Birne Binegar: Okay, so.
Birne Binegar: Let me remind you we've scheduled our second exam next Monday and.
Birne Binegar: we'll do a review a full review on Friday i'll i'll do is we did last time i'll try to give a quick summary of the material we've covered and then we'll work our way through a practice exam that.
Birne Binegar: should be pretty close to the actual exam you would take on Monday, but will do that review.
Birne Binegar: On Friday today i'm actually intended today's lecture is actually do a bit of review as well, but what i'm going to do.
Birne Binegar: Is i'll place the things we've learned about the vector space are in place it in a more general context context, so you see.
Birne Binegar: Why we spend so much time talking about sub spaces and bases, etc, is because now we have a platform where that apparatus becomes useful and, in fact, what we need is is it's actually the language of linear algebra the talk about.
Birne Binegar: linear algebra as it's going to be used in applications beyond this course, so let me remind you quickly.
Birne Binegar: What we did on Monday let's see if I can.
Birne Binegar: rabbit little better okay so.
Birne Binegar: When I get on Monday I introduce the abstract notion of an abstract or general vector space we're now on our vector space it's not necessarily an ordered list of numbers rather at some abstract set could be vibrations, it could be the.
Birne Binegar: States have an elementary particle that many situations that we're going to.
Birne Binegar: Now, be able to handle using the language we've developed or in what we need is a set the.
Birne Binegar: and on on that set we need some notion of scale of multiplication some way to take elements of the set and multiply them by a number to produce a new element of the set and we need some way of adding them together and surprisingly.
Birne Binegar: Maybe it's not a surprise because the first time you've seen it, but it is to me it's kind of surprising that just setting up an abstract set and giving two functions on that on that set gives us the whole structure of linear algebra.
Birne Binegar: So we've been developing that piecemeal throughout the Semester, but i'm.
Birne Binegar: point now is that we.
Birne Binegar: These eight identities are also satisfied once we know how to add elements that are set in once we know how to do scale or multiplication if these properties whole then it's then in effect.
Birne Binegar: are set D is going to behave exactly like our calculable vector space RN and in fact that's the whole purpose of this course is to bring abstract vector spaces into a calculation calculable setting.
Birne Binegar: So what we're going to do today is i'm going to step through the definitions we use for our end but i'm going to place them in the context of a more general vector space.
Birne Binegar: And then, as I move through these definitions partially to prepare you for the exam on Monday, but also the show you why you're actually taking this course, and finally.
Birne Binegar: sort of sad that it's actually at this point in the course that we're actually able to display with linear algebra is good for.
Birne Binegar: OK, so now Let me run quickly through the examples of vector spaces that abstract vector spaces, that I covered on Monday, first of all, a familiar vector space aren't and that consists of ordered list of numbers.
Birne Binegar: Naturally, once we define the usual notions of scale multiplication and vector addition for ordered list of numbers all eight of the proceeding properties they hold automatically them, looking around here.
Birne Binegar: Then i'm.
Birne Binegar: stephanie second example was well once we discuss our end and I talked about sub spaces apart, and in fact subspecies of our end or another example of an abstract vector space because.
Birne Binegar: Because you have closure on the scale of multiplication and deck tradition.
Birne Binegar: You can just by restricting to the vectors that live inside this subset you get maps that take you from.
Birne Binegar: numbers and elements of a subset back to your element to use upset or pairs of elements inside of subspace Matthew back to another element inside the subspace until you get.
Birne Binegar: well defined notions of scale or multiplication in Dec tradition, when you restrict the vectors to live inside the subspace and that gives the subspace the structure of a general vector space.
Birne Binegar: here's a more abstract example that it's actually going to be important today's lecture most of the examples of an abstract vector space, I give today are going to be function spaces, but in fact.
Birne Binegar: anytime you have a function taking values in a in a vector space doesn't even have to be are you could have like the electromagnetic field where you have every point in space taking a value in.
Birne Binegar: in a particular direction you know the direction that the electric field is setting or the gravitational field basically anytime you have a function space.
Birne Binegar: it's going to be it's going to have well defined operations at scale or multiplication in Dec tradition.
Birne Binegar: Except that rather than acting on order list of numbers now scale or multiplication is changing a function or adding two functions produces another function, and so we can now view functions on the real line or.
Birne Binegar: Effectively anything now we really only need to take up the values of the function.
Birne Binegar: Be in some depth or space that we can do these additions and scale or multiplication long as we have that we have another vector space and so something that's amenable to our linear algebra techniques.
Birne Binegar: Then we talk, I talked about the vibrational mode string that's also.
Birne Binegar: Non mathematical non it may be physical me when i'm talking about the library just bringing reality, not the mathematical model that we describe it, but it's still an abstract vector space and polynomials are again another example of the vector space.
Birne Binegar: Okay, so now, the point of this is that.
