Birne Binegar: Okay, so welcome back.
Birne Binegar: So.
Birne Binegar: Before spring break I I talked about having a second exam this week on the Friday before spring break I I relax a little bit thought that was too ambitious Nice.
Birne Binegar: that's the case today as well, I think the second exam not going to be this week at all it's going to be a week from today what we'll do this week as.
Birne Binegar: A quickly review the topics that we recovered just didn't the lecture before everyone went on spring break the notion of linear transformation, there are associated sub spaces, how we calculate those in terms of.
Birne Binegar: matrices getting up to the point where we can describe conditions, about whether or not function linear transformation between two vectors spaces, whether that's vertical or not.
Birne Binegar: That was all covered before spring break it all review that real quickly, maybe the first 10 minutes today's lecture and then, when I want to do is move on to what is essentially a new topic, but one that actually is really nice prior to your second exam because.
Birne Binegar: As i'm going to what i'm going to do today is i'm going to lift consideration.
Birne Binegar: or introduce vector spaces, other than our favorite vector space us are ordered list of numbers and will describe how we use vector space is amorphous.
Birne Binegar: To do calculations in a more general ledger linear algebra setting where.
Birne Binegar: Your vectors may not be quantities, with a magnitude and direction, nor do they need to be ordered list of numbers, they can be.
Birne Binegar: Much more abstract things and so i'll introduce the concept of a general vector space and what i'll try to do is show you how the apparatus that we've developed so far is actually critical to.
Birne Binegar: Understanding vector spaces in this more general context and, in fact, as you leave the course.
Birne Binegar: you're not going to be dealing with vectors so much as ordered list of numbers, but rather in your applications they might be solution sets of differential equations or they might be vibrations of a.
Birne Binegar: Of a string of could be states of the hydrogen atom other such more abstract.
Birne Binegar: rendition to what a vector is so i'll try to explain how that works and then that will that will do for me is on Wednesday, I can show you how the apparatus sub spaces linear transformation how that all comes to play in your applications and now that will serve as sort of review for.
Birne Binegar: Again, to which is basically going to be the calculations under that support sub spaces and linear transformation i'm like.
Birne Binegar: Okay, so um so first of all, are there any a strenuous objections to having the second exam a week from today.
Birne Binegar: Okay, so.
Birne Binegar: that's what we'll do the second exam will be a week from today until Let me now just step through this material that was just touched upon prior to spring break.
Birne Binegar: What we were I was talking about at that point was functions between two vectors spaces specifically linear transformations.
Birne Binegar: Which.
Birne Binegar: If I can get this short linear linear transformation, this is a special kind of function, first of all, it maps you from one ordered list of numbers to another set of order list of numbers from our end and and.
Birne Binegar: What makes it special to linear algebra is the fact that its function behaves well expect to are two fundamental vector space operations.
Birne Binegar: scale or multiplication in Dec tradition effectively what that says is I can do scale multiplication before I applied to transformation or afterwards I get the same result that's special condition.
Birne Binegar: and, similarly, if I add two vectors and apply the linear transformation, I get the same result as if I had delayed adding vectors until after I applied the linear transformation.
Birne Binegar: Now what we saw before spring break was the fact that it's not so hard to make such linear transformations, particularly when we have matrices at our disposal.
Birne Binegar: anytime you have a matrix within rows and columns you can use that matrix to define a function between it m dimensional space and an n dimensional space way that works is simply use matrix multiplication to define that map So when I said T so a is a given matrix.
Birne Binegar: And let's make it even more specific a matrix with em rows and rows and columns, then I can define.
Birne Binegar: A function that will take em dimensional vectors.
Birne Binegar: The N dimensional vectors where the rule it's going to be.
Birne Binegar: affect your ex inside our em it's going to get mapped to effect or a contact now where's this guy going to live.
Birne Binegar: When you have an n by n matrix multiplying a column vector that we going to have him component and.
Birne Binegar: Because of the number of columns of the first factors equal limit the Rose second factor that multiplication is well defined and the results.
Birne Binegar: it's going to be, in my one matrix and use the outside, you mentioned to get the result of the product and so you're going to have any.
