Birne Binegar: Okay, so.
Birne Binegar: i'm remind you.
Birne Binegar: That we've scheduled our second exam for next Monday, and it will cover the material we've gone over and lecture since the first exam so basically notion of Sub spaces bases linear independence and linear transformation.
Birne Binegar: What i'm going to do today is try to put all that material in the in the more general context of what you'll be using it for after you leave the course.
Birne Binegar: So the discussion today is not going to be literally covered on the exam on Monday, but however it's still going to be relevant to that exam because i'll still be fine.
Birne Binegar: Sub spaces bases and all that apparatus, but now, in the context of a more general vector space and i'll try to give you an examples to show you show you why.
Birne Binegar: linear algebra is already relevant that most of my examples are going to deal with spaces of functions So hopefully what you'll do is recognize some of the properties or.
Birne Binegar: way we deal with functions in mathematics and an application, a lot of that is based on linear algebra you just haven't seen the underlying structure, yet so that's going to.
Birne Binegar: be the main point of today's lecture to show you linear algebra in a more general context the way you actually use it after you leave this course.
Birne Binegar: Okay, so.
Birne Binegar: Oh, and I should also mention that there's.
Birne Binegar: we've had two homework sets since the second first exam homework set six and seven, those are also going to be explicitly covered on the exam.
Birne Binegar: And what i'll do on our next lecture on Friday it's all prepare a sample exam and then we'll step through it on Friday So you see precisely how the calculations, in theory, are going to match up on the exam.
Birne Binegar: Okay, so today what we're going to talk about our generalizations of the sector space our end that we've been dealing with since the beginning of the Semester.
yeah.
Birne Binegar: This is just a set.
Birne Binegar: Alright, it verbally ordered list of numbers.
Birne Binegar: ordered.
Birne Binegar: list this.
Birne Binegar: In numbers that seems like an abstract way of defining what we mean by we write something like this and think of it as a vector.
Birne Binegar: We have bigger pieces components, but the point are the courses, as far as to learn how to calculate inside the settings so bringing everything to numbers that we can add multiply and divide etc now that's that's what makes what we've done so far, concrete.
Birne Binegar: What we're now going to do is take the calculations, that we can perform in this arena and now see how it can be applied to more general settings.
Birne Binegar: But that's done by introducing a notion of an abstract vector space we still want to maintain the two fundamental vector space operation scale multiplication and vector edition, but we want to apply it to other sets other than just list of numbers.
Birne Binegar: Nevertheless, this is the set of vectors is going to remain important because, when I talk about a more general vector space, the basic idea of how to deal with it would be transported into this setting where we can do calculations.
Birne Binegar: Well i'm going to describe that process today, but first let's just establish the arena that we're we're we're working in now, we have some abstract set maybe it's my.
Birne Binegar: Operational mode to the string maybe it's a set of functions on which we can nevertheless pick a number.
Birne Binegar: And an element of that set and produce a new element of the set the an operation will call scale or multiplication and certainly like we have two elements of the set, we can add them together in some fashion.
Birne Binegar: Well, we need more than that, we need these operations to behave in our abstract set exactly the way those operations behave when they're defined on ordered list of numbers and it turns out from a purely mathematical point of view, if you have a set of operation.
Birne Binegar: set to operations and scale multiplication effect traditions that satisfy these eight properties, then effectively whatever abstract space you started off with it's going to behave exactly like.
Birne Binegar: The calculable extra vector states are in and what you doing linear algebra typically is.
Birne Binegar: Starting with some space a function, you want to be able to calculate with that explicitly so you transport everything into the calculable setting here get an answer, and then you just map back to your original setting to interpret that answer.
Birne Binegar: Alright, so now we don't want to do them today is give you some review the examples we covered last time and then i'll probably specialized to the case where we're looking at.
Birne Binegar: vector spaces that consist of functions, so that you'll actually see that maybe things you learned in calculus or differential equations how those.
Birne Binegar: Those operations all had a linear algebraic underpinning.
Birne Binegar: Alright, so just some examples of sets that have scale or multiplication and vector edition and.
Birne Binegar: The first one, of course, is just our end but also some spaces of our in they're also going to have no chance of scale or multiplication in Dec tradition, but in this case it's kind of trivial.
Birne Binegar: All you do to get a notion of scale or multiplication in the subspace are in is to use the regular usual skill and multiplication are an unusual vector addition.
Birne Binegar: And simply restrict the input vectors to live inside your subspace because these guys, if you have the subspace scale multiplication and vector addition.