Birne Binegar: We focused on our end because, in fact, and the purpose of our end is it gives us the calculation of platform, by which we can study all these more general examples of the vector space.
Birne Binegar: So, to prepare for that one, I want to do is just step through the definitions that we know concerning.
Birne Binegar: The vector space are and and now just change our end to V and i'll restate the definitions are essentially the same.
Birne Binegar: For example, if you have a abstract vector space, and you have a subset of it that's going to be if it's closed under scale or multiplication and vector edition then we'll call continue to call it a subspace well.
Birne Binegar: The original definitions, where you take a number and an element of your set sorry there's some typos here.
Birne Binegar: And a vector that lives inside your your subset if that scale of multiple remains inside that subset and it's closed under scale or multiplication.
Birne Binegar: and, similarly, if you start off with two vectors living in this subset an abstract set D, but when you add them together, using the rules for addition and in the.
Birne Binegar: End you stay inside that set well that's closure under vector edition, and we say that any subspace it's closed under both these operations is the subspace.
Birne Binegar: Similarly, we can take a set of objects sitting inside this set, we have a way of scale or multiplying them, we have a way of adding him, and so we have a way of taking linear combinations of these guys, so we can take linear combinations of functions, if you will, and if.
Birne Binegar: You do that all the time it's just just you've never thought of it as a linear algebraic action.
Birne Binegar: But if we can take linear combinations of elements of this set, then we can form general linear combinations and in fact create an entire subspace.
Birne Binegar: Wherever this whatever this function space is if we have a list of elements in it, we can take all possible linear combinations of them, and that will create a subspace of this abstracts vector space.
Birne Binegar: Well, and then.
Birne Binegar: For ordered list of numbers, we had a notion of linear independence that was if we had a.
Birne Binegar: list of all the list of numbers, and the only way we can get linear combination of them to equal to zero vector with be by taking all the coefficients separately equal to zero.
Birne Binegar: Well that's What if this condition hold we say that the vectors that we started off with are linearly independent and all i'm doing in this definition is replacing.
Birne Binegar: Our n like this general vector space D that's exactly the same definition it's just the arena in which we're carrying out these operation has changed we're now doing it in an abstract vector space, rather than you are and.
Birne Binegar: And we still have a notion of a basis a basis for subspace in our end with a set of linearly independent vectors that was capable of generating the entire.
Birne Binegar: vector space and that's exactly the definition will use in this more general context, what we require to have a basis for an abstract subspace of a general vector space is.
Birne Binegar: Well, we need to be able to take linear combinations of these generating vectors and and thereby create the entire vector space.
Birne Binegar: And secondly, we need these guys to be linearly independent because the main utility of a basis is to provide coordinates and in the current context that's even more important.
Birne Binegar: Where our vector spaces as us.
Birne Binegar: solution said differential equations or its States have a hydrogen atom vibrations up an oscillating spring I can describe those objects, but I you don't have a clue as to how to calculate with them until you can actually bring them into.
Birne Binegar: A setting where numbers can work and that's exactly what this notion of basis does for it.
Birne Binegar: I introduced it as a way of coordinate icing substances saying, for example, I had a three dimensional subspace.
Birne Binegar: Of attend dimensional space I don't want to use 10 parameters to describe the elements of that three dimensional space, so we introduced this concept of basis as a way of giving.
Birne Binegar: Better coordinates for the elements that actually live in a smaller dimensional subspace of a larger vector space.
Birne Binegar: In the current context, though it's even more important, because when you have an abstract vector space you don't have any way of attaching numbers to it at all until you actually furnish it with a basis.
Birne Binegar: But.
Birne Binegar: Our principal results concerning bases for our N, are going to carry over until bases for these abstract vector spaces, so, in particular, every vector space where the rich or and or more general vector space, it does have a bassist.
Birne Binegar: This this theorem actually.
Birne Binegar: doesn't rely so much on linear algebra rather on abstract set theory it's basically reducible to the actual name of choice and set theory.
Birne Binegar: But the other thing is that common thing that carries over into our more now general setting is that every basis for a vector space has the same number of factors.
Birne Binegar: In the context of our N, the number of faces sectors we use that to ascribe a dimension to our subspace well now what.
Birne Binegar: If every basis for a general vector space always have the same number of vectors now we can ascribe a notion of dimension that even these abstract vector spaces, is a number of elements, you need in order to reach every point in that space and to provide you with good coordinates.
Birne Binegar: All right.
Birne Binegar: So let me give you.
Birne Binegar: An example of how will use bases in a more general vector space basically it.
Birne Binegar: has to do with what I was stressing before the ability to use bases to coordinate eyes subspace.
Birne Binegar: Let me recall that procedure only now, instead of saying subspace i'll just talk about a general vector space so suppose the is a general vector space, it could be a substation more and but now we have more examples of these.
Birne Binegar: General vector spaces and what I want.