Birne Binegar: comments here and that's going to put you give you a vector and side on, and so, since you have a matrix you have a function between these two vectors thing says.
Birne Binegar: And what we also saw is that, conversely, if you have a linear transformation between a vector station aren't am a vector space are, and you can attach to this linear transformation this function between vector space is a particular matrix here's how you construct it.
Birne Binegar: These vectors he subscript I.
Birne Binegar: Well that's just the standard basis factor, where you have zero is everywhere.
Birne Binegar: Except in the ice slot.
Birne Binegar: And we were referred to that is the standard basis vector to construct this matrix you take the function.
Birne Binegar: You apply it to what's going to be an n dimensional vector.
Birne Binegar: that's going to kick out an n dimensional vector and you write the components of this vector as the columns of the matrix and you do that for each standard basis factor.
Birne Binegar: That will produce a matrix with and rows and columns and what it what's going to happen for this particular matrix is I can use that matrix as an alternative way of implementing this map.
Birne Binegar: What will happen if.
Birne Binegar: If a T is equal to that matrix.
Birne Binegar: Then you're going to have he of X is exactly the same thing that matrix timestamps, and so I could implement the function simply be a matrix multiplication.
Birne Binegar: It kind of reproduces this action here, but the difference between these two lines here I start with the matrix and I define a function.
Birne Binegar: In the bottom line I start with a function and I define the corresponding matrix but you see that there you can go back and forth between these two pictures.
Birne Binegar: Okay, so um, the main purpose of introducing this matrix, though, is it.
Birne Binegar: Once you once you actually calculated it's a matrix full of number so it's.
Birne Binegar: Naturally optic that you can do calculations with because all the entries are just numbers.
Birne Binegar: And so, in fact, we use that matrix to actually attached to a linear transformation.
Birne Binegar: Let me define the sub spaces first and then then i'll show you how we act we calculate the corresponding subspecies using the matrix formulation up a linear transformation so first of all.
Birne Binegar: Suppose we have a linear transformation taking him dimensional vectors to n dimensional vectors.
Birne Binegar: there's a subspace of the co domain are in that's defined by.
Birne Binegar: Looking at all the points in this target space that you land on by applying the linear transformation to some point in the domain.
Birne Binegar: that's called the range and it turns out that this subspace is identical to the column space of the associated matrix so the way you actually make this concrete.
Birne Binegar: Avoiding this functional definition is you calculate the corresponding matrix and then you find a, for example, a basis for the columns face, let me remind you.
Birne Binegar: Find.
Birne Binegar: A basis for the columns face.
Birne Binegar: matrix now what is called a matrix a what we do is reduce.
Birne Binegar: The matrix to a row Echelon form.
Birne Binegar: And then you grab.
Birne Binegar: A columns.
Birne Binegar: Of the original matrix a they correspond.
Birne Binegar: To.
Birne Binegar: A columns of The.
Birne Binegar: Real Echelon for form that have pivot.
Birne Binegar: And then, of course, the reason.
Birne Binegar: Why, we want a basis is because, once you have a basis for a subspace, then you have a way of writing every vector net subspace uniquely.
Birne Binegar: So let's call this basis be suppose i've done this i've.
Birne Binegar: taken the matrix a grill reduced it to a row Echelon form, I figure out where the pivot slip and i've gone back to the original call and I grab those factors so maybe B is equal to certain set of vectors the end, then, if this is a basis.
Birne Binegar: So, once you have the basis.
Birne Binegar: Then.
Birne Binegar: Every vector.
Birne Binegar: In your subspace.
Birne Binegar: say the common space of this matrix can be written.
Birne Binegar: uniquely.
Birne Binegar: Say vs equals C.
Birne Binegar: Plus C.
Birne Binegar: And, and you have a unique way of writing a vector and you can use these coefficients C one C n effectively, yes, coordinate where the vectors inside your your subspace so.
Birne Binegar: What this matrix is doing for us it's it takes some sort of abstractly define subspace some subset of the range here it's specified this way, but the way we can actually calculated as we construct the corresponding matrix.
Birne Binegar: find a basis for the column space, and then we can use that basis to express every vector in the range as a linear combination of your given basis vectors so that's how we actually coordinate is the subspace.
