Birne Binegar: Friday.
So.
Birne Binegar: my sister issues about.
Birne Binegar: guess.
Birne Binegar: Okay welcome back so first thing on the agenda today I want to see about scheduling the second exam now.
Birne Binegar: sometime during the week before.
Birne Binegar: spring break, I said that the exam was going to be this week, but I think that's too ambitious, so what i'm proposing is to have the second exam a week from today.
Birne Binegar: Then the plan will be today to remind you of what went on just before we went on spring break.
Birne Binegar: quickly run through this stuff until we get up to vector space is amorphous.
Birne Binegar: But then i'm going to switch gears of the next topic general vector faces will actually help us prepare for the exam.
Birne Binegar: Because what i'm going to do it effectively generalized all the constructs sub spaces linear transformations various sub basis associated with mater see.
Birne Binegar: All that stuff yeah I can show you how to actually applied in general applications once we cover the topic of general vector spaces so i'll introduce that today.
Birne Binegar: And i'll probably spend Wednesday just talking about examples of general vector spaces So you see how all the apparatus that we've developed since first midterm how it can be applied.
Birne Binegar: So it's going to be this material in general vector status not going to be covered explicitly on the them, but my hope is is that.
Birne Binegar: it's close enough to the topics that were covered on the on the exam that in any way, it will be a study for the.
Birne Binegar: For the second exam and then what we would do on Friday is i'll do a general review session that will maybe we'll run through a practice second exam on Friday, and that I think will leave you pretty well prepared for a second exam a week from today.
Birne Binegar: So any questions on that.
Birne Binegar: Okay, so i'm going to tentatively scheduled the exam a week from today, if you guys.
Birne Binegar: I get a bunch of emails saying I have engineering engineering exam on that day, can we please consider moving it, I will consider it but not.
Birne Binegar: Because of a week away is just a week away, I want to announce the exam today, and if we need to reschedule because of conflicts and we can adjust that on on Wednesday, but anyway, that would still be exam A week later than.
Birne Binegar: A week later than your first notice of the exam.
Birne Binegar: Okay, so any questions on that yeah.
Birne Binegar: You could take it if you're getting a bath Monday morning you could take with my afternoon class I thought that would be convenient okay.
Birne Binegar: Okay, so let me start.
Birne Binegar: So, the main thing we focus on just prior to our.
Birne Binegar: well deserved break with functions between vector spaces and there are associated substances so right after the first exam like.
Birne Binegar: reintroduced matrices and.
Birne Binegar: Then simultaneously substations and sub spaces, where cash two matrices by looking at the span of their columns are standard the Rose or the solution set up an associate linear system, I don't want to do today is go through similar stuff but now it's dealing with.
Birne Binegar: linear transformation, rather than matrices but one of the first things i'll show you on the next slide is that linear transformation talk.
Birne Binegar: Naturally, associated with some spaces and, in fact, from a calculation of point of view, if you understand how to find sub spaces for matrices, then you also can find the.
Birne Binegar: Corresponding sub spaces for linear transformation and i'll work through that material until the get to the point of vector space ice amorphous items.
Birne Binegar: These arm are going to be linear transformations between vector spaces so functions that take you from one vector space to another.
Birne Binegar: But to be an ice and mormonism you there's yet another additional condition, and that is that the function should be in vertical so no one can you go from one vector space to another, but you can return to the original vector space, by applying an inverse function.
Birne Binegar: That idea it's going to be useful, we talk about general vector spaces, because what we're going to find is that a general vector space is anyway, going to be ice a morphic to.
Birne Binegar: One of the our ends our calculations vector spaces and so when we once we introduce general extra spaces, the use of ice amorphous and actually will take the general vector spaces into a calculation setting.
Birne Binegar: Okay, so let's see um.
Birne Binegar: Alright, so a linear transformation is a function between two factor spaces that's compatible with our fundamental vector space operations of scale and multiplication and vector addition.
Birne Binegar: These guys can naturally be associated with matrices and the correspondence goes both ways if you start off with an matrix with in rows and columns you can create a function that goes from a n dimensional vector space to an n dimensional vector space simply using.
Birne Binegar: matrix multiplication.
Birne Binegar: So what you see by this formula here is that if a is an n by n matrix it can multiply column vector.