Birne Binegar: be one through be N, is the basis for that what that means is first of all that every element of the set can be written using a linear combination of these given list of elements of the of the.
Birne Binegar: and, moreover, if it's a basis, then these numbers these multipliers see one to see and they're all numbers and, in fact, since I have.
Birne Binegar: ordered list of vectors here i'm going to get an order list of coefficients, then I can now consolidate into an ordered list of n numbers until an element of our end so what this is doing for us.
Birne Binegar: it's it's taking.
Birne Binegar: These abstract vectors think of it as a function or vibration and it's giving you explicit numerical coordinates for that, but.
Birne Binegar: The way you go from abstract vectors to concrete coordinate so that you need this basis in order to get that process started.
Birne Binegar: Alright, so let me show you how have you seen bases before, but it was never explained to you in in the language of linear algebra.
Birne Binegar: For example, in calculus to you learn that every every function that you can differentiate repeatedly on the real line always has unique Taylor expansion.
Birne Binegar: looks like this value function, the point X evaluator expansion point zero it's derivative evaluated that point time displacement from the expansion point and similar terms here's the generic term oops this should have been raised to the power and.
Birne Binegar: And in capitalist to this is introduced this a way of.
Birne Binegar: trunk if you want to approximate a function, you can just truncate this infinite expansion in a finite point, and then you get a polynomial that approximate some otherwise we're complicated function.
Birne Binegar: But what I want to point out here is i'm hoping that you recall, this theorem but what I want to point out is that now, you can understand this situation is saying that the function on the real line that that you can differentiate.
Birne Binegar: They form a vector space with the.
Birne Binegar: usual notions of scale, a multiplication.
Birne Binegar: scale a multiple of a function by a number Lambda is the new function, whose value at the point X is Lambda times the value of the function of effects, so this is how we take a number and a function and we get a new function.
Birne Binegar: and similarly.
Birne Binegar: We can we have two functions, we can add them we create a new function, whose value it X, this is some of the values of of F and G so.
Birne Binegar: These two operations are easy to define give the space of functions, the structure of a vector space and here when i'm saying is this Taylor expansion.
Birne Binegar: What it amounts to is expressing an arbitrary function as a linear combination of a given list of functions, and in fact the tailor expansion is unique.
Birne Binegar: Once you have the function these coefficients it's just a number right so evaluated at a particular point so even though.
Birne Binegar: You may have to use your calculator to get a value for this thing this is just a number, this is just a number, this is just a number as well, and what they're doing is they're multiplying particular functions.
Birne Binegar: The first number is just multiplying function who's constant values equal one, the second number is multiplying the function, whose value is X minus X not so here's.
Birne Binegar: insert the one here, those are your basis factors.
Birne Binegar: Here, and here, and then these are your coefficients.
Birne Binegar: And so um yeah so the tailor theorem is just a theorem that tells you that the vector space of differentiable functions has a natural basis and, in fact.
Birne Binegar: It has many bases, because you can choose your expansion point at will, you can choose to expand around the origin X equals zero you can expand around X equals one So in fact it's not only shows you that.
Birne Binegar: You have a natural basis for functions that you can differentiate but, in fact, you have many, many choices of bases as well, and that was part of the point of talking about substations as well as the fact that.
Birne Binegar: We have many bases in general, what you want, is a good basis that makes.
Birne Binegar: things easier for you, rather than more complicated think of this as just finding a good basis is just equivalent to finding a good set of coordinates.
Birne Binegar: What is a good set of coordinates depends on the sort of problem you're looking at.
Birne Binegar: If you're working on a sphere i'm working outside the context linear algebra but if you're again a sphere, you want to use spherical coordinates if you're working in a play and you'll want to use Cartesian coordinates well.
Birne Binegar: Now, in this more general context when we're working with functions that might be best to simplify matters is to choose a nice basis of function, so that.
Birne Binegar: A problem that we're looking at becomes a little bit easier and then you can understand this Taylor expansion.
Birne Binegar: As saying well we're looking to look at general functions on the real line, but what we're interested in is the behavior functions newer particular point zero.
Birne Binegar: Then it makes sense to look at the functions and think of them in coordinates in which the behavior nearly point zero becomes increasingly important and so you're you're actually you're able to focus on a particular aspect of the function, by choosing a nice basis for this.
Birne Binegar: Now this this idea it's not just limited to Taylor expansions.
Birne Binegar: If you do circuit theory and you want to.
Birne Binegar: Maybe you have a signal electronic signal fluctuating through a circus.
Birne Binegar: Maybe you even have it as a laboratory you put your circuit up to an oscilloscope to see that changing surface surface.
Birne Binegar: Well, typically wave.
Birne Binegar: Electrical engineer will analyze such a situation, if, by doing what's called a 48 expansion, where instead of.
