Birne Binegar: Okay, so.
Birne Binegar: remind you this Friday will be our first exam I will do a full review session on Wednesday, what I want to point out today just the resources that are available to you on canvas so thanks at the top of the page i'm sure you've seen.
Birne Binegar: Once we get past the lectures.
Birne Binegar: lecture notes.
Birne Binegar: towards the bottom so on Wednesday.
Birne Binegar: What i'm going to do is i'm actually going to go through the practice again, do you want to have a look at it beforehand.
Birne Binegar: That might be a good idea, but but anyway i'm going to explicitly work through that practice exam in class on Wednesday, and the other thing that you may not notice but it's there, nonetheless, is that.
Birne Binegar: i've been posting solutions to the web assign homework and you can you can look at those in fact for problem sets four and five, I posted the solutions.
Birne Binegar: Last week, even though the homework set for five or not officially do until today i'll frequently do that because, in fact, if you look at the solutions are not going to match the.
Birne Binegar: problems that web assign gives you exactly because web assign rammed analyzes the numbers, but if you simply want to know how to proceed on a problem you could look at the solutions and you'll see well and analogous problem this worked out in detail.
Birne Binegar: So, have a look at those I mean what what i'm.
Birne Binegar: What I hope to see if I can bring one of these up.
Birne Binegar: comes up.
Birne Binegar: oftentimes canvas is not so good on wireless and so yeah I may not be able to bring this up what I wanted to point out is the whole row reduction process is explained and so.
Birne Binegar: You can look to those examples, and if you ever stuck it's what you do, what do I do next, you can look at the examples enable shorter.
Birne Binegar: leads, you need the way, so to speak, okay Well, this is not coming up afraid so.
Birne Binegar: So much for my in class demonstration.
Birne Binegar: All right, let's cancel.
Birne Binegar: OK, so the intended today's lecture is actually not prepare for the exam will do that on Wednesday, but rather.
Birne Binegar: push forward a little bit with material that will be concentrating on after the exam.
Birne Binegar: So I introduced this notion of Sub spaces last Friday.
Birne Binegar: or reintroduce that and then i'll give you a bunch of examples in counter examples and then come to.
Birne Binegar: A new.
Birne Binegar: A new notion called a basis and i'll try to explain that if I get to it at the very end of lectures, but in fact there were three things that are going to be really critical once we get past the first exam.
Birne Binegar: Knowing how to work with sub spaces and what will happen is that what bases will allow us to do just coordinate ice vectors that live in a subspace and i'm.
Birne Binegar: Do the when I do the examples i'll show you sort of the problems that that you're going to encounter as soon as you try to talk about.
Birne Binegar: say a three dimensional hydroplane embedded inside a 10 dimensional space problem it's going to be, how do you coordinate eyes that three dimensional surface when every object is actually attend dimensional vector.
Birne Binegar: But you don't want to use 10 parameters to parameter is a three dimensional hyper plane until bases are the key part that actually allow us to get good coordinates for vectors that live inside of subspace rather than in more or less general position within a n dimensional space.
Birne Binegar: Okay, so let me get started here.
Birne Binegar: So.
Birne Binegar: Why don't you introduce on Friday it's this notion of an operation, acting on a set leaving that set.
Birne Binegar: Well staying within this that under the action of that operation, for example, when we scale and multiply vector we create a new vector.
Birne Binegar: So the question would be if I start off with vectors that lives in some subset of the vector space or in and I scale and multiply what does it take it outside that set well that can often happen.
Birne Binegar: and
Birne Binegar: On the first examples I gave.
Birne Binegar: On Fridays you look at the point on the inner circle, you know these these points on the inner circle have the fact that they're raising their distance from the origin is always useful the one so you're looking at point in the plane.
Birne Binegar: But with a special conditions that distance from the origin has to equal one.
Birne Binegar: This set is called s one and, by the way, one thing to note, in which we proceed further in the course is my use of this set notation it's going to become.
Birne Binegar: One more frequent notice how constructing this set first I tell you I give you a big set and everything sits in basically just all the vectors inside the two dimensional vector space ordered list of two numbers but I put this Center line inside the set that's to be read as such.
Birne Binegar: So i'm looking at two dimensional vectors such that their link is equal to one or the some of the square for the components is equal to one alright so that's.