Birne Binegar: With em rows so this matrix multiplication would be defined, and it would produce so here, you get the necessarily match the.
Birne Binegar: Mention of the vector that kicked out it's going to be in by one, so you can think of this as going from n dimensional vectors to n dimensional vectors and D, the function defined in that way is.
Birne Binegar: First, and most important example of a linear transformation.
Birne Binegar: This function is going to satisfy these properties automatically just going to reduce the properties matrix multiplication.
Birne Binegar: But you can also go in the opposite way as well, if you have a function between two vector spaces that satisfies these two rules, then you can create a matrix that can implement the action of this function simply be a matrix multiplication here's how you produce that matrix.
Birne Binegar: If he is going from n dimensional vector space to an n dimensional vector states, well then you can use the standard basis factors for the domain, so this one is the vector.
Birne Binegar: The standard basis vector where the only non zero components, the first component similarly for the last one you just apply the function to the standard basis vectors That gives you an n dimensional vector that you write us a column.
Birne Binegar: So if you fill this thing out by explicitly seeing how this function X on the standard basis vectors you're going to get an invite him matrix calling it a ti and what this matrix has a property of is that.
Birne Binegar: can be replaced by at act that is you could compute the action of this function, simply by using matrix multiplication.
Birne Binegar: OK dot matrix define linear transformations and linear transformation defined matrices.
Birne Binegar: fairly equivalent to one another, and in fact.
Birne Binegar: When we talk about sub spaces related to matrices there's going to be a corresponding sub spaces corresponding the linear transformations.
Birne Binegar: So here's the two sub spaces.
Birne Binegar: that are of interest.
Birne Binegar: When we look at functions between vector spaces, one is first base consists of looking at all the points in the co domain that can be reached by applying the function here's some.
Birne Binegar: Precise mathematical statement the range of T is a subset of vectors inside art and so they're living in here, but they all have to be reachable by applying your function to some element of your domain.
Birne Binegar: And it turns out, and we show this.
Birne Binegar: I guess in the preceding lecture was that the range of this about linear transformation can be computed by first calculating its associated matrix and then determining the call the column space of that matrix, let me remind you.
Birne Binegar: For a matrix.
Birne Binegar: hey.
Birne Binegar: You get a basis.
Birne Binegar: For the column space at the matrix by reducing.
Birne Binegar: A to a real Echelon form.
Birne Binegar: And grabbing.
Birne Binegar: columns.
Birne Binegar: Of the original matrix.
Birne Binegar: hey by that correspond.
Birne Binegar: The correspond to the column, to the regression line for him to have pivot.
Birne Binegar: row Echelon forums.
Birne Binegar: contain pivot.
Birne Binegar: Job that's how we actually end up calculating this range we construct the matrix it's on the preceding slide construct this matrix.
Birne Binegar: And then we reduce that matrix to row Echelon form figure out where the pivots live go back to the original matrix and use those as the generators of this subspace range of tea.
Birne Binegar: Right and then a second subspace so this range is a sub.
Birne Binegar: Sub space of the co domain, the kernel of key is a subspace of the domain and it's consistent all the vectors in the domain that gets sent to the zero vector so that that's not a number zero that's zero vector inside.
Birne Binegar: The co domain or in so it's going to be.
Birne Binegar: The vector or you have in zeros all right and that's.
Birne Binegar: This is the subspace and what it's actually computer bubble, as the null space at the associated matrix, in other words it's the.
Birne Binegar: it's the solution set of the corresponding matrix us has a coefficient matrix up a linear homogeneous linear system.
Birne Binegar: Okay, so those are.
Birne Binegar: Two important sub spaces that one can attach to a matrix and i'm part of the utility of this is that.
Birne Binegar: i'll show you later that if the range of T is in fact the entire co domain and the kernel of T consistent only the zero vector.
Birne Binegar: Inside the domain, then, in fact, you know that that linear transformation has an inverse now and.
Birne Binegar: that's That was the main point of the lecture just prior to today's lecture let me.
Birne Binegar: point out a couple other salient back to come came from before our discussion before we for spring break.
Birne Binegar: If we have an n by n matrix remember the number of columns of a that's equal to the rank of a plus and notice on here i'm just reminding you of some definitions.
Birne Binegar: A rank of a well that's dimension aerospace which turned out to be exactly the same as dimension of the column space so what's striking here there's this row space.