Birne Binegar: there's another subspace.
Birne Binegar: The range is the subspace of the co domain there's another subspace that's associated with the domain of the function.
Birne Binegar: And that's consists of all the points in the domain that gets sent to the zero vector inside the code domain, so when you're in the you're in the kernel if, after applying the function T you end up on the zero vector inside the Co domain.
Birne Binegar: And this is a subspace of the co domain, but it's calculable in terms of this auxiliary matrix at one we have one recreate by applying to all the standard basis vectors and arranging those is called so.
Birne Binegar: That subspace associated with a linear transformation is calculable as the no space of this concrete and your miracle matrix, which in turn is course just the solution set a set of linear equations.
Birne Binegar: Alright, so.
Birne Binegar: That much.
Birne Binegar: What was the introduce the primary objects, the lecture just prior to spring break um one thing that falls from this correspondence between.
Birne Binegar: matrices and linear transformation is.
Birne Binegar: Well, we use that the associated matrix to calculate bases for both the range of tea in the kernel of tea.
Birne Binegar: But let me remind you that.
Birne Binegar: If you have an n by n matrix there's a relationship between.
Birne Binegar: The number of columns of the matrix and the dimension of the column spatial dimension of the no space and directional roast beef and the dimension of funnel stage, so I guess I put these definitions up the rank couple matrix.
Birne Binegar: Is alternatively, you can think of it as dimension of the row space of the matrix click all possible linear combination of the rows of a generator subspace support a lot of.
Birne Binegar: See, that would be a sub stage of.
Birne Binegar: Our am.
Birne Binegar: Or let me put this up, because this.
Birne Binegar: What i'm saying here is that.
Birne Binegar: If you haven't.
Birne Binegar: matrix.
Birne Binegar: Then.
Birne Binegar: You get a basis.
Birne Binegar: While I got a basis for the column space here, let me remind you that basis for the row space of that matrix.
Birne Binegar: that's obtained by.
Birne Binegar: Reducing.
Birne Binegar: Again, just as we did here to a row Echelon form.
Birne Binegar: And in this case, instead of grabbing columns of the original matrix what you do is you grab.
Birne Binegar: The Non zero rows.
Birne Binegar: The row Echelon on.
Birne Binegar: And one thing that comes about from these prescriptions about how we find the basis factors is how we.
Birne Binegar: relationship between dimensions.
Birne Binegar: Remember bases, these represent the independent directions, you need to generate the entire space, the number of independent directors, you need it's going to be the dimension, the number of basis factors.
Birne Binegar: And so the statement here is that they dimension up the road spaceship a matrix is always equal to the dimension of the columns based in the matrix that's a bit surprising it first because.
Birne Binegar: The row space.
Birne Binegar: Of a matrix if it's an n by n matrix well look at each row each row house exactly em components, but that's going to be a subspace.
Birne Binegar: of art in meanwhile the columns they have exactly as many entries, if you have rows, and so the columns space of the matrix it's a subset.
Birne Binegar: of our end and so it's a little bit surprising, the subspace is living in different vector spaces actually have the same dimension, but the reason for that is because.
Birne Binegar: You find the number of basis vectors you have well, it reduces to counting the number of pivots you have in a row epsilon form.
Birne Binegar: In this case, you figure out where the pivots are and then you grab the corresponding rose Okay, so you have as many basis vectors for the roast basis you have pivots in the row excellent point and the same thing here that number.
Birne Binegar: Where you actually get the actual columns it's different than the prescription here, but you start off by looking at where the pivots live in the row watch on form and for each one of those you get a basis factor for the columns space.
Birne Binegar: So the total number of factors, you get in either case it's going to equal, the number of pivots you have international long form so that's why the dimensions of those features are the same.
Birne Binegar: Meanwhile, the no space of a matrix or what's called the dimension of the north face or the so called novelty of the matrix.
Birne Binegar: that's the solution set, but when you use the matrix as a coefficient matrix of the linear system and you saw the linear system.
Birne Binegar: And the number three parameters it's going to be the dimension of that solution set, but the free parameters.