Birne Binegar: a subset it's every vector in it actually fits inside our two it's a subset of our two.
Birne Binegar: But what's going to happen if I steal a multiply vector say this vector by two I get a vector it's twice as long and I no longer lying on this circle, so the simple operation that scale and multiplication has taken me outside the circle.
Birne Binegar: So that's the sort of geometric phenomenon what we're trying to focus on.
Birne Binegar: And not only are we worried about what scale or multiplication does to element of the set, we also worried about how how they behave with respect to vector addition.
Birne Binegar: We wanted to have two vectors that start off inside your son set we add them together, we want to stay inside that subset so I can give you another example here I start off with these two vectors they have coordinates.
Birne Binegar: One zero and this vector here as coordinates 01 when I add them together one zero Plus.
Birne Binegar: 01 is going to equal the vector one one and here one squared plus one squared to some of the squares of the components is equal to two.
Birne Binegar: not equal to one, so if I start off with these two vectors I add them together, I get a vector that's about out here it's a it's late this now the square root of two rather than one.
Birne Binegar: Okay, so.
Birne Binegar: In general, when you have a sub set of a set of vectors it's not going to be well behavior perspective are two fundamental operations and.
Birne Binegar: Really linear algebra I stress this, over and over again it's compatibility with scale and multiplication and vector addition that really drives everything in linear algebra.
Birne Binegar: So what we did on Fridays, we introduced the definition of special subsets that we're going to focus on in this course are called sub spaces and as the nomenclature indicates, you can think of them as a little vector space.
Birne Binegar: A sub sub vector space sitting inside a larger vector space.
Birne Binegar: And all we require of these guys is that.
Birne Binegar: In such a set if we scale the multiply any vector inside the set we don't leave this set and.
Birne Binegar: If we add two vectors in the set we don't leave the set.
Birne Binegar: and
Birne Binegar: it's actually these two conditions that really make.
Birne Binegar: a subset behave like a vector space and then not only do they look like that your spaces, but they're also going to behave exactly like that your spaces, because they're compatible with are two fundamental vector space operations.
Birne Binegar: Alright, so I just went over the unit circle and showed you that when you scale and multiply, a point on the circle you leave the circle, when you add points on the circle you leave the circle, and so the unit circle this subset of the set of two dimensional vectors it's not a subspace.
Birne Binegar: and
Birne Binegar: here's another set that I went over on Friday I take element in a two dimensional vector space where I restrict the entropy component here to be integers.
Birne Binegar: Then, in fact, that set easy to see is closed under back tradition, because if I take any two vectors that live inside the set well the components have to be integers so i'll just write in one and two and one and two and the requirement is that those components.
Birne Binegar: Are integer so minus 1123 any the usual whole numbers and their negatives.
Birne Binegar: When I add to that through together, then i'm adding an integer to an integer i'm going to get an integer similarly for the second component and so.
Birne Binegar: adding two vectors of this of these types produces another vector where the components are still integers until it is closed, under that tradition.
Birne Binegar: But it's that it's not closed under scale and multiplication because the rule for scale multiplication is you can multiply by any real number and stay inside the set.
Birne Binegar: So if I happen to choose an irrational numbers, like the square root of two and I take a vector that lives inside the set.
Birne Binegar: yeah noticed I have integer components, so I live inside that set and once I multiply by the square root of to I get a vector who's component is first component it's no longer manager, so it takes me outside that set.
Birne Binegar: go in fact it's easy to destroy you can set up sets that maybe you think.
Birne Binegar: Have a good possibility of being a subspace because, at least, this one is closed under vector edition, but it's easily seem to not be closed under scale and multiplication.
Birne Binegar: And so it fails to be a subspace and from our point of view will be stressing later on, if it's not a subspace we're not interested in it for the most part.
Birne Binegar: All right, so.
Birne Binegar: Let me give you a third example.
Birne Binegar: I talked a little.
Birne Binegar: bit about this on Friday.
Birne Binegar: let's look consider a hyper plane, now the nice thing about hyper planes and say at least look like that perspective right if I draw a.
Birne Binegar: Maybe a three dimensional picture.
Birne Binegar: You will have to use your imagination to like drawing is not that great I have a three dimensional space, I can put.