Birne Binegar: You can actually think of the oscillations of that circus is forming a vector space and the way you get a basis for that is by.
Birne Binegar: enumerating the natural vibrational modes yeah you can think of it as.
Birne Binegar: Having a simple sign function or.
Birne Binegar: Maybe a harmonic associated with that simple sign function.
Birne Binegar: You think of this, the signal you see on on your seller scope what you can do that, you can take.
Birne Binegar: I want you to your functions, and you can expand it.
Birne Binegar: For example, C and cosine.
Birne Binegar: and
Birne Binegar: You can expand it in terms of these functions that naturally oscillate and when you do, that what you what that is called it's a 48 extension.
Birne Binegar: And it's used to analyze.
Birne Binegar: The signals in a.
Birne Binegar: Maybe a musical signal or other electronic signals use to analyze now in terms of simpler based vibration, so to speak, but what it is it's a bunch of coefficients times particular functions and there again and you're seeing the use of a basis, this time not.
Birne Binegar: Simple powers of X minus X not now what we're using as a basis for this, the space of functions is a bunch of trigonometric functions, but they'll still play the same role linear algebra.
Birne Binegar: They allow us to express an arbitrary function as a linear combination of a given list of functions and then what the engineers do is they regard these numbers here as the coordinates that will actually reproduce a given a given signal.
Birne Binegar: Okay, so for all this apparatus that we've been developing it really comes into fruition.
Birne Binegar: When we start looking at vector spaces, other than or in yet are in remained the calculation of platform that we strive to bring in more general vector space, such as a space a function, we try to bring it into that calculable rena.
Birne Binegar: Okay, and that's what basis do for us bases allow us once we we've got a bases identified, then we can expand every function in terms of them, and we can use the coefficients of those expansions to get a good numerical handle on the more abstract object we're dealing with.
Birne Binegar: All right, yeah yeah.
Birne Binegar: yeah well yeah and, in fact, what what you'll find in your applications is your first problem and actually the hardest problem that you have it all in your analysis is to find that basis, so, for example, when you.
Birne Binegar: When you saw the second order linear differential equation why double prime P of X lifetime value of X, Y is equal to zero, so if you've had.
Birne Binegar: math 20 to 33 the whole second half of the course is dealing with differential equations like this and what you know about in math 20 to 33 is that the solution side of this.
Birne Binegar: Well, the general solution is always expressed as a linear combination of two.
Birne Binegar: Two solutions that satisfy the right condition, what does this really saying it's saying that.
Birne Binegar: Every every the solutions of this or functions and back we're looking at a subspace of the function, we met the sub.
Birne Binegar: The functions that actually satisfy this differential equation, so we get a sub space at the space of functions, but that's still.
Birne Binegar: A vector space and here we're this statement that the solution must be this form is equivalent of saying that solutions have a linear differential equation.
Birne Binegar: Second order linear differential equation or always linear combinations of two basic solutions.
Birne Binegar: And these are your so called fundamental solutions, but what they are, they are basis for the solution set.
Birne Binegar: And the statement that every solution of this has this form is really equivalent to the linear algebraic statement that your two solutions are actually sufficient to generate every solution.
Birne Binegar: But to your question, the hard part about solving a differential equation is finding those two solutions yeah so so.
Birne Binegar: Effectively, all i'm really telling you is is that in a more general context where you have a more abstract vector space is that yeah you need a basis to get started that's the first major problem that you're going to run into.
Birne Binegar: But once you have that now once you have your solutions is now, you can think of them as vectors and you can be able to you can manipulate them into vectors.
Birne Binegar: And it will you'll actually have a whole language for dealing with solutions have them have a differential equations that you'd like before you had linear algebra.
Birne Binegar: Okay where's my clicker here.
Birne Binegar: Alright, so.
Birne Binegar: Another thing we had in the context of the vector space are and what's the notion of functions that we take you from one are and to another.
Birne Binegar: We call those linear transformation if the functions would be compatible with scale multiplication and vector addition well like just changing these domain and co domain from R and R m to more abstract vector spaces D and w same definition holds.
Birne Binegar: And in fact.
Birne Binegar: We can also define the range of linear transformation and it's Colonel.
Birne Binegar: The range, well before it was a set of vectors in the co domain that could be reached by applying your linear transformation and that's exactly the same definition will use.
Birne Binegar: In now this context of a linear transformation between two general vector spaces and the Colonel sorry went off screen here the Colonel is the same it's just.
Birne Binegar: it's the set of vectors in the domain.
Birne Binegar: They get mapped to the zero vector over here and sorry, this is a typo as well, it should be the zero vector inside w.
Birne Binegar: And, but this.
Birne Binegar: These are two substances that we can now attach to any linear transformation between two abstract vector spaces Okay, let me show you what these are good for.
Birne Binegar: In the context of them back your space of functions.
Birne Binegar: Well, first of all.
