Birne Binegar: awesome right foot.
Birne Binegar: forward.
Birne Binegar: happy today.
Birne Binegar: Hard harder and harder.
Birne Binegar: than.
Birne Binegar: I did okay.
Birne Binegar: Okay, I guess, we can get started so we've lost a bunch of days from snow I was still actually over a week.
Birne Binegar: Of classes.
Birne Binegar: Already this semester, while i'm just going to continue marching through the lectures, so what may be at the end very end of the semester we don't get you all 41 of the supposes lectures for this course, but we'll just proceed it at the at the pace that sort of outline already with the.
Birne Binegar: With prepared lectures, of course, what i'm doing is before every lecture i'm looking back what I did last Semester and and re editing things, but at least i'm changing the day.
Birne Binegar: So you know if you see a current date there then you'll know that this is some sort of an updated version of the lecture notes.
Birne Binegar: What we're going to do today is, first of all I need since we haven't talked in a week i'll recap what we've talked about a week ago about sub spaces.
Birne Binegar: How to coordinate them that were basis come in and how to find a good basis that's where linear independence comes in.
Birne Binegar: And then today we'll look at a particular example we're looking at sub spaces that are generated by.
Birne Binegar: You, the Rosa that matrix and well i'll show you how you actually explicitly find a basis for those kinds of substances and then maybe at the end we'll find them and bases for other sub spaces or let one can attach to a matrix.
Birne Binegar: Alright, so the principal thing we've been talking about since the first exam has been the notion of what a subspace is.
Birne Binegar: I told you it's a little vector space sitting inside a larger vector space, but there's more to it than that, and in fact i'm.
Birne Binegar: The essential part that makes what makes a subspace important it's not just a set of vectors but rather it's a set of vectors that behave well respect to.
Birne Binegar: The fundamental vector operations scale multiplication and depth tradition, so if you have a set of vectors.
Birne Binegar: That closed under scale of multiplication, meaning that when you multiply by a number any vector that lives in that set you don't leave the set.
Birne Binegar: or when you add two vectors that start off in the set you still end up in the set then that's when the conditions, you have to have a subspace and these turned out to be really the fundamental.
Birne Binegar: objects and linear algebra typically when you're dealing with linear algebra and an application you're not only interested in doctors doctors themselves, but the substances, for example when you're solving a differential equation.
you're.
Birne Binegar: Generally speaking, you can find one solution that would be equivalent planning one vector but you want to get all the solutions and back that what will.
Birne Binegar: eventually see is that when you solve the differential equation you're looking for a subspace and functions that actually satisfy.
Birne Binegar: The differential equation, and that they will be a subspace in the vector space of functions so up faces are really important they are typically how we encapsulate.
Birne Binegar: Application problems it's figuring out how to coordinate is a particular subspace.
Birne Binegar: So there's two ways that we can make substations really easily one, you should just take a list of vectors and you just start combining them using our vector space operations scale or multiplication.
Birne Binegar: And Dexter addition when you've done that in every possible way for a given list of vectors that's when you create what's called a span of a set of vectors so this construction here when I let these coefficients very over the entire real life, it generate.
Birne Binegar: A big space in fact it's going to be a hyper plane of some sort there'll be a hyper plane that passes through the origin.
Birne Binegar: But another way, we can create a set of vectors that's closed under scale and multiplication and vector vision is to require the vectors X, to be the solution to have a linear system.
Birne Binegar: Solutions of linear systems are also going to be closed under scale or multiplication vector addition because.
Birne Binegar: If a X is equal to zero, then.
Birne Binegar: A time Lambda X well, you can take that scale of multiplication and you can delay it until after you apply the vector but then, if a concept, because equal to zero.
Birne Binegar: that's going to mean Lambda time eight X is equal to zero, so that means Lambda times zero vector equals zero vector oh.
Birne Binegar: If you have a solution than any multiple that solution by numbers still going to be a solution, and similarly if you have two solutions and you add them together, you get eight X.
Birne Binegar: plus a why, but then you just get 00 if both X and y are solutions of this equations so now, one way to make a subspace is to take a given list of actors and just.
Birne Binegar: Consider all possible linear combinations of them, but a second independent way of creating a subspace is to demand that the.
Birne Binegar: vectors in that subspace satisfy particular homogeneous linear equations.
Birne Binegar: That word homogeneous, meaning that on the right hand side, we have the zero vector rather than some non zero vector.
Birne Binegar: OK, so those are the that's the objects we're going to be looking at, and the first task at hand, we.
Birne Binegar: started to talk about last week was well given a subspace, how do we actually coordinate, is it, how do we actually I mean you would like to think of this maybe as a katie mentioned a lot of space, because you have a directions, from which.
Birne Binegar: to depart from the origin, in order to get all these points.
Birne Binegar: And so, but what we'd want to do if this is what katie mentioned subspace we don't want to use Andy mentioned to specify the individual vectors so I guess what's relevant here is to see a picture of what's going on alright so.
Birne Binegar: I did this a three dimensional cube it's supposed to represent you think of this imagine it to be an arbitrarily dimension are in our directors are living.