Birne Binegar: A two dimensional hydroplane it sort of slices through it well, if you think of this as an.
Birne Binegar: end the edges off to infinity you can think of this as a two dimensional plane embedded in the three dimensional room or in well that kind of looks like a little vector space sitting inside a larger vector space because, after all, you could coordinate ISIS plane and then.
Birne Binegar: Great vectors it's like.
Birne Binegar: Life flat plane or surface nice natural vector space, but in fact in general.
Birne Binegar: hyper planes, even though they look like vector spaces.
Birne Binegar: sort of intuitive they turn out not to be sub spaces so they're not really proper.
Birne Binegar: clothes close subsets of a vector space they just they just visually look like, but they the key property that we're interested in is how they behave with respect to scale and multiplication in that tradition.
Birne Binegar: So the statement i've want to show you that just that, if you have a hyper plenty.
Birne Binegar: Then, if it doesn't contain the zero vector it cannot be a subspace and that's easy to see.
Birne Binegar: To be a subspace, we have to be closed under scale or multiplication so whenever I multiply a vector inside this hyper plane.
Birne Binegar: I multiply by a number I should stay inside the hyper plane, but if my hypothesis is that this hyper play somehow avoids the the the zero vector that's right at the origin.
Birne Binegar: Maybe be easier to see with a line.
Birne Binegar: we've had here for two dimensional space and here's a one dimensional line or one dimensional hyperplasia path, if this line does not contain the origin, then it cannot possibly be a subspace because.
Birne Binegar: Most clients multiplying any vector inside your hyper plane by zero is going to produce the zero vector.
Birne Binegar: From this picture you see that this line does not contain the zero vector so multiplication by zero is going to take every point on this line to a point that doesn't live on the line, and so it takes me outside of that space.
Birne Binegar: And so hyper planes in general they're not going to be substances, even though visually they appear to be substations they look like lines plane to.
Birne Binegar: Other very simple geometric objects that you would think of as being a vector space.
Birne Binegar: Alright, so.
Birne Binegar: Before getting more counter examples that last.
Birne Binegar: example where I had a hyper plane, and if it did not contain the zero vector I said it, it could not be.
Birne Binegar: A subspace well that's true not only a Piper plane, but anytime you have a subset of our end if the zero vector is not in that set, then it can't possibly be a subspace so that's usually one quick way of detecting whether a given subset.
Birne Binegar: is possibly could possibly be a subspace if it doesn't contain the zero vector than throw it out it it's not going to work because.
Birne Binegar: You have to be close under scale or multiplication and multiplication by the number zero is always going to take you to the zero vector So if you start off in a set that doesn't.
Birne Binegar: contain the zero vector multiplication by zero always always kind of give you a vector that does is zero vector until we take you outside your set.
Birne Binegar: All right.
Birne Binegar: yeah.
Birne Binegar: little bit of a problem, let me go back to where we consider span of a second well yeah.
Birne Binegar: What could happen, I mean if this P zero, which is zero vector yeah then then simply by setting the numbers, so one two X K equal to zero.
Birne Binegar: You can at this is the zero vector then by setting these numbers equal to zero, you see that zero vector is an element of eight.
Birne Binegar: On the other hand, you could have an offset here but and still be able What, then, what you need to some linear combination of these factors to cancel this term.
Birne Binegar: And there may be special values of the parameters s one SK such that well the individual terms are not equal to zero there some is equal to zero know it's a little bit this P zero is there.
Birne Binegar: And that's, the only thing you know it's not necessary, like true that he is not as subspace but it's a little bit stronger condition because.
Birne Binegar: If this set with P zero w wk all non zero vectors it could still be a subspace as long as it contains the zero vector and so that would be some choice of the numbers F1 to SK that's that.
Birne Binegar: This linear combination athlete works out to be zero and then he would contain the zero vector, on the other hand.
Birne Binegar: For most P zero they're going to be unrelated to these factors here, and so, most of the time, if you have this offset here it's not going to be subspace.
Birne Binegar: This caveat that i'm.
Birne Binegar: Sure, explaining that these two these terms could end up canceling this, and so you would get the zero vector that's a pretty special condition and it's not going to happen, usually.
Birne Binegar: So, in short, if the p zero was not appearing there, then you definitely have a subspace if it does appear here well 99% of the time you don't have a subspace but if there's a special cancellation that occurs and yet you could possibly have some states.