Birne Binegar: You have a function which you can take as many derivatives as you want you're going to get an operator that's going to take you from.
Birne Binegar: differentiable functions to differentiable functions, I simply differentiate and what I want to point out is that action.
Birne Binegar: When you take the derivative of a scale are multiple function you just get scale or multiple of the derivative of that function, and when you take the derivative of the summer functions, you get a some of the derivatives let's just.
Birne Binegar: Different differentiation ignores confidence and it distributes over additional functions well these two conditions.
Birne Binegar: tell you precisely that the act of taking a derivative is a linear transformation on the space of functions.
Birne Binegar: you're the two properties, we want to be able to multiply do scale multiplication either before or after we apply the derivative and get the same result well.
Birne Binegar: that's compatibility with scale or multiplication in here we can add two functions before or after we apply this.
Birne Binegar: derivative operation and we still going to get the same result well that's compatibility with that tradition, and so this operation taking your derivatives going to take function to a new function well it's actually a linear transformation on this on this space of functions, and moreover.
Birne Binegar: skipping a little bit ahead, not only is taking a single derivative a linear transformation, but if you take multiple derivatives it's just like composing a linear transformation with another linear transformation, you get another linear transformation, so, in fact, not only is dx or dx.
Birne Binegar: Not only is that a linear transformation, but if you take arbitrarily high derivatives.
Birne Binegar: it's still a linear transformation and, in fact, you can do things like take a function of X.
Birne Binegar: and take a higher derivative and this thing is still going to act like a linear transformation on your space of functions.
Birne Binegar: Okay, so why is this up yeah.
Birne Binegar: Yes, yes that's my next was my next example talk about laplace transforms and you'll see that that's also a linear transformation.
Birne Binegar: Okay, so.
Birne Binegar: So if all these guys are linear transformations well, we can ask well, what is the range of the linear transformation, what is the current in this context, the kernel is going to be more interesting.
Birne Binegar: suppose I construct a linear transformation that Center function F.
Birne Binegar: So a new linear transformation on the functions that we can differentiate will send F two.
Birne Binegar: I don't know.
Birne Binegar: X squared.
Birne Binegar: GD X squared plus x dx plus.
Birne Binegar: Plus, maybe one all have that lie to me.
Birne Binegar: So suppose suppose that taught linear transformation my claim is that this guy.
Birne Binegar: is going to be seen to have the same property, so I can do scale or multiplication or vector addition.
Birne Binegar: Before after I do this operation on it on a function i'll get the same result so suppose, this is a linear transformation let's call it T and the kernel of tea remember this is defined it's going to be the set of functions in your domain.
Birne Binegar: Such that he applied to that function is going to equal the zero function.
Birne Binegar: What I want to call it.
Birne Binegar: I call it F zero.
Birne Binegar: inside.
Birne Binegar: See infinity you are well here what I mean is the function that behave like the vector zero in what when see infinity of ours regarded as a vector states, but what this is it's actually this.
Birne Binegar: is going to be a set of functions.
Birne Binegar: F and C infinities.
Birne Binegar: are such that.
Birne Binegar: X.
Birne Binegar: X squared GD X squared plus x the X plus one times X is equal to zero, and so the kernel.
Birne Binegar: Of the linear transformation, corresponding to this differential operator well it's Colonel that's exactly the solution, set up a differential equation that corresponds to this differential operator.
Birne Binegar: And, in fact, if you go on if you've had linear algebra this has been working in the background, all the time, but if you go on to intermediate differential equations or partial differential equations this point of view regarding.
Birne Binegar: linear differential operators is being linear transformation that's really important and, in fact.
Birne Binegar: The whole apparatus of linear algebra is then brought to the forefront, where you typically.
Birne Binegar: Well, for example, when you have a differential equation you can't solve by simple means What do you do, you might try a power series.
Birne Binegar: What is the power series method well it's just expanding it your functions in terms of the basis of one X squared X X, to the third X, to the fourth and then trying to figure out what those coefficients are so you're just trying to figure out.
Birne Binegar: How to how to get coordinates of a given function with respect to the standard basis, consisting of function one X X squared X, to the third etc.
Birne Binegar: So all that all that apparatus that you.
Birne Binegar: Were seeing purely in the context of differential equations you can now see, in fact, is an application of linear algebra to the setting of differential equations.
Birne Binegar: And all you need for that to happen, is to have this abstract vector space to work with, and then, once you're in that setting now all the linear algebraic apparatus comes into play all right, and there was another.
Birne Binegar: example alluded to suppose I define.
Birne Binegar: A function l that's going to take me from.
Birne Binegar: Oh, I don't understand the infinity of our although.
Birne Binegar: I should be more careful I mean now what I need is.
Birne Binegar: I don't even need seen any of our when I need are integral functions, but they don't have good notation for that.