Birne Binegar: Just correspond to the number of columns that dumb contain pivots when you've got the real Echelon format, the matrix So if you add these numbers up number of columns with pivots and you add that to the number of columns without pivots you get the total number of columns and so.
Birne Binegar: that's where this identity comes from.
Birne Binegar: Alright, so that was.
Birne Binegar: A relationship between two sub spaces associated to a matrix but because major cities.
Birne Binegar: And linear transformations are so closely connected, you should expect that might be a connection between the dimension of the range of a matrix of a linear transformation and the dimension of the kernel and in fact that's true.
Birne Binegar: If you have a linear transformation, then the dimension of the domain yeah it's going to equal to the some of the dimension of the range of the linear transformation and dimension of the Colonel.
Birne Binegar: Remember, this is, this is the set of points in the co domain that are reachable by the wind on your transformation and the Colonel if if the.
Birne Binegar: subspace up the domain that gets sent to the zero vector when you apply it there, they live in different decter spaces, or the different vector spaces in general, but there are dimensions have to add up the M and the reason for that is because.
Birne Binegar: The range of tea is effectively the same thing as the columns based at the associated matrix and the kernel of tea is no space of the associated matrix and we saw from the last slide that them the rank of a matrix and its novelty have to add up to the number of columns and so.
Birne Binegar: You find this relationship.
Birne Binegar: Okay.
Birne Binegar: Then.
Birne Binegar: I guess, I want to point out one other thing that we talked about before on spring break and it gave.
Birne Binegar: One immediate reason why we're interested in these two sub spaces, to begin with, why we're interested in the range of a linear transformation or a kernel.
Birne Binegar: And that is a theorem.
Birne Binegar: Before.
Birne Binegar: spring break.
Birne Binegar: That said, that.
Birne Binegar: If.
Birne Binegar: First of all.
Birne Binegar: Mr m.
Birne Binegar: To our end is a linear transformation.
Birne Binegar: Then.
Birne Binegar: He has an inverse.
Birne Binegar: T the miners first that's going to go in the opposite direction from our end back to our em.
Birne Binegar: If, and only if the.
Birne Binegar: first condition is that the range of tea.
Birne Binegar: This particular subspace.
Birne Binegar: That has to equal exactly.
Birne Binegar: The entire co domain or sorry up here tire co domain are in.
Birne Binegar: And second of all what you need to see kernel of tea.
Birne Binegar: Instead of pointing the domain that gets sent to the zero vector that has to equal just the zero vector inside are in.
Birne Binegar: So this will be a zero dimensional subspace because it consists of a single vector zero vector.
Birne Binegar: Alright, so.
Birne Binegar: If I asked you, then I gave you a linear transformation and I asked you, is there an inverse linear transformation you don't have to invert a matrix or anything like that to actually calculate it, you can verify that you do have an inverse simply by finding basis for the.
Birne Binegar: row space and cons space sorry buying bases for the range and kernel of tea that's how you get these dimensions but.
Birne Binegar: More specifically what you'd want to show.
Birne Binegar: Is the range of tea is the entire space and that you would get by.
Birne Binegar: calculating the corresponding matrix for this and then verifying that the columns of that matrix are all linearly independent that will make the column space and dimensional.
Birne Binegar: And to verify that the kernel is equal to zero, you simply solve the linear system, a subscript T times X is equal to zero, if you only have one solution the zero vector, then the kernel of T is going to equal that and so.
Birne Binegar: Both of these conditions, because that the substations can be calculated once you've constructed the matrix corresponding to this, and so these are readily verified conditions.
Birne Binegar: And they don't involve actually writing down this inverse function that takes you back if the statement is only about there is some function that will it will take you directly back to your original domain, so long as these two conditions are satisfied.
Birne Binegar: The existence of this inverse function is really important, though, and what's going to happen today, because what it means is not only can I.
Birne Binegar: go from one subspace to another, but I can go back to here, and so you can think of information here as being moved over to this setting or in vice versa information here can be moved back into this setting now that doesn't look like very profitable discussion.
Birne Binegar: In the context of vector spaces, consisting of order list of numbers, but the point of that is going to be is when.
Birne Binegar: We have a concrete calculation a vector space and we have some more general vector space here, then that ice amorphous ISM is going to be more meaningful, because what it's going to mean is that we can take.