Birne Binegar: are closed operation on that sub space as soon as you have a number in the vector you can get another vector inside the same vector space, so you get this scale and multiplication now.
Birne Binegar: and, similarly, you have two vectors you add them you get another vector so you get a map between pairs of vectors and vector space, these two operations give him your intrinsic.
Birne Binegar: notions of scalable cation Gala multiplication factor addition on your subspace and in fact they'll automatically satisfy the eight axioms that the vectors and North and already satisfied so maybe this case it's not so interesting because it's basically.
Birne Binegar: The same vector we've been dealing with sub spaces for quite some time now, so more interesting example, might be functions on the real line in here.
Birne Binegar: Your notion of steel and multiplication is again taking a number and an element of the set and producing a new element, but now what you do.
Birne Binegar: To create the new function which will call them to F that I get like taking a number and a function already in this set I simply define.
Birne Binegar: value of function Lambda F at the point X I define it to be Lambda times the value of the original function at that point, until this define the notion of scale and multiplication on the set of functions.
Birne Binegar: Similarly, if I have two functions in this set I can add them together well I need to define the function so i'll tell you what it's values on.
Birne Binegar: The value of this function F G is the some of the values that have a function F and G, at the same point so thief provide the definitions that giving you the two operations, you need.
Birne Binegar: And once you also given zero vector you can verify all eight axioms and claim is then does that this space of functions behaves exactly like a vector space.
Birne Binegar: At least with respect to these operations.
Birne Binegar: I also talked last time about you can think of the vibrational modes of a stretch string as being a vector space will steal a multiplication now either changes the amplitude of the.
Birne Binegar: vibrations, or maybe it reverses the face, etc, and Paul know meals etc alright, so that this much I actually talked about on Monday, the main point of today's actually show you how you go back and forth between.
Birne Binegar: The concrete vector space where we can do calculations and the more abstract vector spaces that are going to be interesting to your applications.
Birne Binegar: Alright So the first thing I need to do is perform a slight revision of all the basic definitions we had for.
Birne Binegar: linear algebraic apparatus associated with this vector states we had sub spaces linear transformation bases etc well let's redo those definitions.
Birne Binegar: What i'm going to effectively do is simply replace the vector space aren't in now with a more general vector space, so I subspace.
Birne Binegar: of our end it's a set of subset of art and that's closed under scale or multiplication in addition well to get a subspace of the general vector space.
Birne Binegar: You just need to subspace that set a subset of that set and you need it to be closed under scale and multiplication vector addition meaning if you do scale or multiplication you don't need to set or, if you do vector edition you stay within that set.
Birne Binegar: Until it's exactly what we had before just that the arena which will working is this is labeled differently Similarly, the span of a set of factors.
Birne Binegar: Well, as long as I can do scale or multiplication and vector addition I have a way of taking a linear combination of elements of an abstract vector space.
Birne Binegar: And when I look at totality of all the vectors like you reach by applying operation to scalable application and degradation that creates.
Birne Binegar: An example if the subspace and it continues to be called the span of instead of vectors only now the factors are living in an abstract space, rather than our end.
Birne Binegar: We have a notion of linear independence in our and what vectors are linearly independent if we can, the only linear combination that equals zero vectors the one we're all all coefficients or equal to zero.
Birne Binegar: same definition here is just that, on the right hand side here I don't have the zero vector the factor, whose components, the old religion of numbers which every number is zero rather we have.
Birne Binegar: A zero vector that specific to the vector space we're look working in but other NAB it's the same definition.
Birne Binegar: list of elements of an abstract vector space are linearly independent if the only linear combination that produces the zero vector inside that abstract set if the linear combination we're all the coefficients or zero.
Birne Binegar: Then, if you have a set of linear independent vectors that in fact generate the entire vector space that's what we call a basis, so what we need is a set of vectors that we have sufficiently many directions to reach every point in my set and.
Birne Binegar: We need them to be literally independent because the idea of a basis if we want to coordinate ties or vector space we will actually want to get numerical coordinates for every point inside this abstract vector space and.
Birne Binegar: In order for that to happen, we need these.
Birne Binegar: Elements of column vectors I need them to be linearly independent, you need to satisfy that condition, so that.
Birne Binegar: When I.
Birne Binegar: want to talk about a vector w living in a abstract vector space so abstract.
Birne Binegar: vector space.
Birne Binegar: If we have a basis for this.