Birne Binegar: might help, so, if any, would contact support maybe I don't know anyway, what I want to describe just let's think of this is just going some some nice class functions.
Birne Binegar: Nice class functions what it's going to do.
Birne Binegar: it's going to take a function.
Birne Binegar: And it's going to produce a new function.
Birne Binegar: By integrating zero to infinity F of X integrating against a decaying exponential function and here you have a new parameter.
Birne Binegar: And we use that for each value of this parameter so we're going to get.
Birne Binegar: When this is a number, this is just an integral of a function on the entire.
Birne Binegar: Positive no real axis and i'll get a number of this so for each value of the parameter so i'm going to get a particular number, and that gives me a function of the new parameter s.
Birne Binegar: turned out that this is a effectively a linear transformation that takes me from functions on the real line with variable edge to a new a new function or now the underlying variable is S, and indeed.
Birne Binegar: Lambda.
Birne Binegar: We take laplace transforms the skills are multiple function you just going to get Lambda time to scale up multiple because.
Birne Binegar: A replace F by Lambda times, if I can just drag that constant through the through the integral sign, because the integration ignores Constance and we're also going to have.
Birne Binegar: laplace transforms.
Birne Binegar: Have a summer functions it's going to equal the sum of their transcripts.
Birne Binegar: reason for that is that when I replace F, I have some assumptions, I can then integrate turn by turn and then i'll see both both terms are.
Birne Binegar: You basically class transforms.
Birne Binegar: So in fact there's a whole lot of apparatus that you've been learning since you've been in college is now being brought into.
Birne Binegar: This arena of linear algebra.
Birne Binegar: One purpose of.
Birne Binegar: Doing this transformation.
Birne Binegar: Was is to actually solve differential equations and what you're doing is you're taking.
Birne Binegar: A given presentation of a function you're transporting in into a better calculate trouble setting doing a calculation there and then of course remember in.
Birne Binegar: The hard part of differential equations it's not doing the laplace transforms but doing the inverse whiplash transform but it's there, and although we don't necessarily use linear algebra.
to calculate.
Birne Binegar: These will flock transformed it's certainly there in the background playing an active role.
Birne Binegar: Okay.
Birne Binegar: Oh let's see.
Birne Binegar: So there's one other thing I wanted to say today, and my that way now today's lectures is it's kind of informal nothing that.
Birne Binegar: I say today's lecture is actually going to appear on the exam However, what i'm hoping to do is to make the materials exam a little bit more palatable by giving you a wider context, I do want to warn you, though, is that when you think about a basis don't think about.
Birne Binegar: tailoring Taylor expansions or.
Birne Binegar: The very end when you're looking for examples that give you some confidence that you're thinking.
Birne Binegar: about it the right way, maybe that's the time to think about Taylor expansions.
Birne Binegar: vibrational mode, the thing to remember, while you're in this course, is it for the most part vectors are just ordered list of numbers and even when I venture, as we can now venture out the consider more general vector spaces.
Birne Binegar: Our goal is actually to transport whenever we're looking at into that calculable setting of our end until.
Birne Binegar: So the discussion today is just to show you how you're going to be using linear algebra once you leave the course but.
Birne Binegar: While you're in this course think of linear algebra and how it's been taught us for that we're primarily interested in understanding how to work with ordered list of numbers.
Birne Binegar: Alright, so.
Birne Binegar: Let me.
Birne Binegar: talk a little bit about vector space is a more systems, so what are they they are.
Birne Binegar: There linear transformations between factor spaces, but what, in addition, what we require is there in vertical meaning once I have a linear transformation that takes me from Dita w.
Birne Binegar: If it's a nice amorphous that means there's another linear transformation that will take me from w back to be, and it will have the property that it just reverses exactly what the original linear transformation does not, I can now go from btw and I can go from w back to V.
Birne Binegar: Alright, so this notion of ice amorphous is actually the real reason why.
Birne Binegar: All this apparatus and general vector spaces, how it actually works.
Birne Binegar: So um.
Birne Binegar: Let me remind you that.
Birne Binegar: A function between two vector space is a function between to set it's going to have an inverse if, and only if that function is objective and subjective meaning.
Birne Binegar: I searched activity meanings that this map completely covers the Co domain and injected means that two points here get mapped to distinct points over here, if you have those two conditions and you'll always have.
Birne Binegar: At least the existence of an inverse you may not have it in your hand, but at least you know that the inverse exists well in linear algebra a context those two.
Birne Binegar: Those two properties injected it in subjectivity well, they have a.
Birne Binegar: There their equivalent to the following subjectivity the ability to reach every point over here that's equivalent to saying that the range of your linear transformation is actually the w.
Birne Binegar: And that may be not something that's readily see in these abstract vector stations and, in fact, remember that when we had.
Birne Binegar: A vector stakes are em in a vector space aren't in.
Birne Binegar: The range of linear that linear transformation just didn't.