Birne Binegar: Think of this as the abstract vector space we can bring it into a calculation of setting do a calculation and but because we have the inverse function as well, we can take the answer back, and in fact that's how you do calculations.
Birne Binegar: In general linear algebra where your vectors are no longer.
Birne Binegar: ordered list of numbers, but more general objects that could be the solutions of a differential equation, they could be.
Birne Binegar: A family of polynomials they could be know the vibrational modes of the string.
Birne Binegar: We will deal with them from a cash relational level, this will try to establish a nice amorphous and that takes our abstract vectors into the concrete calculation of setting do calculations there and then take our answers back so this idea of ice amorphous.
Birne Binegar: This is, this is what we need to check and make sure we have a nice and warm but, once we have that much we've established a nice amorphous between two vectors faces, we can readily move back and forth between them.
Birne Binegar: Okay, so.
Birne Binegar: That brings me to the today's topic, which is what are these more general abstract vector spaces.
Birne Binegar: How do we set them up.
Birne Binegar: Alright, so.
Birne Binegar: here's how we do it.
Birne Binegar: A vector space over our weekly call this a general vector space over the real numbers now.
Birne Binegar: In math.
Birne Binegar: We only consider vector spaces over the real numbers, but there are alternatives, you can consider you can use the complex numbers and you'll get a bit up a vector space with essentially the same.
Birne Binegar: All the same, definitions, except you substitute for the real numbers you step super complex numbers, or you could use the rational numbers here, or you could use.
Birne Binegar: The two or any finite field there's but keep things simple in math 3013 we only work over the real numbers, but other possibilities are also available to us.
Birne Binegar: But will will will confine ourselves to this city, what we need to get a abstract vector space over the real numbers is some notion of scalable application.
Birne Binegar: Which means some function that takes a number and a vector or an object in the set and produces another a new object inside that set.
Birne Binegar: Second, of all once some analog of vector addition meaning we start off with a pair of objects in this slide slide That said, we have a way of adding them together to produce a new member of that set.
Birne Binegar: Moreover, well just having these these two functions it's not enough, in addition, what we're going to require that these two the scale of application and vector addition that would be fine on the set has to satisfy eight axioms.
Birne Binegar: Each of which.
Birne Binegar: corresponds to a well known property of vectors over our end.
Birne Binegar: So the first one, if I have.
Birne Binegar: Three vectors and I want to add them together.
Birne Binegar: there's two different ways, you could do that.
Birne Binegar: My rule for adding factors.
Birne Binegar: Given a scale or multiplication vector addition my rule only tells me how to combine two vectors don't want to combine three vectors.
Birne Binegar: I use the rule for combining two vectors twice and this side, first, I can buy a new and D and then I combine the results, one with this, but it each day to that i'm only using two vectors at a time, but alternatively, I could combine the last two vectors and then add it to the first director.
Birne Binegar: What the requirement we demand this function that maps pairs of factors back into the original set D is that it should make any difference as to how we add three vectors together.
Birne Binegar: No matter how we compute it, we should get the same result.
Birne Binegar: Moreover, we don't just like adding vectors in our end we don't want the order of the individual terms, to make a difference in the result that's conductivity.
Birne Binegar: We need a thorough vector and that's simply a vector that lives inside not your set of objects, such that when you add it to any other vector you get back what you started off with you get back the original be you started off with.
Birne Binegar: You on you need additive in versus, meaning that, for every vector inside your set there's another vector you can add to it that will produce the zero vector.
Birne Binegar: You need.
Birne Binegar: scale and multiplication to distribute over vector some so here on scaling multiplying some the vector some of these two vectors and I want that to be the same as.
Birne Binegar: As if I.
Birne Binegar: thought of this is distributing multiplication over addition.
Birne Binegar: And Similarly, I can hear the difference between these two lines here the addition on the left is addition inside edition of vectors and line six here i'm adding numbers together and then multiplying and vector by that but.
Birne Binegar: Either way, I get the same result here on distributing.
Birne Binegar: The sum of numbers over the scale or multiplication.
