume that an alpha their coefficient is zero and similar statements can be made for beta as well So that is we can we can grow their supports to support both of these things right here And so we can assume they have the same set that is there a linear combination over the same finite set So alpha would look something like this beta would look something like this the vectors are the same in each of these situations I'm not sure why one started at one and one started zero So we'll just put those to be one there And so when you add these together you add like terms the V1's add together the V2's the V3's all the way to the Vn's add together And you get the following situation there. And so the span will be closed under addition What about skill and multiplication? Well, that one's even easier, right? If you take r times alpha, then you use times each of the coefficients, which again Not sure where these zeros are coming from You just times each of the coefficients by r and this then gives you a linear combination that belongs to s so we do get that spans are Submodules in the space in the in the category of vector spaces. These will be subspaces and in math 2270 linear algebra You most likely talked about spans of vectors and these do in fact give you vector subspaces All right, and this is what I said earlier Alternatively the span can be defined as the intersection of all sub modules that contain the set s Which intersections of sub modules can be proven to be sub modules So that gives you necessarily that it's a sub module from a different point of view And so then one last thing I want to mention before we close up this lecture 22 is the idea of a spanning set or a generating set If w is a sub module of v and s is a subset of w We say that s is a spanning set or sometimes it's called a generating set of W if w is in fact equal to the span of s now I want you to be aware of that given a sub module there is not there's not one generating set They're gonna be multiple generating sets in particular w is equal to the span of w That's sort of like a trivial generating set because if you have everything it generates everything. Voila Clearly we would want something much smaller in particular We'd be interested in sort of like a minimums generating set could s be minimum in Some category and some meaning of that sense will in linear algebra a minimum generating set does exact it does exist And it's will be referred to as a basis for a subspace in general module theory We don't necessarily have minimum generating sets a minimal span set And so a basis is a concept that exists of course in linear algebra for vector spaces But not necessarily in general, but fortunately vector space is where we want to be so we do get we do get bases That is a minimal spanning set. You can also define a basis as a maximum Linear independent set, but that's a topic we'll review in a later lecture But let's end this video with an example consider the vector space q would join the square root of 2 Which when you look at that we can see that this is a vector space because Every element of q would join 2 is a rational number plus a rational number times the square root of 2 And so if you take the numbers 1 and the square root of 2 this forms a spanning set for q would join 2 In fact q would join 2 is written as the span of 1 and the square root of 2 These spanning sets are not necessarily unique But this is sort of like the standard basis for this field over that field And this is a concept we'll do many many times as we proceed forward with our study of rings But for now that ends our lecture 22 Reviewing the topics of vector spaces and sub spaces although we brought in the category to modules and sub modules I appreciate you watching if you learned anything from linear algebra or if you learned anything from about module theory Please like these videos subscribe to the channel to see more things like this in the future And if you have any questions, please post them in the comments below