Birne Binegar: Anyway, one of the things we're going to work out.
Birne Binegar: A little bit later, is how to choose these sectors appropriately and so you could make that distinction, as long as the doctors w wk or in somehow independent directions, then in fact yeah yeah.
Birne Binegar: You wouldn't be able to actually be kept the size, the dimension of the space, and you can tell the test whether it passes through the origin or not.
Birne Binegar: Okay, so.
Birne Binegar: let's see that counterexample so Huber alright so.
Birne Binegar: This next set, I want to talk about is exactly related to your question on constructing basically a hyper plane, but notice there's no offset anything.
Birne Binegar: that's the p zero that in our construction of hyper planes it's not occurring here.
Birne Binegar: We have a special name for these kind of subsets call them the span of a given list of vectors the way you create them is you're given a list of K directors.
Birne Binegar: And then you take every possible linear combination of those vectors up great multiplying them by various real numbers and then show me the thing up.
Birne Binegar: And it turned out well, this is a special case of a hyper plane.
Birne Binegar: To special case.
Birne Binegar: Of a hyper plane, where the initial starting point.
Birne Binegar: is zero vector.
Birne Binegar: So.
Birne Binegar: And what I want to show you is that sets of this form the special hyper planes they're always substances and it's really easy to see.
Birne Binegar: I have to show you two things that if I start off with vectors that live inside the span of a given list of vectors then it's closed under scale or multiplication and vector addition.
Birne Binegar: let's look what happens when you do scale and multiplication to an arbitrary vector that lives inside this set so i'm just grabbing this form here.
Birne Binegar: And thinking of it as representing an arbitrary vector all I have to do is specify particular numbers F1 through SK to get a specific example.
Birne Binegar: But this is sort of generic factor sitting inside That said, when I do scale a multiplication well i'll just distribute that altercation across that vector some and now you see what you've got as a result, the scale multiplication of this vector by any number Lambda.
Birne Binegar: Is a new vector that's basically a bunch of numbers times the vectors w wk so this guy has the same form is every element to set a bunch of numbers times the given list of vectors.
Birne Binegar: Alright, so Lambda V have the correct form to live inside this set and so not set F is closed under scale multiplication.
Birne Binegar: separately, if I start off with two vectors that live inside the span of a given list of that there's each of them has to be a linear combination of the given list w one through wk when I add them together i'll can collect terms same.
Birne Binegar: vectors and now we've got some of two vectors that sitting inside capital S it's expressed as a bunch of numbers times the individual vectors w wk add it up, so the one plus the two it's a linear combination of w one two wk, and so it also is an element of F.
Birne Binegar: Alright, so.
Birne Binegar: Now the set S, and this slide is shown to be closed under vector addition adding two vectors gives you a vector of the same type.
Birne Binegar: All right, and so, because.
Birne Binegar: On the preceding two slides we've seen that the span of vectors is closed under scale and multiplication and then the preceding slide it's also closed under vector addition that makes the span of a set of vectors always has subspace.
Birne Binegar: And in fact it's one of the most important ways fundamental ways in which we sub spaces arise and linear algebra.
Birne Binegar: When you have some such a common occurrence, we have some Oh, we have some nomenclature for this, I think I have enough later slide that explains it, but let me point it out now.
Birne Binegar: We think of.
Birne Binegar: Taking a list of vectors and then taking all possible linear combinations we think of that is generating some space we often call the vectors w wk the generators of the subspace that's produced a taking all possible linear combinations of those factors.
Birne Binegar: All right, let me show you another way that.
Birne Binegar: Sub spaces arise, this is another really important example.
Birne Binegar: Well, first i'll start off with the counter example we've seen that.
Birne Binegar: When you have linear systems and their solution sets well for a single linear system, a single equation, and in a note is always a hyper plane.
Birne Binegar: have one less you mentioned them, the number of variable she has embedded in if you have.
Birne Binegar: A list of em variables and you impose single equation on it, what you get is N minus one dimensional hyper plane, but more generally.
Birne Binegar: solution sets of linear systems are hyper planes or intersections of hyper planes, first of all, but then intersections of hyper planes are just hyper planes of smaller dimension so.