Birne Binegar: And lastly, we have these other compatibility properties, if I do a sequence of scale and multiplication first i'm scale a multiply by you, and then I multiply by Lambda I get the same result as if I multiply Lambda times me first and then combine that with me.
Birne Binegar: And last one is we want one time equal be now.
Birne Binegar: In the concepts context of our in each of these properties is really trivially verified.
Birne Binegar: So, for example.
Birne Binegar: In our in Lambda times, an order list of numbers.
Birne Binegar: equals Lambda X one is this way is defined.
Birne Binegar: Lambda.
Birne Binegar: Times X and last component and we add two vectors.
Birne Binegar: let's call it, why one why and and that's going to be equal to X one plus one one.
Birne Binegar: SEC and plus y and we have an in these two definitions of the fundamental vector operation and each of these.
Birne Binegar: Each of these axioms is trivially verified, for example, this zero element.
Birne Binegar: All right, zero subscript are and that's going to be.
Birne Binegar: The vector which all the components are equal to zero, if this is the way the zero element is defined then adding one factor to another.
Birne Binegar: Adding one factor to the zero vector is just going to produce the original vector because all you're doing just sub two zero for the components of why here and you'll get X one plus zero.
Birne Binegar: X two plus zero X n plus zero and and so indeed this this particular vector is going to act like an additive identity for the vectors in this numerical vector space.
Birne Binegar: So, in the context of our in.
Birne Binegar: Each of these guys it's sort of trivial to verify, but the sort of surprising facts is that.
Birne Binegar: We don't.
Birne Binegar: have to limit ourselves to simply order list of numbers, we can find other sets that have a notion that scale multiplication and the notion of vector addition and it's appropriate zero vector.
Birne Binegar: Such that all these axioms are satisfied and the point is is that that means that abstract set behaves from a.
Birne Binegar: point of view is linear algebra exactly like one of our favorite vector spaces or any place where we can do all our calculations so it's by virtue of having a set that will bases rules that allows us to treat it as if it were it's elements were elements of our end and thereby calculable.
Birne Binegar: Alright, so let me give you some examples.
Birne Binegar: So the first one well here's what we need to verify the axes.
Birne Binegar: You have an abstract set you can regarded as the vector stage once we've defined scale or mobile application.
Birne Binegar: That is something rule by which we take a number and an element of the set and we get another element of the set.
Birne Binegar: To get back tradition, we take two elements of the set, you have a way of combining them that will call decter addition, but the result is supposed to be another element set and we need to zero vector once we have.
Birne Binegar: These three things specified, then we can go back to these axioms and manually check to make sure that those identities hold.
Birne Binegar: So.
Birne Binegar: In the case, the first example is actually the prototype of the whole setup The first example is when we regard this abstract set it's actually consisting of order list of numbers.
Birne Binegar: And we use the usual notions of scale or multiplication and vector edition, and the usual notion of the zero vector.
Birne Binegar: With these definitions.
Birne Binegar: These three these eight properties are automatically satisfied, and so are in is.
Birne Binegar: Not only our first vector space we considered it's, the first thing example of an abstract vector space.
Birne Binegar: But there's another situation that where.
Birne Binegar: We can regard sub spaces as being vector spaces, so that seems a little bit strange but remember subspace if if we've handled them, thus far they've always been subsets of a larger vector space.
Birne Binegar: And so they haven't been vector spaces themselves up to this point, what I want to point out is that once you have a subspace about what set of ordered list of numbers, then it's because of the subspace it's closed under both scale or multiplication in depth tradition.
Birne Binegar: If you look at these rules here.
Birne Binegar: I have a number, and I have a vector and I get a result that's another vector inside capital D, or I have two vectors that start off in D.
Birne Binegar: I have a rule for adding them together and produces another vector and he now, in fact, the scale or multiplication and this vector edition is exactly the rules that we use for our end with the exception that we sort of restrict the vectors that we're considering to live inside D.
Birne Binegar: So, in other words the scale or multiplication appropriate to the subspace.
Birne Binegar: What we need is a map that takes real numbers and and vectors in the subspace and produces vectors inside that subspace what i'll use to get this rule it's just the.
Birne Binegar: multiplication on the original vector space and they'll just restrict it's arguments to this subspace.
Birne Binegar: So.