Birne Binegar: If that's the basis well it's gonna be a set of linearly independent vectors that all live in here, but the main point is is that.
Birne Binegar: Every.
Birne Binegar: vector.
Birne Binegar: In your abstract vector space.
Birne Binegar: can be.
Birne Binegar: unique uniquely written.
Birne Binegar: say the equals some coefficient times your first beta factor some coefficient times your second base effect or.
Birne Binegar: chatter some base coefficient times your last basis vector with that so that uniqueness means that these numbers these coefficients do you want to see em are not arbitrary numbers they're actually fixed by your choice of V.
Birne Binegar: And in fact they're just they're just distinct choices and what then what you do if you use.
Birne Binegar: This order list of coefficients.
Birne Binegar: As coordinates.
Birne Binegar: For V and that's how we deal with the abstract vector space we actually coordinate ties it, and so we get numerical coordinates that we can now compete with.
Birne Binegar: Alright, so.
Birne Binegar: Just like for our end if you haven't a general vector space it's always going to have a basis.
Birne Binegar: The proof of this theorem I don't do it, yet in this course, because in fact it actually requires some sophisticated set theory that doesn't even require linear algebra it's basically.
Birne Binegar: dependent on the axiom of choice, but this can be proved every vector space has a basis and it's also not hard to show it all this does require linear algebra the show that every basis for vector space always have the same number vectors.
Birne Binegar: But.
Birne Binegar: If we know that every vector space has a basis and that every basis has the same number of vectors, then we can describe a certain number to a vector space, the so called dimension.
Birne Binegar: yeah you have a vector space that has a basis and every basis has the same number of factors that number of vectors will call the dimension of the vector space.
Birne Binegar: And it makes sense, according to our usual idea what dimension is it's going to be the number of independent directions, you have that will allow you to fill out that space.
Birne Binegar: Alright, so now let me give you some examples, because when we think of.
Birne Binegar: An example of an abstract vector space is being some space of functions.
Birne Binegar: What is the basis in this space of function how How would we recognize it.
Birne Binegar: well.
Birne Binegar: First, let me remind you what a basis does for you, so if you have any abstract vector space, and you have the basis for.
Birne Binegar: That means that you can write every vector as a linear combination i'm just reproducing this argument here, you can associate to this vector abstract vector a concrete numerical vector that you can actually perform calculation.
Birne Binegar: And so.
Birne Binegar: that's the idea, but now let me show you the application you calculus to you learn about Taylor expansions that's the statement that as long as you can.
Birne Binegar: have no difficulty in differentiating a function of my smooth I just mean you can compute derivatives, without any obstruction.
Birne Binegar: Then such a function is always going to have a tailor expansion.
Birne Binegar: That some value of the function at a point X zero value of its derivatives times the displacement from your expansion point and then higher terms that amount to.
Birne Binegar: computing higher derivative lighting by a corresponding factorial and then multiply that should be X minus X not raised to the power and.
Birne Binegar: But you have this expansion function in terms of well Constance linear functions quadratic functions, etc, the idea here, but I want to.
Birne Binegar: impress upon you, with is that this is in fact the expression of an arbitrary function in a vector states have functions with respect to a particular basis at basis so here, I have numbers.
Birne Binegar: gotten evaluating these derivatives at a particular point until they're going to be pure numbers until I have pure numbers time specific.
Birne Binegar: A specific family function but it's the same family of function, no matter what function function I started off looking at powers of X minus X not X minus X, not to the zero power X minus X, not the first power X minus X, not the second power etc so.
Birne Binegar: These basis functions are not changing yet every function that's not on the real line that's a smooth function can always be represented as a linear combination of these basis functions.
Birne Binegar: So yeah So these are exactly your basis functions and what This example shows you, in addition, how many you have a large variety of basis functions, because I can.
Birne Binegar: Even thinking of Taylor expansions it's been my application of linear algebra to function theory I have lots of different choices for this initial value X zero, and so this is not the I simply bearing X zero I get a whole family of functions and.
Birne Binegar: Here i'll make a remark.
Birne Binegar: Generally speaking, you always have an infinite number of possible bases were a vector space.
Birne Binegar: The one you actually use an application so it's going to be the one that's most convenient for the application at hand.
Birne Binegar: And in this example we're using a basis this particular basis could probably what we're interested in is the behavior of these functions near the point X zero and these higher terms.
Birne Binegar: You usually end up ignoring and just treating the first couple terms as good polynomial approximation so there's a utility to using this basis.