Birne Binegar: The thing about you look at solution sets of linear systems, you always get hyper planes but it's like shown on previous slide hyper planes are not always sub spaces there only sub spaces when they contain the zero vector.
Birne Binegar: So what i'm going to show you is that now there's a special kind of linear system that always produces the subspace but in general they they are not some spaces or solution sets.
Birne Binegar: let's let's look at this example here, I have some coefficient matrix a and some.
Birne Binegar: home vector be here and regard those is fixed and therefore defining this set as the solution set of that equation.
Birne Binegar: As you Barry why to.
Birne Binegar: n dimensional space well.
Birne Binegar: If I scale and multiply a solution of this equation by Lambda what happens well here i'm checking to see whether or not a times Lambda why satisfies the same equation is y.
Birne Binegar: At scale a multiplication can be delayed until after I do the matrix multiplication but a time to why by hypothesis is equal to be until I get Lambda times B, but.
Birne Binegar: If Lambda is not equal to one and remember, we have to be closed under scaling multiplication by all real numbers, not just special ones.
Birne Binegar: And so, if Lambda is not equal to one then eight times Lambda why it does satisfying equation, but it's not the same equation that we started off with and so.
Birne Binegar: The set is not closed under scale or multiplication because scale and multiplication takes our solutions to the solutions have a different equation.
Birne Binegar: and North it closed under vector addition.
Birne Binegar: There, let me take a pair of solution, so I have a y is equal to be and also going to have a w is equal to be they both are going to satisfy the equations that define that set capital S.
Birne Binegar: When I check to see if this thing also satisfies the equation, well, I can distribute that matrix multiplication across the vector summoned.
Birne Binegar: And what happens is eight times, why it's going to eat eat because it's solution, a w is also going to equal be because w is supposed to be a solution.
Birne Binegar: And so what we get is that a apply to the sum of wine w doesn't equal be rather equals two times D it satisfies a different equation so in general solution sets of linear systems are not going to be sub spaces there neither closed under scale and multiplication or vector addition.
Birne Binegar: All right, well.
Birne Binegar: there's a special family of linear systems, which are always sub spaces in the head of distinction, we make here, instead of having an arbitrary column vector be here i'm putting the zero vector on the right hand side.
Birne Binegar: I had to be there, I would just call it a linear system, but the fact that I have zero on the right or the zero vector on the right hand side that's what we call a homogeneous linear system.
Birne Binegar: And i'll show you real quickly that solution set of equations like this are always closed under scale the multiplication and vector addition.
Birne Binegar: I so here i'll do a scale a multiplication of solution of this equation, well, I can pull that scale and multiplication past the matrix multiplication and then a time for why.
Birne Binegar: It sounds like this equation so it's going to equal to zero vector Lambda times the zero vector is equal to zero vector so any scale and multiple with a solution satisfies the same set of equations the same matrix equation.
Birne Binegar: And certainly if I have.
Birne Binegar: Some of two solutions of this equation I add them together and then apply the coefficient matrix a to that, then, well, I can distribute the matrix multiplication.
Birne Binegar: Then, because, why is the solution, a y is equal to zero w solution aws equals 00 plus zero is equal to zero, and now I see that I have to start off with two solutions and I add them together, I still have a solution of the same equation.
Birne Binegar: Alright, so what that tells us is that.
Birne Binegar: The solution set up.
Birne Binegar: A usual linear system it's not going to be a safe space, but if, on the right hand side, we have the zero vector then it's always going to be a subspace and, in fact.
Birne Binegar: That those two.
Birne Binegar: Those two possibilities are decisive either.
Birne Binegar: If you have any non serial number here then it's not going to be a subspace, but if you have a zero there will always be a sub state, so you can detect whether or not a solution of the home.
Birne Binegar: Of a linear system is a subspace or not simply by looking at that right hand vector vector on the right hand side.
Birne Binegar: All right, so um there's a connection between the.
Birne Binegar: Two examples, because we know that hyper planes correspond to solutions of linear systems and so and well i've shown you some hyper planes are.
Birne Binegar: Sub spaces, those that they to be a subspace they have to contain the zero vector but usually a hyper plane is not going to be a substation and, similarly, I related shown that when you consider.
Birne Binegar: solution sets well.