Birne Binegar: pop out of thin air, rather, what we had to do is find a matrix that was associated to this matrix and then find a basis for column space, for example, the athlete get a legitimate hands on what this subspace is well that's that's going to stay the same.
Birne Binegar: So.
Birne Binegar: What if we want to show that we actually have a nice and more system that we can go back and forth between two vectors spaces, then we need to show is that the subspace associated.
Birne Binegar: With the range of linear transformation that actually coincides with the Co domain and if the colonel's linear transformation.
Birne Binegar: subspace of factors in the domain that gets sent to the zero vector here if that consist of only the zero vector then in fact we're guarantee that this inverse transformation would exist.
Birne Binegar: Alright, so let me, let me show you.
Birne Binegar: yeah I guess what I wanna wanna show you, I just want to prove.
Birne Binegar: The following thing and it's really simple but but it's, the key to why bases are are important and that is the fact that every finite dimensional vector space is ice and morphic to are in this is actually what's really crucial in the end, so suppose I have the abstract.
Birne Binegar: vector space.
Birne Binegar: And suppose I now have a an ice amorphous That takes me from be to.
Birne Binegar: Our end well if it's a nice and mormonism.
Birne Binegar: The inverse map is also going to exist, and I can go back and forth between this abstract vector space in our in and, in fact, everything every linear algebra question, I can ask over here can be transported over here, you can calculate.
Birne Binegar: get an answer.
Birne Binegar: And then you can take your answer back to your original set and that's what you do with linear algebra and essential to this coordination process is of course the basis.
Birne Binegar: So what i'm going to show you now as soon as you have a basis, you have a map that will take you from your APP stack based the calculation of setting and back again.
Birne Binegar: it's.
Birne Binegar: So.
Birne Binegar: Suppose we have a basis.
Birne Binegar: For an abstract vector space i'm going to show you i'm going to write down explicitly the map that will take elements of the end give you vectors order list of numbers inside our end.
Birne Binegar: All you i'm gonna give you affect the function between the abstract vector space in the concrete vector space I.
Birne Binegar: Well, using the fact that we have a basis we always can expand factors.
Birne Binegar: Once we expand a vector we have an identifiable list of coefficients and of course we'll talk basis.
Birne Binegar: you're not only our guarantee that you have such an expansion, but also that it's unique there's only one choice up number see you want to see in that will allow you in this fashion to represent a given vector, and so we can take our vectors and assign them coordinates.
Birne Binegar: What i'm going to show you is that this function is a ice amorphous we are it's a linear transformation.
Birne Binegar: And in fact.
Birne Binegar: I guess first thing to point out is that I can reach every point in the co domain, so this function ID has taken me from abstract vectors to.
Birne Binegar: Numerical vectors way you're doing it, I use the fact that, if I have a basis, I always have an expansion.
Birne Binegar: The coefficients and that extension is unique, so I have a way of going from an Amtrak vector D to a list of numbers well that's my map the egos to coefficients with respect to this expansion.
Birne Binegar: Well it's clear that I can reach every point in this in this code of me simply by adjusting the number see want to see in in over here, because every vector can be written this way, and so I can completely cover the vectors here like just changing these coefficients.
Birne Binegar: And that that tells me that the range of this function, going from the the art, and that is range is exactly equal to our in I can get any vector I want by using a suitable linear combination alright so that's.
Birne Binegar: let's go back here.
Birne Binegar: that's showing me so here we're going from our abstract vector space to our end and we're not just argue that, for this organization map that goes.
Birne Binegar: Once you have the basis, you can go from the.
Birne Binegar: coordinate two vectors and be what i've shown you is that in fact the range of this guy is exactly equal to the entire co domain.
Birne Binegar: And if I can show also that the kernel of this linear transformation consistent only the zero vector, then I will have established that I have an ice amorphous between these two vector spaces, and so I can go back and forth between them well to do that.
Birne Binegar: We just have to look at what's going on.
Birne Binegar: Here we go if we have if we apply our linear transformation to an arbitrary vector and we get zero well the rule is.
Birne Binegar: What this guy what this linear transformation does is he just grabs the coefficients.
Birne Binegar: So this guy is going to get mapped to the zero vector i'm going to need see one two equals zero i'm going to need see two is equal to zero, etc, so the only.
Birne Binegar: vector inside our abstract vectors paste that gets sent to the zero vector is the one we're all the coefficients or equal to zero and that's exactly the zero vector so, in other words.
Birne Binegar: The kernel of this coordination map is exactly the zero vector in the domain and so we've got the range is the entire co domain, the Colonel it consists of only the zero vector Those are the two conditions that guarantee that this function between two vectors spaces, has an inverse.
Birne Binegar: And so what that shown you is that anytime you have a basis for a vector space, you have a way of coordinate time, seeing it, and in fact it's more or less unique there's a unique way of going back and forth between your abstract vectors in their concrete numerical coordinates.