Birne Binegar: We have an n by n matrix each row is going to have exactly am entry, so this is actually subspace.
Birne Binegar: of our em well the colon space of a that's the set of all linear combinations of the column vectors but the column vectors are going to have an entry shirt in rows so that's subspace.
Birne Binegar: or in so surprising fact was that these two subspace is live in different vectors places, yet they have the same dimension which what really means is that the number of factors, you need to create a basis and in either of these cases is the same.
Birne Binegar: But the reason why we know it's the same as the way we find bases for these two sub spaces.
Birne Binegar: So the rank of.
Birne Binegar: Either one of these numbers just count the number of basis vectors you get for the row space it's going to be.
Birne Binegar: The number of non zero rose, you have in a row Echelon form of a well that's going to be the same as the number of columns of the real actual form that have pivots because a non zero is going to have a non zero first entry.
Birne Binegar: Here rural forgetting, of the basis directors, for the calm state issue again look at the ro s on on the a you figure out where the pivots live.
Birne Binegar: And then you go back to the original matrix and the original matrix and you grab the corresponding call, but the number of columns you grab depends on how many pivots you have.
Birne Binegar: In euro Echelon form, so if these numbers, the number of factors, you get it's going to be the same because they both correspond to pivots in a row Echelon porn.
Birne Binegar: and similarly.
Birne Binegar: The no space of a matrix well that's.
Birne Binegar: The dimension of have no space of the matrix is called a novelty it's the dimension of the no spaces solution set of a linear system and.
Birne Binegar: Page.
Birne Binegar: Is.
Birne Binegar: equal to the number of free parameters in the solution set the dimension of that solution set basically just going to equal, the number of free parameters.
Birne Binegar: But number of free parameters is the number of columns about pivots, and so this observation here the serum is just based on accounting a pivot.
Birne Binegar: The rank is columns with pivots in the row excellent form that dimension of the no space is equal to the number of columns about pivots well when I combine these two, I get the total number of calls of the matrix and so that's how that identity POPs out.
Birne Binegar: But what is that identity also tells us.
Birne Binegar: it's a relationship between the dimensions of Sub spaces attached to a linear transformation.
Birne Binegar: And that is that the range of a linear transformation.
Birne Binegar: subspace over here now up all the points that are reachable by applying the linear transformation, we take the dimension of that subspace.
Birne Binegar: And we add to it to dimension of all the vectors in the domain that get sent to the zero vector over here well those two numbers have to add up to that dimension of the domain.
Birne Binegar: Then that's because.
Birne Binegar: Both the range of tea in the kernel of key can be computed in terms of the associated matrix.
Birne Binegar: and
Birne Binegar: here's the account, we did on the preceding slide with all we need it, we have columns of a row actual on phone they have pivots and columns.
Birne Binegar: rush off on that don't have pivots but to these two numbers add up to the total number of columns, on the other hand.
Birne Binegar: The column space of this matrix is exactly equal to the range of tea and the no space of this matrix is exactly equal to the Colonel team until this identity now POPs out.
Birne Binegar: Okay, so.
Birne Binegar: i'll remind you again that the reason for looking at the range and colonel, is it it's going to tell us when a given linear transformation can be inverted when can I not only go from our md and.
Birne Binegar: When can I follow that back to the written take Dr schirmer in and map them back the original domain or in.
Birne Binegar: Alright, so i'm gonna.
Birne Binegar: yeah I guess i'm a statement I didn't put on the slides, but I want to.
Birne Binegar: Its result from last time.
Birne Binegar: We had.
Birne Binegar: linear transformation.
Birne Binegar: Yes, well, by definition, it means that.
Birne Binegar: That done he if there exists a function.
Birne Binegar: Key the miners first to go take a picture in the opposite direction from R and R m.
Birne Binegar: And you want this to be such that he composed with T minus first.
Birne Binegar: Just is the identity function on this argument and that to the minus first compost tea.
Birne Binegar: Maybe I should have read it this way, why.
Birne Binegar: Why.
Birne Binegar: X.
Birne Binegar: equals X, so why is going to be an element of our an.
Birne Binegar: X is going to be an element of our em but either way you compose these either P composed with Tina miners first or T shirts composed with T they just.