Birne Binegar: Because of these numbers as long as X minus X nine is the number, less than one these these functions, here it gets smaller and smaller as you as you approach X not and so you can at least approximate functions, using a.
Birne Binegar: finite list of functions as a basis, rather than the entire set by the way, this also shows you that.
Birne Binegar: In general, the set of smooth functions it's still the best vector space, because we can do scale or multiplication effect tradition but it's infinite dimensional because there's no bound on.
Birne Binegar: On these powers here unless we know that F is actually a polynomial there's usually no bound on that.
Birne Binegar: But.
Birne Binegar: If you go on you either linear algebra or in the theory of differential equations you quickly learned that.
Birne Binegar: You could handle infinite dimensional vector space is using the same rules that you have for why not dimensional vector spaces.
Birne Binegar: There are points in which things become a little bit more tenuous you have to make sure things converge in the like, if you have an infinite sum up terms.
Birne Binegar: Does that really make sense, because you can't actually compute this infinite some, but you might be able to take a limit or something like this, so you have problems with interpretation in an infinite dimensional vector space, but the linear algebra that series, the same.
Birne Binegar: Okay, so.
Birne Binegar: Aside from.
Birne Binegar: Having a.
Birne Binegar: Broader class of vectors thanks is, we can also talk about functions between vector spaces and the relevant ones are going to be what we call linear transformation.
Birne Binegar: they're going to be functions between vector spaces that behave, naturally with respect to the two fundamental vector operations.
Birne Binegar: We want it to be compatible with scalable location so whether we perform scale and multiplication before.
Birne Binegar: or after we play a function that takes us from one vector space to another, we want that to not make any difference.
Birne Binegar: and, similarly, when we add vectors we want to be able to either perform that addition in the original domaine de or performance afterwards after we apply the function sorry that the specter some in the co domain w we want to get the same result.
Birne Binegar: functions that have these two properties are called linear transformation.
Birne Binegar: Whether or not be as an ordered list of numbers, or a more general vector space.
Birne Binegar: All right, then.
Birne Binegar: When we were looking at linear transformations from say R and R m, we had to.
Birne Binegar: Stop spaces, we are associated with such linear transformation, the range.
Birne Binegar: And the current of a linear transformation now.
Birne Binegar: Those are defined.
Birne Binegar: you're saying, let me right linear algebra.
Birne Binegar: In fact they're defined as sub spaces, although i'm giving i'm giving you the definition that sets it's easy to check that both the range and the kernel of a linear transformation or continued to be closed under stealer multiplication and vector addition.
Birne Binegar: it's just a generalization of what happened for our end So these are both going to be sub spaces and now, finally, what I have.
Birne Binegar: Given that we can now look at more general vector station, so now, and I can give you a more natural interpretation of why we're interested in knowing, in particular the kernel of a linear transformation between two abstract vector spaces.
Birne Binegar: So let me give you.
Birne Binegar: A real critical example of a linear transformation on a vector space of functions.
Birne Binegar: If I have a number and a function that we can differentiate, then the thing about derivatives is a more or less ignore constant so I can.
Birne Binegar: multiply the function, before I differentiate it or multiply it by Lambda after I differentiate it you get the same thing well that's our compatibility with scale or multiplication property that we need to define the linear transformation.
Birne Binegar: right away and, similarly, we have a two functions, I can add them to create a new function.
Birne Binegar: property of this derivative that takes you from a differentiable function to another differentiable function is that it's compatible with this vector the addition vector is this additional functions.
Birne Binegar: And so, in fact, the derivative defined a function that goes from differentiable functions to differentiable functions.
Birne Binegar: It has a property that is compatible at the scale or multiplication on this set and the vector addition on this set so it's, in fact, a linear transformation.
Birne Binegar: Now.
Birne Binegar: In fact it's not just.
Birne Binegar: The derivative that has this property.
Birne Binegar: So.
Birne Binegar: The derivative.
Birne Binegar: Yes, a linear transformation.
Birne Binegar: The space of functions that in fact the vector space.
Birne Binegar: But.
Birne Binegar: But so is.
Birne Binegar: Well, the composition, when you take a linear transformation you go to a new vector space and you do another linear transformation that takes you.
Birne Binegar: To a third vector space, you can compose those linear transformation and they stay the compositions stays a linear transformation well Similarly, I can compose the derivative with another derivative and i'll get another linear transformation and so, in fact, you can.
Birne Binegar: Take higher derivatives that's just applying this operation multiple times and still a linear transformation.