Birne Binegar: I guess here's the distinction, I may 1 in general general hyper plane is not going to be a subspace but if it has this form, where P zero doesn't occur, then it's always going to be a safe space, but you you don't like here is this offset here that's what destroys the ability.
Birne Binegar: To classify disguises subspace.
Birne Binegar: The existence of this guy in a way that can't be counseled by this guy's going to mean that the zero vector that's not live inside of your your hyper plane, and so it can't be a subspace.
Birne Binegar: But similarly.
Birne Binegar: So linear systems correspond to hyper planes know these guys.
Birne Binegar: are not going to be hype substations in general, nor our solution sets of linear systems, but there are two special cases there's a special case of a hyper plane where.
Birne Binegar: You don't have this offset and then it will be a sub space and there's a special case of homogeneous linear systems where you don't have a non zero vector here.
Birne Binegar: Now all we do have the zero vector on the right hand side that's always going to be a subspace so what I hoping, you see, is that a hyper planes is this offset that causes the probably makes it.
Birne Binegar: A general hyperplasia not a substrate.
Birne Binegar: And then for linear systems is this a non zero vector on the right hand side that would make it not.
Birne Binegar: Not a subspace but as long as we have the zero vector there yeah it would be a subspace.
Birne Binegar: dope in some sense.
Birne Binegar: hyper planes solution sets of general linear systems they're close to being vector spaces, because when you actually plot their solution sets they look like vector spaces.
Birne Binegar: are always going to be hyper planes embedded in a in a larger dimensions digital space.
Birne Binegar: But from the geometric point of view hyperplasia it's not going to be a subspace unless it passes through the origin.
Birne Binegar: So only special hyper planes are going to be suspicious and similarly solution set that these guys are always hyper planes, but they don't always pass through the Oregon.
Birne Binegar: However, i'm solution sets of the equation like this always going to contain the origin because one solution of this is simply taking the vector X, to be the zero vector and then you would have any matrix time to zero vector is always going to produce the zero vector.
Birne Binegar: Okay, so.
Birne Binegar: These two examples.
Birne Binegar: My racing these two examples.
Birne Binegar: or kind of fundamental is that will have lots of different sub spaces that are going to rise in this course, but what you'll find.
Birne Binegar: Is we proceed now after the midterm is that anytime I introduced the subspace.
Birne Binegar: It may be because i'm going to solve differential equations maybe because i'm doing something that doesn't look anything like the context working here.
Birne Binegar: When you look at the notes nuts and bolts of the construction of that subspace it's usually going to be either span of a given list of that year, so i'm taking linear combinations of vectors to get a set now or it's going to be a solution, set of some equations and.
Birne Binegar: When it's a homogeneous set of equations then i'll get us up face, it is not a homogeneous equations I won't get us up states so.
Birne Binegar: Just for the time being, just realize that there's two different standard constructions of a subspace.
Birne Binegar: First, one is just to take a list of vectors you consider all possible linear combination with no offset.
Birne Binegar: That will always give you a substation this will be one of the principal ways we construct sub spaces in the future, but another way that substations can arise, is a solutions of homogeneous linear systems where the number on the right hand side there is zero.
Birne Binegar: By the way, I want to show you something because it's related to what will be on the exam.
Birne Binegar: When you have a linear system like this.
Birne Binegar: You can solve it, without even introducing the augmented matrix, in fact, let me show you what goes on in this case, when you write a you form this augmented matrix corresponding to a homogeneous linear system, so the vector on the right hand side is zero vector well, whatever it is.
Birne Binegar: In this last column.
Birne Binegar: you're only going to have 00 and all your subsequent row operations are not going to change this last call if, like interchange two rows I just flipped two zeros.
Birne Binegar: If I multiply a row by a number, I have a number of times zero is equal to zero, or if I add a multiple of one row to another row i'm adding zero to zero, so what happens.
Birne Binegar: For these homogeneous linear systems is this last column never changes and throughout the production and in fact we.
Birne Binegar: will be solving lots of linear systems of this form and then typically what we do is i'll just ignore that last column and i'll only reduce the coefficient part because nothing ever happens in this last call.
Birne Binegar: So, not only are they.
Birne Binegar: homogeneous linear systems.
Birne Binegar: Always sub spaces in some sense they're easier to solve than the general case because I don't have to worry about what's going on in that last call.