Birne Binegar: That composition always produces the identity function that takes a given vector to itself so that's what it means to have an inverse and what I showed you last time is that T from.
Birne Binegar: Our m O R n is convertible.
Birne Binegar: If, and only if to condition told me one is you need the range.
Birne Binegar: of tea to equal the entire code of May.
Birne Binegar: And the second thing you need.
Birne Binegar: Is you need the.
Birne Binegar: Colonel.
Birne Binegar: of your linear transformation that has to consist of only the zero vector inside the domain, so what the subspace really only consists of the zero vector So this was the theorem that we proved before and.
Birne Binegar: convertible linear transformations are called ICO more systems, so another way of saying that the linear transformation it's a nice amorphous if these to hold and what that will mean is that we can pass back and forth between our two vector spaces.
Birne Binegar: Because.
Birne Binegar: If if we have a nice amorphous, we know that the inverse function athlete exists.
Birne Binegar: Okay now so now, what I want to do that doesn't that much us review of what we covered just prior to spring break what i'm going to do is sort of build on on these topics by moving a little bit forward in the course.
Birne Binegar: we're now going to consider vector spaces over sets sets that might not be ordered list of numbers but somehow are going to behave like our in our standard vector space, so what we're going to need for this.
Birne Binegar: is first of all set of objects and just think of them as objects, not as vectors, yet we just we need some set caning a bunch of objects and what we need to be able to do.
Birne Binegar: Put a vector space structure on this set is to have some notion of scale and multiplication meaning we we have.
Birne Binegar: A rule for taking your number and an element of our set and then producing a new set.
Birne Binegar: yeah that takes a number and a vector and produces the scale or multiple this new vector and citizens oil this new object that sits inside that set and similarly.
Birne Binegar: Where every pair of objects in this set, we have a rule that will map pairs of elements of V to a new element of the that still lives inside be itself.
Birne Binegar: And so, when you think of the scale of multiplication that's a function that takes a number in the vector and produces a new vector and vector edition is a function that takes two vectors and produces a new function.
Birne Binegar: But.
Birne Binegar: This is sort of the setup for vector space over our what we also impose together a bona fide vector states over the real numbers.
Birne Binegar: is to you man that these two operations actually satisfy eight distinct axioms that are quite natural.
Birne Binegar: consequences when our vector spaces, is just ordered list of numbers, but now we're extending the ideas of our end to this new setting.
Birne Binegar: So what we want is just like vectors and ordered list of numbers, we add three of the objects in a set be using its rule protector addition.
Birne Binegar: It shouldn't make any difference when we need to some three vectors whether we some the first 2% and then the last factor, or if we add the first vector to someone's life do directors in order to do that computation of three vectors.
Birne Binegar: Are rule for adding vectors only works for two vectors and so.
Birne Binegar: If we're going to add three vectors there's two ways of doing it and the statement is it doesn't make any difference, how we how we actually combine three vectors be a vector addition.
Birne Binegar: Secondly, we want the multiplication commutative so the order in which we add things doesn't make any difference we want to zero vector vector that lives inside the set such that, when we add it to any other vector inside the set we get back what we started off with.
Birne Binegar: Are the more we're going to want every vector and D we're going to want to sort of an additive inverse a vector such that when I add it to a given vector I always get the zero vector as a result.
Birne Binegar: We need scale multiplication we need that to distribute over to vector edition, so this will be true if you and D were numbers, but here they're just objects have a set and so.
Birne Binegar: This is something given a rule for scale multiplication that would have to be verified this short associates compatibility of the rule for adding objects is compatible with the rule for scalable playing objects in a natural way.
Birne Binegar: and, similarly, if we add two numbers together before multiplying, we ought to be able to.
Birne Binegar: delay that addition until afterwards, but notice here i'm adding numbers, while, on the right hand side i'm adding vectors, so there is something there is some content to this formula it's not just distributing.
Birne Binegar: Well, we have to postulate that that distributed rule actually holds.
Birne Binegar: And then we have.
Birne Binegar: A natural way of doing multiple scalable applications, if I do to scale and multiplication two in a row that could be carried out by multiplying the two factors together first and then doing the scale multiplication and then lastly last.
Birne Binegar: Statement so screen is is that when you multiply scale and multiply and object in your set D by the number one you get back what you started off with now, in the case of the vectors inside our n.