Birne Binegar: And in fact you can take.
Birne Binegar: functions.
Birne Binegar: Times derivatives, and this is still, it will be interpret bowl as a linear transformation.
Birne Binegar: On the space of function is still going to have the property that if you act on a function that's differentiable by differentiating a certain number of times and then.
Birne Binegar: multiplying it by some function of X it's still gonna basically be compatible with scale multiplication in vector edition, so these kind of guys are also going to be interpreted as linear transformations but now.
Birne Binegar: The kernel of a linear transformation remember this was defined let's say our linear transformation is going from our.
Birne Binegar: All i'm gonna write it in the abstract some the CW well the kernel of such a function between two vector space is the set of point the in the domain, such that when I apply that function to it, I always get the zero, I get the zero vector inside the Co domain.
Birne Binegar: Right something like that Okay, and so.
Birne Binegar: If I have.
Birne Binegar: A linear transformation on this vector space of functions, safe and see if anyone symphony of art symphony do are and maybe What it does is it sends a function to.
Birne Binegar: let's say X squared.
Birne Binegar: GD X squared plus x the F X plus.
Birne Binegar: Two F.
Birne Binegar: Suppose it does takes a function differentiated twice multiply by X squared differentiate one multiply by X etc all turns out that this action F going to this guy is it going to be your transformation and in turn.
Birne Binegar: interpret the kernel of this is.
Birne Binegar: A kernel of that linear transformation it's going to be well let's just substitute in linear transformation is going to be the functions F.
Birne Binegar: inside our.
Birne Binegar: vector space of functions such that X squared dd X squared plus.
Birne Binegar: X E, F G X plus two F equals well the zero vector inside this space a function is the set of function that organically equal to zero for all values of X i'll just substitute in.
Birne Binegar: The number zero, but what I really mean here is the function of X, whose value is always equal to zero, and then.
Birne Binegar: So that's that function that's the zero vector inside this face and what i'm not telling you is that finding the kernel of this particular differential operator.
Birne Binegar: got others the linear transformation is actually equivalent to solving the original differential equation a differential equation, you have linear transformation operating in an arbitrary vector producing zero and that.
Birne Binegar: Is the young yeah that's that's.
Birne Binegar: Solving this differential equation is in linear algebra equivalent of finding the kernel of a particular differential operator.
Birne Binegar: Alright, so.
Birne Binegar: What I may do after the first exam is to show you how you can actually solve differential equations without doing any integrals or other things to undo derivatives, but rather by.
Birne Binegar: realizing a differential equation as as being equivalent to salt finding the solving differential equation is the equivalent the payment Colonel and then find the colonel, we can use our.
Birne Binegar: linear algebra techniques, we represent the linear transformation isn't matrix and we find the null space of that matrix and then that will give us the solution set.
Birne Binegar: That process doesn't involve doing any integrals at all it's just a matter of employing the linear algebra which comes from the setting that we can now regard faces the functions as being analogous to be.
Birne Binegar: Concrete numerical vector spaces.
Birne Binegar: Alright, so.
Birne Binegar: So.
Birne Binegar: After I introduce linear transformation and I talked a bit about ice amorphous.
Birne Binegar: So, and I some more system.
Birne Binegar: From.
Birne Binegar: Our and our and our.
Birne Binegar: Yes.
Birne Binegar: Just.
Birne Binegar: Any vertical linear transformation.
Birne Binegar: So it's more than just a function between two vector spaces it's a function for which you can go.
Birne Binegar: forward to a new vector space or and and you can also take a vector here and pull it back here.
Birne Binegar: back these dimensions are going to have to be the same, but that's something to prove but i'm not work, I some more system, I simply me function between to back your spaces, but which you have a functional inverse a way of going back to the original domain.
Birne Binegar: Now, when we were talking about.
Birne Binegar: there's two ways of viewing this one is in terms of linear algebra.
Birne Binegar: and the other is in terms of just pure function theory.
Birne Binegar: So, in terms of pure function theory, I have a set a in a set be and I have a function F that goes from A to B.
Birne Binegar: In order to get a function that goes in the opposite direction from the back to a.
Birne Binegar: There are two things that have to happen first of all, if you want to have this function.
Birne Binegar: Every point here has this is going to now be the domain, so I have to have a rule that tells me exactly where this guy goes it's like follow it back to the original domain, a well.
Birne Binegar: For what I can't have is a single point going back to two different points here right, because then the rule that tells me what the inverse of this point is it's going to be ambiguous.