Birne Binegar: Of the corresponding augmented matrix.
Birne Binegar: Okay, so.
Birne Binegar: let's see.
Birne Binegar: Oh yeah so here's where i'm.
Birne Binegar: Introducing some nomenclature for dealing with.
Birne Binegar: This will be a frequent construction will take a given list of factors and take all possible linear combinations of them.
Birne Binegar: I guess I should write this out in detail, this is the set.
Birne Binegar: That I create by taking for a given list of vectors I just take all possible many combinations of those that's done by multiplying the vectors by various numbers.
Birne Binegar: And then, allowing those numerical coefficients.
Birne Binegar: The range over the entire real line alright so frequently right span, but you should always bear in mind what that means is taking all possible linear combinations have you Given this, the vectors.
Birne Binegar: We say that this list of vectors do you want to pay generates the subspace it generates it via this.
Birne Binegar: Creation of new backers by taking linear combinations of the given lists.
Birne Binegar: Sometimes we also call the individual vectors do you want to be K we call those the generators of the subspace.
Birne Binegar: But.
Birne Binegar: One of the things we're gonna have to deal with, so if I create a sub space in this fashion, how do I compare it with other stuff spaces and the problem that we're going to find, is it.
Birne Binegar: You could have different vectors different choices of the want to bk that generate the same subspace, but if you just stare at the.
Birne Binegar: at how they perform that not going to be at all obvious in fact.
Birne Binegar: phenomenon that there is that.
Birne Binegar: suppose I want to.
Birne Binegar: Think of the blackboard is my set s and I start off with two vectors.
Birne Binegar: Well, I am That said, as well as long as these are independent directions, I can reach any point in this plane or any point on this.
Birne Binegar: subspace so by taking linear combinations of these two vectors but i'm not restricted to just using say.
Birne Binegar: V1 and V2 to generate this space, I can also choose maybe w one and another vectors w two and take all possible linear combinations of these actors, I still generate the blackboard, and so the set is the problem and it's going to be this one so specified as say.
Birne Binegar: The span of V1 and V2.
Birne Binegar: And you have another set that's the span of a different set of vectors w one w two how to tell.
Birne Binegar: If he has an s are the same.
Birne Binegar: So being able to create.
Birne Binegar: Sub spaces it's kind of easy, but our problem that's going to actually going to be, how do we compare some spaces when they were generated in different fashions Another thing I could have done is I could look at the set es generated by a span of vectors and I have another.
Birne Binegar: Face that's the solution set of a homogeneous linear system, I don't want detect if they're the same well, these are all questions that we're going to address after the exam.
Birne Binegar: What i'm doing before the exam just pointing to the problem we're gonna have to face.
Birne Binegar: Alright, so.
Birne Binegar: first observation is even though this is going to be a common construction it's not going to be unique construction, even when you have a particular subspace fix up space in mind.
Birne Binegar: So.
Birne Binegar: The first problem we're going to address we study sub spaces further is well.
Birne Binegar: it's hard to tell whether to sub spaces are different and even.
Birne Binegar: Even better than that is how do we actually coordinate eyes the vectors that live inside of subspace that's kind of a different issue, let me see slide.
Birne Binegar: Well, this slide sorter gets to the point.
Birne Binegar: But let me, let me state it.
Birne Binegar: that's another problem.
Birne Binegar: With.
Birne Binegar: Understanding sub spaces.
Birne Binegar: So actually the first problem I just alluded to, is that.
Birne Binegar: it's hard to tell.
Birne Binegar: If.
Birne Binegar: to subscribe.
Birne Binegar: are equal, because they can be made different ways.
Birne Binegar: What you'd want to do if you had two sets up the word of the same you want to show that each element, and once it also lives in the other, but when they.
Birne Binegar: If one set is formed by solving a system of equation, and another set is formed by taking a span of a given set of vectors well how do you make that comparison you certainly cannot compare.
Birne Binegar: an infinite number of vectors in one set with an infinite number of vectors in the other set that's that's too hard to do, and so we will be looking for alternative means.
Birne Binegar: But.
Birne Binegar: Another problem that we have is that typically we have say the spanned.
Birne Binegar: So this is a second problem if say you have the span of three vectors V1 V2 v3 and say, this is a subset.