Birne Binegar: Each of these identities is Tripoli verified I may do a couple of those on the next slide but the point of these axioms is if we have any set D.
Birne Binegar: With some notion of scale multiplication and some notion of vector edition effectively is going to behave exactly like one of our familiar vector spaces are in.
Birne Binegar: And in fact the whole apparatus that i've been developing so far is just so that we can take abstract vector spaces and bring them to a calculation of setting.
Birne Binegar: Is where i'm going so, in fact, in the end way we deal with these general vector spaces, is by transporting questions about them into the questions inside our in will we can do calculations and then we'll take the answer back yeah.
Birne Binegar: yeah it's an abstract at your face and i'll give you some examples So you see how.
Birne Binegar: What it means to be an abstract vector space right now.
Birne Binegar: All our familiarity and i've really stressed this since the start of the course is thinking of vectors is ordered list of numbers because that's what our calculations are based upon.
Birne Binegar: But now what i'm proposing now is will consider other steps, other than order list of numbers, but the calculation or apparatus that we've been developed, will be able to be applied to these more general situations.
Birne Binegar: Alright, so let me give you some examples, so the first one first thing.
Birne Binegar: If this were approved course.
Birne Binegar: One of the homework assignments, I would make is that, in each of the examples verify that all eight axioms whole.
Birne Binegar: But i'm not going to do that because there's a lot of axioms that check and I will have five distinct examples as well i'm not going to do 40 demonstrations of trivial rules but.
Birne Binegar: I do want to show you.
Birne Binegar: At least, how you would start verifying these eight actions in print in practice what you need is a rule for doing the scale multiplication of rule for vector edition.
Birne Binegar: So some way of taking an element year set and a number and producing a new element that would be scalable application some way of combining two elements set to create a new moment and that to be interpreted that tradition and last thing you need in order to satisfy.
Birne Binegar: The third axiom if you need a zero vector inside your your vector space.
Birne Binegar: But once you have this data once you've decided what the zero vector is and what scale of multiplication what vector edition is then it's just a matter of sitting down with a pad of paper and making sure that.
Birne Binegar: These axioms are all satisfied.
Birne Binegar: Alright, so let me start with a basic example where are our vector space is a set of ordered list of numbers and then scale multiplication is defined.
Birne Binegar: yourself a number and an ordered list of numbers and here's a new order list of numbers, so that defined schism cation vector edition, and the zero vector these things we've been using throughout the course.
Birne Binegar: All right now.
Birne Binegar: Next example shows you why I was so interested in sub spaces, first of all because sub spaces take place.
Birne Binegar: At least up to this point in the course they always takes places special sets of vectors inside set of ordered list of numbers right So if you have a subspace of RN.
Birne Binegar: What it is it's a set of vectors that live in here, but to be a subspace, it has to be closed under scale multiplication injector edition so soon as you have a sub space of our end, you know that these two properties whole and, in fact.
Birne Binegar: These two properties what they what they will do for us now if they will serve defined a notion of scale or multiplication in Dec tradition that occurs on D rather inside of our end.
Birne Binegar: The way we get those rules, the way we get a rule that takes a number in a vector and produces another vector inside this abstract said he is well, we can just take.
Birne Binegar: The rules inside our end and just restrict them the vectors that live inside the subs the subset D because because it's a subset I know that if I take a number and a vector.
Birne Binegar: I can scale and multiply inside our end and get another vector, but it also lives each ID so I can take a number in a vector and get a vector and D, that will be my notion of.
Birne Binegar: scale up scale multiplication in D, but all it is, is the scale or multiplication and your original vector states are in that's been restricted to this, the subs the subset domain, so that the main.
Birne Binegar: scale multiplication in our end is our cross our end, but now I just restrict our cross on the subset the end and similarly.
Birne Binegar: Because when I add two vectors in the eye because visa subspace I always end up in D that's furnishes me with a mapping from the to be the Cross de de de and that.
Birne Binegar: will be my notion of vector addition that now, I can regard is working purely inside the vector states D inside the subspace D.
Birne Binegar: and, moreover, do you have a subspace one thing we know about subspace is that they always must contain the zero vector of the vector space in which they live.
Birne Binegar: And so we can just grab the zero vector here it automatically lives inside our subspace D and we can use that as the zero vector for the so now i've got.