Birne Binegar: Here the you think of this as a function that takes a number and a vector and produces the new vector the domain, if this is.
Birne Binegar: All possible real numbers and all possible vectors here all i'm saying here that have considering all possible or your list of numbers, I only have to consider.
Birne Binegar: The vectors inside the subspace, but I still get a map that takes numbers and vectors and produce inspectors and, similarly, using the audition in our end.
Birne Binegar: Because some spaces are closed under back tradition, I get a map that takes pairs of vectors living in D and producing another vector and the data I can think as high.
Birne Binegar: As a vector addition.
Birne Binegar: For this set.
Birne Binegar: and
Birne Binegar: Those two operations.
Birne Binegar: they're naturally going to satisfy all axiom.
Birne Binegar: The third thing I need to establish a subs.
Birne Binegar: A vector space, not only do I have to define scale multiplication vector addition we also need to zero vector but remember that.
Birne Binegar: If you have a subspace that subspace always contains the zero vector because.
Birne Binegar: It has to be closed under scale and multiplication if you multiply by zero you always get the zero vector so the zero vector always lives inside the subspace and we'll just use this guy at the zero vector for smaller subspace.
Birne Binegar: So we've already dealt with an abstract vector states it's just that we dealt with it in the guys at the subspace.
Birne Binegar: All right, so.
Birne Binegar: Those are.
Birne Binegar: Both are in in substances are in we've considered in this course and they're now presented as examples of a more abstract kind of vector space.
Birne Binegar: One where you have scalable application and vector edition defined in which satisfy eight actions here's some more interesting ones.
Birne Binegar: Suppose consider the functions on the real line, now that doesn't look like any vector we've considered before.
Birne Binegar: they're not quantities with magnitude and direction they're not ordered list of numbers, but nevertheless I can define to operations on them.
Birne Binegar: scale multiplication so that should take a number and an evolution of my set and she produce a new element my set so I take a number, and I take a function and I define a new function by telling you what the value is at any point X well its value was just going to be.
Birne Binegar: A given function evaluated X and then I multiply by that number Lambda well that gives me a value for this function Lambda X at the point X and so what defines that function.
Birne Binegar: And certainly I can add function together like.
Birne Binegar: here's the guy want so think of business, not just the name of what it means to add function F to function g of describe what that is I tell you what it's die you is at any point X.
Birne Binegar: And what i'll use is simply the value of F of X now add to that, as a number, the value of g of X and that would give me a new function it's going to take X to a new number.
Birne Binegar: All right, and then zero vector.
Birne Binegar: The zero vector is it's not the number zero, it has to be an element of the set that behaves like the number zero.
Birne Binegar: So what we do for this particular vector stays is we define a function F subscript zero, which is defined by the by its values, we demand that F subscript zero evaluate it X always equals zero for all X and X, so that defines it as a function.
Birne Binegar: And then we observe.
Birne Binegar: That we take any function G inside the.
Birne Binegar: So F.
Birne Binegar: So this is going to be.
Birne Binegar: Well it's the set of all functions.
Birne Binegar: From.
Birne Binegar: The real line to realign and.
Birne Binegar: If I take g.
Birne Binegar: And I added to this this zero vector inside this space well, what did our values it's going to be g of X plus zero X using this rule for carrying out back tradition but.
Birne Binegar: This function we're using it, the zero vector has a property that is value in any X is equal to zero.
Birne Binegar: And so what we see is.
Birne Binegar: The functional some of G and the zero vector inside, be it is equivalent to the original function F, so this act like an additive identity.
Birne Binegar: But it's important to note, though, that this zero vector is not the number zero it's a vector that lives inside that's that set.
Birne Binegar: But now i've got it defined and.
Birne Binegar: If you adopt these three definitions scale multiplication vector edition and zero vector you can with patient still readily verify that all eight actions are automatically satisfied.
Birne Binegar: Alright, so those are.
Birne Binegar: For three examples were all examples of vector spaces in mathematics here's an example for vector space in physics or engineering, where you maybe consider the vibrational modes of a stretch string or maybe on a rod extended Rod, whatever it is.
Birne Binegar: Think of it as the set of all possible vibrational mode.