Birne Binegar: function gives you a unique value for every every point in the domain, so if this is going to give you unique values, you can have multiple paths back to the original starting set that condition is called enjoy activity.
Birne Binegar: That means that F of a equals F of a prime only if.
Birne Binegar: You started from the same point A is equal to a prime and that's necessary if you want to have this inverse function, but the other thing you want.
Birne Binegar: In order for this inverse function to be defined, is that the idea is you're going to follow whatever he comes over here you're going to follow it back to get the value of the inverse function, but if you have points in your set be.
Birne Binegar: That don't live inside the image of this function, then these guys have no place to go, so what you need.
Birne Binegar: For this inverse map to exist, you need this to be the entire image of the set a and that's called subjectivity.
Birne Binegar: You want the.
Birne Binegar: set of points that are reachable by F to actually equal the entire co domain now what happens in linear algebra.
Birne Binegar: If these two function theory condition for.
Birne Binegar: an inverse to exist, can be stated in linear algebraic terms.
Birne Binegar: What happens.
Birne Binegar: Is that.
Birne Binegar: If he is a linear transformation now.
Birne Binegar: Then.
Birne Binegar: We have two associated subsets subspecies first one is the subspace up the Co domain.
Birne Binegar: And let me just write it it's really just the image.
Birne Binegar: it's all the points in w that can be reached by applying your transformation two vectors and ID.
Birne Binegar: And the Colonel.
Birne Binegar: team Well, this is the subspace of the of the domain and it is just equal to the.
Birne Binegar: point in the domain so little thing inside capital, the touch that T a B is equal or not the number zero when he maps into the sector space.
Birne Binegar: w, and so what I really need is the vector inside w that we interpret this zero vector right so that's.
Birne Binegar: The two sub spaces we've been calculating in the context of our end.
Birne Binegar: And what we showed in case of our and it's also true in this more general setting is that a linear transformation.
Birne Binegar: Is inverted.
Birne Binegar: Meaning he the minus first so if he is going from btw he to mind first is going from w back to be.
Birne Binegar: yeah if.
Birne Binegar: And only if.
Birne Binegar: well.
Birne Binegar: These two function theoretical properties have to hold, we have to cover the entire co domain and distinct point have to map to the State point.
Birne Binegar: Well, we can express those two conditions in terms of this linear algebraic language so.
Birne Binegar: The range of T.
Birne Binegar: equals w that's equivalent to.
Birne Binegar: subjectivity.
Birne Binegar: And the second condition.
Birne Binegar: Is is that the Colonel T.
Birne Binegar: defined linear algebra basically.
Birne Binegar: it's the same thing at the zero vector inside the domain consistent only the zero vector domain and in fact this.
Birne Binegar: let's straightforward the show this corresponds to inject him at.
Birne Binegar: The only way, you can reach zero vector is by applying your linear transformation exactly to the zero vector well that's what we showed in the context of our end that's equivalent to this condition that distinct vectors.
Birne Binegar: tenafly on this didn't land on the same point.
Birne Binegar: Okay, so.
Birne Binegar: So here we have a linear algebraic way of checking to make sure that an inverse function actually exists.
Birne Binegar: And so, if our linear transformation.
Birne Binegar: was actually say the action of differentiation on a vector space a function well, this gives us some concrete.
Birne Binegar: Things to check to make sure that we can actually solve that differential equation, what we want to do is undo the derivative, and so we we looking for some operation that's going to take us.
Birne Binegar: do in order to solve the differential equation, we have to undo derivatives, and so we need some inverted transformation and this gives us a linear algebraic way of checking yet.
Birne Binegar: right but it's actually a little bit deeper than that.
Birne Binegar: next series of topics, I want to.
Birne Binegar: Talk about is.
Birne Binegar: Well bases yeah how we use bases and i'm in first The first point is actually.
Birne Binegar: The following.
Birne Binegar: Every finite dimensional vector space is ice a morphic the some are in now is a morphic meaning there's a linear transformation that takes D to our end but there's also a linear transformation that takes.
Birne Binegar: In dimensional vector and interventional the miracle vectors that takes it back here, so what i'm saying here is there is there a way of core digitizing all the vectors and be in terms of ordered list of numbers.
Birne Binegar: don't.
Birne Binegar: This seems like it would be hard to prove.
Birne Binegar: abstract vector stacy's you can think of oscillations or states have a hyphen Adam how we actually supposed to associate with such abstract vectors these concrete numerical coordinates.