Birne Binegar: A sub sub sub a set it's actually a subset apart and that's a which I have operations of scale and multiplication and vector addition, in a zero vector all defined and what one would do is verify that all eight axioms of the vector space are satisfied.
Birne Binegar: These notions of scale multiplication vector edition, and this zero vector yeah.
Birne Binegar: Are xp xp.
Birne Binegar: Oh so and yeah i'm just i'm just trying to distinguish between where I put the subscript here and so that's have to be a i'm thinking if star.
Birne Binegar: As going from real numbers cross.
Birne Binegar: And and producing a vector inside our usual scale of multiplication.
Birne Binegar: In our in and the rule is that Lambda times X one X and.
Birne Binegar: Yes, Lambda X one Lambda X and and, similarly, we have.
Birne Binegar: vector edition inside or in.
Birne Binegar: takes two vectors and it produces another vector now what i'm now distinct from this situation.
Birne Binegar: This when I have a subset of vectors inside our end it actually have subspace what I want to do is, I want to define.
Birne Binegar: analogs of scale and multiplication and vector addition appropriate to this new subspace well there's got to be maps.
Birne Binegar: That takes me from numbers and element to the subspace to element of the subspace or takes pairs of elements and substation produces the new subspace.
Birne Binegar: that's how i'm going to define these two functions that are going to be interpreted double a scale or multiplication in D and vector addition in D.
Birne Binegar: So that's that's like but their subscription I don't want, I want to distinguish this from the two basic operations that are living in the sort of parents state parent vector space our end.
Birne Binegar: So yeah so that's the reason for the sub scripts and in fact here you notice that.
Birne Binegar: I usually do no zero vectors by simply a bold fit both on zero, but now I have to worry about which vector space that zero vectors living i'm pretty sometimes pretty subscription that make that clear.
Birne Binegar: Okay, so this zero vector.
Birne Binegar: It actually sits inside here but.
Birne Binegar: substations always contain the zero vector of the of the vector space in which they live, so it also also lives in D and then I can use this guy as the zero vector inside the set because it's an element of the set and it's going to act like zero respect to vector addition.
Birne Binegar: All right now what's nice about this.
Birne Binegar: The fact that substances or an example of vector spaces is.
Birne Binegar: A lot of the developing the course, for example, my whole spiel that took me several weeks, talking about bases and sub spaces it's now going to be more relevant because we worked hard to find bases for sub spaces, because they allow us to coordinate I sub spaces well.
Birne Binegar: We have a subspace we can usually figure out what vectors are living in that substance anyway, if we solve some equations or we look at the span of the set of vectors but now when we're dealing with abstract vector spaces.
Birne Binegar: Sub spaces provide sort of a non trivial example of how do you find good coordinates for for a substrate that question is now going to.
Birne Binegar: Or the answer to that question, how do we find a good set of coordinates for a subspace it's now going to have that occasion, is how do we find good coordinate for an abstract vector space.
Birne Binegar: And so.
Birne Binegar: So this is sort of Nice is that.
Birne Binegar: We not only had the basic example of a vector space, since the start of the course but another example, the general vector space is simply a substation of a vector space and so we're already somewhat familiar with dealing with vector spaces that are distinct.
Birne Binegar: From from our end.
Birne Binegar: Alright, so.
Birne Binegar: Now let's do some more examples here's a real principle one probably the reason why you're in this course.
Birne Binegar: We do require kapsalis in this course, but you know it's we don't do any differentiation or or integration or anything like that what we.
Birne Binegar: What we really required for us to certain Mount of functional or mathematical sophistication then that what that functions, you can use them, you can create a set of functions.
Birne Binegar: be taking differential equations you're often interested in set of functions that satisfy differential equations so.
Birne Binegar: that's the functions are going to be really common in your applications and what I want to show you now is how to interpret such a set.
Birne Binegar: No, no we're close to an ordered list of numbers, however, nevertheless regarded as a vector space well, what we need is a set.
Birne Binegar: Some notion of scale or multiplication so here when i'm suggesting is that, if I have a number Lambda and a function.
Birne Binegar: On the real line did I can create a new function i'll label it Lambda F, but this is going to be defined by the fact that its value at an arbitrary paint X is just the value of F at that point time parameter Lambda times the number Lambda.