Birne Binegar: And then embedded inside this large vector space it's going to be a hyper plane that passes through the origin that's going to be our subspace.
Birne Binegar: don't it's this flat sheet this grid sheet the tackling cutting through this cube and then the individual vectors of the subspace.
Birne Binegar: Well, think of them as the one through bk there are a bunch of directions, but by simply taking linear combination these vectors I can reach any point in the subspace.
Birne Binegar: So this is the picture of.
Birne Binegar: An overall vector space are in subspace F that's embedded inside that vector space and then the individual vectors so these individual vectors they live.
Birne Binegar: Open the vector space, but they're also confined to this hyper plane and what we're trying to get around is how do we coordinate ISIS hyper plane without using the original coordinates which would be.
Birne Binegar: Well, we would need n coordinates to specify a vector and dimensional space, but these factors are confined to a lower dimensional things so we're looking for better coordinates to describe those.
Birne Binegar: Right so that's where bases come in bases are the are the gadgets that will allow you to find good coordinates for vectors living inside of subspace.
Birne Binegar: The idea is is that.
Birne Binegar: If every vector near subspace is a linear combination of a given list of vectors, why not just use these numbers do you want to take you're going to need is K of them to.
Birne Binegar: To create a vector that lives inside this set, however.
Birne Binegar: problem is is that if these vectors don't represent independent directions, then there's going to be multiple ways of picking number C one C K expanding.
Birne Binegar: As in the right hand side here and getting a given w so there could be two different ways and I the examples I gave last week I showed you explicitly single vector that can be written two different ways, given using an expansion like this.
Birne Binegar: it's all so what I began to talk about last week was well what conditions can I put on these generators, to make sure that.
Birne Binegar: Once I write a vectors a linear combination of my generators that these numbers see one two ck are unique.
Birne Binegar: If they're unique, then I have a way of specifying this sector by specifying these numbers now have good coordinate, but if they're not unique, then these these numbers see one two ck well they're kind of ambiguous and they won't do much good.
Birne Binegar: Alright, so now.
Birne Binegar: Here again, is the pitches.
Birne Binegar: And overall that your space subspace sitting inside that vector space a bunch of vectors that are generating nets upstate, but the problem remained is that.
Birne Binegar: For an arbitrary set of vectors do you want to bk these guys are not going to give you good coordinates because there's too many ways.
Birne Binegar: of reaching a given point so you think of this point here, I can maybe realize it it's a linear combination of just these two vectors or maybe I can.
Birne Binegar: realize as a linear combination of these two vectors are these two vectors they'll be multiple ways of writing.
Birne Binegar: A given vectors a linear combination of these look like look like, I have a total of six factors, so I hear you have a two dimensional plane six factors in it, you only need two independent directions reach every point on that plane.
Birne Binegar: Our problem in general is that we're not going to have this picture of our disposal, what we're going to see is a list of vectors and we're going to wonder well how do we coordinate ice, they all the vectors that we can.
Birne Binegar: generate using these guys as our generators well you don't have this geometric picture in place you're just staring it list of numbers and trying to figure out how they're going to use those records.
Birne Binegar: So.
Birne Binegar: We find our notion of basis accordingly would say that a set of factors is linearly independent if they satisfy what will seem like a strange condition.
Birne Binegar: What we demand of these factors is that the only way they could sum the zero is by taking all the coefficients equal to zero.
Birne Binegar: Now that's a trivial solution to this equation right if I said X one is equal to zero and X all of the coefficients equal to zero, I automatically have zero is equal to zero doesn't put any conditions on the vectors themselves.
Birne Binegar: But as soon as I have.
Birne Binegar: An equation like this well one of these coefficients X one or it could be fk if one of those coefficients is non zero then i'll be able to effectively remove one of these factors from the list.
Birne Binegar: suppose I had three V one plus to be two plus D3.
Birne Binegar: is equal to zero vectors so the equation of this form, but where I haven't set all the coefficients equal to zero, well then I can use it for example.
Birne Binegar: So I guess one third times minus two the two.
Birne Binegar: minus one third, the three I could use this equation to solve for one of the factors in terms of the other two and so V one.
Birne Binegar: depends on the other two vectors me to MP3 by demanding that the vectors do not satisfy an equation like this unless these coefficients or equal to zero i'm eliminating this possibility i'm eliminating the The ability for us to replace.
Birne Binegar: One of the vectors in this list by an expression in terms of the remaining doctors.
Birne Binegar: All right now, it turns out that's exactly the condition, we need to make sure that if we expand if vectors are linearly independent and we create a sub sub space by kicking linear combination of these these guys will be a basis, he will automatically give us the good coordinates.
Birne Binegar: Yes, yeah that that's how we're actually calculate them so i'm treating this the formal part first and then we'll get into how we actually find bases in calculus but you're right the key thing will be to look at the factors and regard them as the rows of the matrix.
Birne Binegar: Right so.
Birne Binegar: If you have a set of vectors you can always use them to make a subspace but it's not clear that the directions, are all independent one and now they're slowly might not be.