Birne Binegar: Well, the answer to that, and the reason why we emphasize bases in this course is that, in order to relate, you have tracked back your space to a concrete vegetable space where you can do calculation what's essential if the existence of a basis.
Birne Binegar: So when we were dealing with subspace is what we needed was a set of vectors living in the subspace that we're capable of writing every vector in the subspace as a linear combination then will you do those expansion coefficients to give us coordinates will use the same idea here.
Birne Binegar: What i'm going to show you is there's a linear transformation that will take your abstract vectors and map them to this calculation will setting.
Birne Binegar: But it's going to be an inverted in your transformation, so not only can we bring abstract vectors into a calculation of setting we can get answers here and take our answer that just.
Birne Binegar: mapping back to the original setting.
Birne Binegar: So let me show you why that's true suppose you have a basis for an abstract vector space that means every doctor in your space has unique expansion in terms of your basis factors well.
Birne Binegar: here's the linear transformation that's going to take us from are abstract vector space to the concrete vector space or in we take a vector and D.
Birne Binegar: My definitions and be as a basis, it has this unique expansion so, given the unique expansion, we have unique coefficients see one to see em.
Birne Binegar: I just write them as an ordered list of numbers and that creates an element of our end, and so I can go once I have a basis, I have a way of transporting abstract vectors through these concrete numerical vector.
Birne Binegar: and
Birne Binegar: What i'll show you that in fact this function between our abstract vector space and the concrete vector space is actually an ice and more efficient so it's going to have an inverse and we can go back and forth.
Birne Binegar: Alright, so first the show that a linear linear transformation is convertible we look at two things in linear algebra we verified that the range of that linear transformation is the entire code domain.
Birne Binegar: So.
Birne Binegar: This function ID here's the domain D and here's The co domain so to be subjective I need this map to completely cover the target space and you see you can do that.
Birne Binegar: To get this particular point in the co domain, all you have to do is start with this vector here and then this map which is going to grab the coefficients is going to reproduce the specter So in fact every vector inside the Co domain can be regarded as the image.
Birne Binegar: of a particular vector inside the until.
Birne Binegar: So this function between two factor spaces is subjective.
Birne Binegar: it's also easy to see that the that math that takes extract vectors to concrete numerical list of numbers is also has Colonel zero y.
Birne Binegar: All right, alright so here's the zero vector inside or in if i'm going to land on that zero vector by applying this function.
Birne Binegar: Well, what it does, what does it do this function just grabs the coefficients so if this.
Birne Binegar: If this factor is going to land on this point i'm going to have to have see one is equal to zero see two is equal to zero C three is equal to zero all the coefficients have to banish but if all the coefficient Spanish and the vector I started off with had to be the zero vector.
Birne Binegar: And so the shows you that the only vector that gets mapped to the zero vector is going to be the zero vector inside my original vector space, and so we have.
Birne Binegar: A linear transformation between your abstract vectors things D in your concrete that your stakes aren't and but which.
Birne Binegar: Which which is convertible because its range is the entire vector space on our end and it's Colonel is just the zero vector.
Birne Binegar: Alright, so we now we have this ice amorphous.
Birne Binegar: and
Birne Binegar: Your I talked about.
Birne Binegar: How R n has a natural basis but we've talked about this before here's your natural basis, but this coordination map doesn't really do anything in this.
Birne Binegar: In this context, because it's so trivial as soon as I expand with respect to this basis, and then I grabbed the coefficients I get back to the original vector I started off with so there's not much utility of.
Birne Binegar: For this notion of faces inside our end you already have a perfectly good basis.
Birne Binegar: But, in general, when you're dealing with an abstract vector space you can't even get started until you choose a basis.
Birne Binegar: The reason is, is that.
Birne Binegar: Think of these guys as being just a set of function well how am I going to manipulate these functions how am I can actually figure out.
Birne Binegar: What value of dysfunction I need in order to have a certain property I need the way of computing and then an abstract vector space, you have.
Birne Binegar: abstract object, but they're going to be quite different than numbers, something that you can explicitly compete with.
Birne Binegar: And so what one uses is this organization map that not only allows you to take your abstract vectors and reinterpret them in terms of the coordinates and our end, but it also allows you to take whatever answer you got here back to your original setting.
Birne Binegar: And so here's how the calculations scheme works in a more general vector space.
Birne Binegar: You have something you want to compute.
Birne Binegar: You map it over into our end where you can actually carry out the computation you get an answer, and then you just somehow map it back, I mean you'll you'll get a specific vector here that will be written as a linear combination of the basis vectors but what.