Birne Binegar: Well, you can do with an analog scale and multiplication and what that is in this context is changing the amplitude of the vibrations.
Birne Binegar: Increasing the displacement from its equilibrium position as the vibration occurs, so if you simply change the amplitude by a positive factor that's be correspond to.
Birne Binegar: multiplication by a positive number Lambda but, in addition, you can do another operation as well, let me sort of.
Birne Binegar: This is meant to be a purely physical example but it's a scribe when i'm saying suppose.
Birne Binegar: Let this be T going off this way, this is the thing is vibrating so it's amplitude it's it's displacement from the equilibrium is changing, with respect to time.
Birne Binegar: So here i'm sort of giving you a graphical picture of what this vibration should look like maybe on the telescope where you actually measure it.
Birne Binegar: What scale or multiplication would you say bye to we just increase the amplitude of these oscillations.
Birne Binegar: know what you really scale every point by a factor of the every displacement by a factor of two that might be so the call this.
Birne Binegar: call this vibration D at might be Lambda V and.
Birne Binegar: So a given migration is converted to a new vibration by simply changing amplitude.
Birne Binegar: Another thing we could do it, given this initial vibration, we could also reverse it saves.
Birne Binegar: So much happening here is this positive displacement here is being mapped to its negative and that would this would be.
Birne Binegar: The action of a negative minus one times the original vibration.
Birne Binegar: Now, this one is maybe not so readily performed if you're a conductor, you can always tell violinist to play louder and you can change the amplitude it's not so easy to get a violinist to change the face that's coming out of their.
Birne Binegar: instrument, but if you have a guitar or something like that, or you have some electronics and ball it's relatively easy the flip the flip the voltages and thereby.
Birne Binegar: achieve the negative of a vibration or effectively the anti vibration, because what happens, this is the way noise cancelling headphones work if you add.
Birne Binegar: A vibration to it negative you get the zero vibration and so.
Birne Binegar: So yeah in this thing i'm keeping the frequencies constant but i'm just saying, if I super impose So if you have one sound so worth making one and another one with exactly out of phase.
Birne Binegar: This what happens inside your earphones they cancel out you don't hear it, and in fact that's exactly how noise cancelling headphones work.
Birne Binegar: You know you have you have the headphones, but they also have a small microphone it's pinned to the outside, so if you're in an airplane all that.
Birne Binegar: All that noise is picked up by the microphone electronics invert the signal and then it combines that with the signal coming from you MP3 player, or whatever, and so that the noise get capital, but the the other signal remains.
Birne Binegar: So yeah so the scale multiplication actually has a lot of practical utility.
Birne Binegar: And then now.
Birne Binegar: The vector edition that's where you actually super impose things.
Birne Binegar: Maybe.
Birne Binegar: Now log if you have an orchestra you hear you can hear individual violins if you listen to acutely but, in general, you hear both simultaneously and that's coast, corresponding to super super imposing vibrational mortgage.
Birne Binegar: And then the zero vector in this situation is just silence that's that's your.
Birne Binegar: The zero vibration.
Birne Binegar: Then, finally, let me give you one more example because this is sort of the in between, you can think of polynomials is functions.
Birne Binegar: Or you can think of them as the way you construct them if you simply take numbers times powers of X and add them up.
Birne Binegar: What, however, you think of polynomials they have a natural scale multiplication.
Birne Binegar: it's just the way you would do multiplication of polynomials you just multiply every coefficient by the same number in the way you add polynomials issue collect together the coefficients have the same power action and them together, but what you get.
Birne Binegar: For this role of combining is another expression that basically numbers times different powers of X and then the zero vector is just a polynomial that's identically zero so here what you've seen in this lecture it's five different ways that we can.
Birne Binegar: generalize past order list of numbers and, of course, the reason why you're in this course is because your applications, you can find many more applications.
Birne Binegar: What many more ways in which the structure of a vector space of our end can be replicated by the phenomenon that you're investigating and that's what we're trying to take advantage of in this course is to give you the tools to carry out that analysis.
Birne Binegar: Okay, so.
Birne Binegar: i'll pick up with this topic again on Wednesday on Friday of this week, that will be a review session for the exam a week from today okay.