Birne Binegar: And Similarly, I can add two functions if F is a function between R and R amp G is a function between our our I can create a new function that maps from.
Birne Binegar: Real numbers to real numbers by simply taking the values these component functions and adding them together to get the value some F G at X.
Birne Binegar: And then the last thing we need.
Birne Binegar: To establish a vector space on this set is we need some sort of zero vector something that when we add it to another vector always just reproduce it that vector so what I want as a function.
Birne Binegar: g that would have the property when I added to any function F I get get back that function F and that can be.
Birne Binegar: Such a function would be the zero function, the function.
Birne Binegar: whose value with any X is always equal to zero, so it just the constant function, where the.
Birne Binegar: For all X, the value of the function is equal to zero, and then that function is going to have the property F of zero Plus any function g.
Birne Binegar: H evaluated at a point X well, according to this rule, this would be F zero evaluated X plus G evaluated X, but.
Birne Binegar: If this is the function F zero is defined to be zero for all values of X is going to be zero Plus T effects it's going to equal to you at so this function added to any function in this set just reproduces that that function G and so here I verify.
Birne Binegar: The existence of an additive identity inside this set it just happens to be the trivial function, where the value of any point is just equal to zero.
Birne Binegar: And, in fact, not only do we have an additive identity.
Birne Binegar: For this set with these operations, and this is zero vector they all they're going to satisfy all eight axioms that can be that can be verified by simply working out the details my hand.
Birne Binegar: Alright, so.
Birne Binegar: Now we're really talking about an abstract vector space.
Birne Binegar: don't really correspond to directions or order list of numbers or quantities with magnitude and direction that's something completely different in this example, yet it's very, very prominent in.
Birne Binegar: In your applications.
Birne Binegar: here's another one.
Birne Binegar: In this rendition of a vector space.
Birne Binegar: I want you to think even outside the context of matt mattox simply want you to imagine a vibrating string.
Birne Binegar: And so it's something a vibrating string we can actually treat as an actor, how can we do so well scale or multiplication now, first of all, we have this set all the vibrational mode to say up piano wire violin string we have all its vibrational mode, but we can.
Birne Binegar: send a vibrational mode to new vibrational mode I change the amplitude by rescaling know.
Birne Binegar: The displacement of the corresponding vibrations and if you do that by a positive factor that will just change the amplitude, but you can also.
Birne Binegar: Add, in addition, you can multiply by negative numbers, where the factor of minus one is interpretive reversing the phase, so in general.
Birne Binegar: You have some.
Birne Binegar: vibration going on.
Birne Binegar: If you do scale a multiplication by a positive number, all that will do is change the amplitude sorry this should follow the path of this but it changes the amplitude of the vibration.
Birne Binegar: So that we multiplication by a positive number multiplication by another a negative number would have, in addition, this would get converted to well.
Birne Binegar: This would be minus.
Birne Binegar: This is, this is the then minus the.
Birne Binegar: would be the vibration that were instead of having a hilltop here you have a valley it's the same displacement from the from the horizontal axis.
Birne Binegar: So.
Birne Binegar: You can implement these you can implement these.
Birne Binegar: Well, by instructions to your.
Birne Binegar: For your orchestra to be loud or soft, this is a harder to pull off within an orchestra, although if you have guitar players, they often have a switch that they can use to switch the phase of their their signal.
Birne Binegar: But these can be implemented purely physical not move any intervening mathematics and vector addition well for you have to vibrations, you can often hear them simultaneously, and in fact that's the way we would consider.
Birne Binegar: vibrational modes, the added deserve they occur simultaneously.
Birne Binegar: And then the zero vector well, that would just correspond to the string this just silence and stringing you would get that, like, for example, scale and multiplying by the number zero and what you take a function and you multiply by zero you just get zero okay.
Birne Binegar: All right, so there's an example.
Birne Binegar: Of a physical situation where we can define notions of scale and multiplication definition and even a zero vector.
Birne Binegar: With no intervening mathematics, except for the the size of these scale are factors.
Birne Binegar: And then.
Birne Binegar: here's another example polynomials.
Birne Binegar: Save degree in.
Birne Binegar: So we have scale multiplication what is it well it's the same way, we always multiply scalable polynomials we just multiply all the coefficients.