Birne Binegar: Okay, so was there a question yeah.
Birne Binegar: or.
Birne Binegar: Specific there's only one.
Birne Binegar: yeah yeah so the reason for the subscript be is because it does depend on that basis, if you look at it look at this map see we once we have these basis factors, so that.
Birne Binegar: that's contained in this set be once I have these guys, then this code, because by definition of what a basis is each coefficients have to be unique, but it does give me a function that.
Birne Binegar: takes an arbitrary vector and gives me a list of numbers and what we've shown in on last series applied was that this function between.
Birne Binegar: The abstract vector space and the concrete vector space is actually an ice amorphous and it's the existence of that, like some Wharf ISM that allows us to do the calculations, we can take.
Birne Binegar: concepts in our abstract vector space bring them over to the calculation of setting do the calculation take the answer back, but it's flowing through that coordination nap and that organization map doesn't exist until you have a basis.
Birne Binegar: Okay alright so i'll stop here today and on Friday what we'll do is we'll review for the exam that will be on Monday.
kuwaiti’s iPhone: Sir so.
kuwaiti’s iPhone: Can you hear me, sir.
Birne Binegar: yeah talking to me.
kuwaiti’s iPhone: Yes, yeah.
kuwaiti’s iPhone: um so the exam will be on next Monday, I guess, once you make sure.
Birne Binegar: that's right the exam will be on on Monday.
kuwaiti’s iPhone: Okay, thank you very much, sir.
Okay.
Birne Binegar: i'm sorry, you had a question.
Birne Binegar: That the no space yeah yeah the no space of the associated matrix if the same thing if the Colonel linear transformation.
Birne Binegar: So.
Birne Binegar: If you look at that posted lecture slide there's an example where I saw the differential equation just using.
Birne Binegar: linear algebra what I do is I take that linear transformation I associate a matrix to it, and then I calculate the north face of that matrix.
Birne Binegar: And then i'd have to take the answer there and map it back the functions, but yeah you can solve differential equations using just make your season and these ice and mortars a match.
Birne Binegar: yeah.
Birne Binegar: That a linear transformation is isn't it yes.
Birne Binegar: He is equal to.
Birne Binegar: Well, we did this before spring break, but when I was talking about linear transformation yeah.
Birne Binegar: It.
Birne Binegar: is projected proved that right, so that that was at least by activity, you were two conditions, you have they're completely covered the target set and you had to make sure that.
Birne Binegar: Two different points didn't didn't land on the same point because of the land on the same point you wouldn't have a unique function backwards.
Birne Binegar: So that's what that's doing and then making sure that you completely cover the targets that that's making sure that you have a function, so that when you do try to apply to invert function, everything has something were to go back to.
Birne Binegar: Right so here everything here results from being landing here, then you can always go back to some point here, and then the interactivity makes that whatever preceded this point makes it unique and then you get the function that goes back and forth what this.
Birne Binegar: The range was it was a different way of saying that the image of your your function is the entire code of me, and they inject the Colonel equals zero that turned out to be let the same that two different vectors that don't end up on the thing spot.
Birne Binegar: what's your take.
Birne Binegar: And injected it is yeah then showing that the two conditions of the range of the criminal or the equivalent to those for now they're.
Birne Binegar: Not like more efficient and now i'm want you to think I saw more so that means me it's essentially the same vector space, but we, we have better coordinates when we think of it as being like our end rather than in its original abstract setting.
Birne Binegar: So what does ice and morrison do it allows you to take the whole vector space and work with it here rather than its in its original setting will we be able to.
Birne Binegar: Will we be able to show examples that are not as trivial as.
Birne Binegar: Well, so now i'm not i'm not gonna require you to prove it or any prevent this is supposed to be expository rather than.
Birne Binegar: You know, formal part of the course that you need to know how these proofs work, I just want to show you what that what what this it's good for and what the language is good for.
Birne Binegar: A.
Birne Binegar: free to take a linear transformation.
Birne Binegar: You know.
Birne Binegar: The.
Birne Binegar: that's usually the hard part so, for example, in the context of differential equation, the way you get that basis well effectively, you have to find two independent solutions.
Birne Binegar: And then, once you've got those now you've got the basis, and you have a way of talking about the whole solution set.
Birne Binegar: But your principal problem is to find that basis that's always going to be the case so, even though it looks like Oh, all you know, I have a now I can always take things into our end and do my calculations yeah yes provided you have a basis, and so that you really do have a function.
Birne Binegar: So the basis gives you the way to coordinate eyes and then, then the whole game of applying linear algebra comes into play, now, you can talk about your abstracts it concretely.
Birne Binegar: I ID is that is the way we it's the coordination map you what you have are.
Birne Binegar: You have a don't have anything up on the board everyone i'm sorry morning over me yeah i'll take this conversation outside.