Birne Binegar: I need you to understand that this stage of the game is this whole system because they're being able to do computations and D and get answers.
Birne Binegar: You need a basis to shut off and that's why basis is we're always stressed up to this point in the course because, as soon as we leave the comfortable setting up or in you can't do anything without a basis.
Birne Binegar: In fact.
Birne Binegar: I got two minutes left, let me give you an example of how have you seen this before.
Birne Binegar: In math 2233 the second half of the course you're looking at.
Birne Binegar: you're looking at linear second order differential equations so.
Birne Binegar: differential equations of the form, why double prime plus P of X, Y prime plus Q of X, Y is equal to zero.
Birne Binegar: And what you're learning in differential equations is that if you have a differential equation of this form.
Birne Binegar: The general solution.
Birne Binegar: Is of the form.
Birne Binegar: y X equals some constant times one solution, see to.
Birne Binegar: find another solution where.
Birne Binegar: The only condition life on these.
Birne Binegar: options why one in why to is that why one and why to satisfy the differential equation.
Birne Binegar: and
Birne Binegar: they're wrong skin.
Birne Binegar: Which is defined as why and why to prime minus one prime like to that does not equal zero.
Birne Binegar: So.
Birne Binegar: This is what happens in the second half of 22 three you you're interested in solving differential equation of this form.
Birne Binegar: And you expect answers to have this functional form, but now we can interpret this linear out linear algebra actually.
Birne Binegar: This is basically a linear transformation, acting on a function y so solve the differential equation is going to be equivalent defining the Colonel.
Birne Binegar: differential operator, corresponding to the left hand side of this differential equation, and the answer it's going to be a linear combination of two special functions.
Birne Binegar: First of all, they have to live in the space of solutions so that's why they have to satisfy the differential equation, and in fact what we're saying here is that every solution.
Birne Binegar: can be written as a linear combination of just two solutions and then so, in fact, the functions why one in like two are going to provide a basis for this solution set and, moreover, we have this peculiar RON skin condition.
Birne Binegar: What this roski condition actually means linear algebra thankfully that these two functions why one lie to their linearly independent in the sense that the only way I can get this linear combination that equals zero is to take both see one and see two separately equal to zero.
Birne Binegar: The way you see actually see that is you.
Birne Binegar: Look at this and think of it as a differential first order differential equation for saying why to in terms of the function why one and what you'll find is when you solve that differential equation.
Birne Binegar: If this combination equals zero, then it must be that why to is a constant multiple of why one, so the only way.
Birne Binegar: they'll run scheme of two functions equals zero is one the two functions are linearly dependent in the linear algebra makes sense.
Birne Binegar: So what's happening now, and I hope I don't confuse you too much before the second exam is now suddenly the arena for which we can do linear algebra has broadened dramatically.
Birne Binegar: don't worry too much at this stage again at the stage exactly how we're going to interpret differential operators as linear transformation.
Birne Binegar: i'm just trying to motivate you why it's an interesting question we're actually never going to really address that in this course, but as soon as you leave this course.
Birne Binegar: Things are going to be stated linear algebra when you try to solve the differential equation, you should think of it.
Birne Binegar: should say you do a power series solution, you can think of that i'm trying to express my solution with respect to this basis of one X X squared X, to the third etc, and things will start to.
Birne Binegar: coalesce a little bit for you both.
Birne Binegar: Your.
Birne Binegar: Your other subject differential equation you'll see how linear algebra actually supports that supports that point of view and and actually becomes vital when you start solving more difficult differential equations.
Birne Binegar: Pardon why don't they have us take this course.
Birne Binegar: um.
Birne Binegar: Well, I mean, for example, you can go through all the first order differential equations not do any linear algebra and, in fact.
Birne Binegar: This two dimensional vector spaces, is just a little bit more complicated than a one dimensional lecture states are.
Birne Binegar: So it's like a baby steps towards you this idea, but really as soon as you leave this course, then you combine differential equations linear algebra now you're in a.
Birne Binegar: Much brighter setting you're sort of in a position to understand what 48 series are now and other more complicated things but.
Birne Binegar: And in fact i'm not sure we still do this, we used to teach a course that was linear algebra and differential equation that does exactly like that except we can't cover very much because there's so much to talk about.
Birne Binegar: Okay, so i'm sorry ran over so i'll stop here today and on Friday what we'll do is a review session for the exam on Monday.