Birne Binegar: In each term I common factor Lambda that would be scaling multiplication and we know how to add polynomials we just collect together the coefficients have the same power X and then we can add them together.
Birne Binegar: Yet either way if we think of a polynomial as just being an expression, where we have certain numbers times distinct powers of X what we're getting for both these operations is new polynomials created from old polynomials either by this scale a multiplication or the sector edition.
Birne Binegar: And the zero vector well, that would be the zero polynomial all coefficients equal to zero.
Birne Binegar: So now you've seen our five different sets that are.
Birne Binegar: functionally are going to behave exactly like are in.
Birne Binegar: And in fact.
Birne Binegar: This is probably the first course where you actually seen mathematics as mathematics and calculus.
Birne Binegar: yeah you take calculus because you need to know how to compute derivatives need to know how to compute integrals because that's what your application need.
Birne Binegar: But what you see in this course is we've gotten up to this course we're over halfway through the course and i've only now introducing.
Birne Binegar: How you're going to use it after the course because what I had to do prior to introducing the calculations that you're going to use I had to introduce the language that.
Birne Binegar: The more these calculations take place and those big surprises is that abstract sets if you have you know vibrational modes of them.
Birne Binegar: Have a string to the states of hydrogen atom or other things, these are abstract sets that I evil you don't see how to calculate.
Birne Binegar: what's going to be important is be able to take these abstract that your status bring them into the calculation of setting of our and what we can actually.
Birne Binegar: Do calculations and make prediction, and then we go back, but you know that's calculation.
Birne Binegar: that's true it's the important part of linear algebra and we spent the first half of the course describing how to do calculations, but now we have all that in our hand now, and now we can actually apply them through the more general setting for which this course is supposed to.
Birne Binegar: Teach you.
Birne Binegar: All right, so now main thing to remember this day to the game is that while we're widening the arena over which we can apply linear algebra in the end.
Birne Binegar: This our end is going to remain fundamental because, even when we're looking at more general vector space is the idea we're doing concrete calculation is going to take that general that you're states and somehow realize it, you know inside these calculations, that your spaces are in.
Birne Binegar: Alright, so um.
Birne Binegar: I guess see I only have.
Birne Binegar: Two minutes left.
Birne Binegar: Before I actually show you how we do calculations in general vector spaces, I was going to run through the definitions that we've already seen for our end and show you how they're modified.
Birne Binegar: In this new setting and effectively they're not modified much at all, all I have to do is everywhere, I see an art and in the preceding definitions I replace it by a.
Birne Binegar: General vector space D and we have exactly the same definitions so, for example, a subspace of the general vector space it's a subset of that vector space that's closed under scale of multiplication and addition meeting, the same thing is it did inside our end except now.
Birne Binegar: Who is going to be a subspace up in a subset of an abstract vector space, rather than our end but it's the same two conditions compatibility with scale and multiplication and vector addition.
Birne Binegar: We have.
Birne Binegar: sort of element living inside of general vector space, then we can create a subspace.
Birne Binegar: By simply taking all possible linear combinations of them, and that, indeed, is going to be a subspace of about abstract vector space.
Birne Binegar: We have a set of vectors inside of general that your space will continue to call them linearly independent as if they were living in type R N, meaning that the only way they can some to the zero vector in five this abstract vector space is by taking all the coefficients equal to zero.
Birne Binegar: And finally, maybe this is enough for today a basis for subspace of a general vector space is simply a set of vectors such that.
Birne Binegar: You can reach every point in the vector space or taking linear combination to these guys and such that the factors are linearly independent meaning they have this property, the only way they add to zero just by taking all the coefficients equals zero so what's.
Birne Binegar: what's going to happen now is that simply by replacing.
Birne Binegar: Our in my.
Birne Binegar: generals vectors they D, but which scale multiplication vector edition and zero vector has been defined, we can replicate all the theory all the theory, we had about our end is going to carry over into this new setting, including the definitions by simply modifying.
Birne Binegar: The vector that the name of the vector space we're working it.
Birne Binegar: OK so i'll stop here and.
Birne Binegar: What I hope to do on Wednesday is show you how, for example, sub spaces and calculating no spaces of matrix are no spaces are our kernels of linear transformation how that all look.
Birne Binegar: How that all looks inside this new setting this new framework.