Birne Binegar: of our tank well what they said is, of course, is to set up all possible linear combinations.
Birne Binegar: Of the three given vectors 33 the 321 P two P three bearing over the real lie alright so.
Birne Binegar: here's another sort of conceptual problem that we're going to face.
Birne Binegar: These factors I say are are generating a subspace of our 10 well, that means that the one the two in D3 are ordered lists of 10 numbers right there.
Birne Binegar: elements of the set of all ordered list of 10 numbers, on the other hand, when you look at how the set was formed, it looks like it's you only have three.
Birne Binegar: Three parameters needed to specify every point in this set so in some sense this thing looks three dimensional because there's.
Birne Binegar: Three things we can vary to reach every point yet every each one of these vectors he's actually attend dimensional vector because i'm taking linear combinations of 10 dimensional vectors.
Birne Binegar: So our intuition gets kind of messed up, am I talking about a three dimensional object or i'm talking about a collection of 10 dimensional objects and it that that's.
Birne Binegar: The problem i'm going to rephrase now has given us some space.
Birne Binegar: How do we find good coordinates for it and then this makes a little bit more sense here we have a set of 10 dimensional vectors.
Birne Binegar: But we only need three parameters to reach one, so it makes better sense to think of tier one tier two and three three as be good coordinates for the elements of the set.
Birne Binegar: Even though the individual vectors live in attend dimensional space, the natural way to coordinate is this shit to be to use just three numbers do you want to to MP3.
Birne Binegar: So that's construction.
Birne Binegar: I which we take a little vector space sitting inside a larger vector space and we're able to use coordinates of the same dimension at the size of that space, not the individual vectors, but rather the.
Birne Binegar: the totality of the space that they generate these linear combinations that's what we're trying to coordinate ties.
Birne Binegar: And the way you do that.
Birne Binegar: Well here's another example here, I have three two dimensional vectors.
Birne Binegar: and
Birne Binegar: What I want to show you on this slide.
Birne Binegar: is yet another problem we face.
Birne Binegar: So this vector.
Birne Binegar: there's typically how you create a vector i'll use three different number C one C to C three because i'm taking linear combination these sectors like specify.
Birne Binegar: A value for see one value for see two and values for C three I get a vector and clearly belongs to the span of the given this the vectors because it's just a linear combination of sectors but notice that.
Birne Binegar: C one C to C three so make this choice and I construct the corresponding linear combination, I get the vector after doing some algebra arithmetic I get the vector to zero.
Birne Binegar: If I choose another set of.
Birne Binegar: coefficient see one equals zero Z to is equal to zero and see three is equal to two I construct that linear combination calculated, I get the same vector so what happened here, even though.
Birne Binegar: I would like to think of tier one tier two and tier three as be good coordinates for this three dimensional space of vectors here, we see that.
Birne Binegar: A scale or multiples.
Birne Binegar: You can choose different different coordinates and end up at the same vector, and so we have a problem with duplicity again is that if.
Birne Binegar: This entire set is generated from three factors.
Birne Binegar: And that's what we did we might think of coordinate tightening the element in the span is just by grabbing the coefficients in the expansion, but what does example shows you is that well that's not good enough, because you can have vectors with two different coordinates.
Birne Binegar: corresponding to the same two dimensional vector, and so what you want, for coordinates is a unique map that takes you from values of your coordinates.
Birne Binegar: To unique points in your face that's how you coordinate times you don't want to have a choice of two different coordinates for the same point.
Birne Binegar: Alright, so maybe this is a good place to stop because i'll introduce the definition that will begin discussing in detail, after the exam.
Birne Binegar: What we're going to look for when we have a subspace is a set of vectors such that not only can we write every vector inside ourselves face as a linear combination of factors, but so that these.
Birne Binegar: coefficients see want to see cake and actually be good coordinates we want this to only work for exactly one choice of the coordinates then.
Birne Binegar: Just by specifying these key numbers, then we get a unique factor that sitting inside this up face, and we have successfully coordinate Tyson vm the substance and that's how to construct these guys is going to be our first topic after the exam.
Birne Binegar: Okay, so on Wednesday we'll do a review session, you might look at the practice exam beforehand, but I will go over each class.
Birne Binegar: And then, of course, if you have any questions on on how to do reduction in other things we can discuss that on Wednesday as well.