Birne Binegar: give you a basis from which you can coordinate ISIS subspace.
Birne Binegar: However, what we proved last time is if if the vectors are linearly dependent, so if the only way we can get them to add zero is by taking all the coefficients equal to zero well that's going to mean that the vectors are basis until they will provide the ordinance.
Birne Binegar: Alright, so let me.
Birne Binegar: update my definition of what a basis is the original definition which was practically I wanted a set of factors that could.
Birne Binegar: make a subspace at the same time would also give me good coordinate, so I want to be able to write every vector in my subspace.
Birne Binegar: As a particular linear combination of the given list of vectors, then I can use the number C one C ck s coordinate for the vector w that lives inside my substation that lives inside my larger vector space.
Birne Binegar: And so here's what I want will have a basis if we can reach every point in the subspace like taking linear combination and.
Birne Binegar: The numbers he wants tk are necessarily unique.
Birne Binegar: Okay, so that was our definition of what a basis is but with this notion of linear independence, we can revise that definition we can say a basis it's a set of vectors.
Birne Binegar: That you have enough of them to reach every point in your subspace at this conditions, the whole space is created by taking linear combination of your given list of vectors but.
Birne Binegar: we're also we also demand that the factors are literally dependent and that's what will make these linear combinations unique.
Birne Binegar: So yeah and what we have, if you take a set of factors that are the set is large enough to generate the entire subspace.
Birne Binegar: And you also know that the only way, you can get them to add together is to produce the zero vector is by taking all the coefficients equal to zero that's linear independence, these can do to conditions hold that, in fact, you can coordinate ties all the vectors inside of substation.
Birne Binegar: As soon as you issue right affect your down it lives until I dishpan, it has to be a linear combination of these factors, but then as soon as you write it down, then these numbers, so you want to pay can be used as coordinates.
Birne Binegar: In fact here's how you see that katie mentioned subspace.
Birne Binegar: You see what i'm saying, and since they're all equal to zero linearly independent that's right.
Birne Binegar: that's right that's right so yeah we can always make the subspace, but the problem is to find good coordinate so we needed a Nick a condition on the generator.
Birne Binegar: It turns out that the right condition is to demand that the only way these sectors can add together and produce the zero vector is by taking these numbers equal to zero.
Birne Binegar: By the way, I should point out, I should have pointed this out a long time ago that you're.
Birne Binegar: you're familiar with the notion of basis from your physics classes.
Birne Binegar: there.
Birne Binegar: Is a notion of standard basis for three dimensional space or three the so called I.
Birne Binegar: 100 J.
Birne Binegar: 010 and OK equals 001 in three dimensional space, we have the standard basis vectors that point along the extraction one the y direction one dizzy direction and every vector.
Birne Binegar: Say 123 can be written as one times the first standard basis director I plus two times second standard basis director.
Birne Binegar: Three times how the third standard basis Okay, and so any vector inside or three can be written as a linear combination of the standard basis, after this construction and used to, and in fact that's what we're trying to mimic.
Birne Binegar: Once we know that a vector can be expressed in terms of these days at vectors, then I can get its coordinates.
Birne Binegar: what's kill you about.
Birne Binegar: Our use of our three though and and the sort of typical where we are at this stage of the course is that.
Birne Binegar: Since we already have this vector written it's ordered list of numbers, this expansion just look makes it look more complicated.
Birne Binegar: But when we're dealing with subspace is it's going to be through expansions of the sort that we actually understand what we're dealing with we're not dealing with arbitrary doctors insider stuff they were only looking at the vectors that can be constructed using linear combinations.
Birne Binegar: Okay, but we're all.
Birne Binegar: of our basis vectors if we tried to be coordinate ISIS subspace is exactly analogous to way these standard basis vectors are used to coordinate eyes are three.
Birne Binegar: Okay, so.
Birne Binegar: So here's some more be fine pitcher again the box is supposed to represent a general vector space or in the flat two dimensional plane embedded inside that box that's supposed to be represented subspace that were generated by a list of vectors and.
Birne Binegar: Here I have a look like I have eight vectors drawn your old imagine they're all living inside this planes.
Birne Binegar: here's our basic problem when we when we create a sub space like taking all possible linear combinations these guys don't provide good coordinate.
Birne Binegar: What we want it's a smaller set of vectors that will provide good coordinate it so those that smaller said i've indicated, with the dread Arabs.
Birne Binegar: and, hopefully, you see how on coordinate typing the space now now this vector V one is giving me all these horizontal directions and the vector P two is giving me.
Birne Binegar: These directions and by taking.
Birne Binegar: A multiple this factor in a multiple this doctor I can reach any point in the plane, and in fact it's pretty clear from this grid here that this point here there's going to be a unique multiple be one and a unique multiple be to that will lead me to this point.
Birne Binegar: Okay, so what basis it's going to be generally a smaller set of years which every other vector in the subspace can be created using linear combination.
Birne Binegar: We want that set to be small enough so that there's no duplicity and the way we represent vectors it always has to be large enough, though, so that we can reach every point in the subspace.
Birne Binegar: going to turn out that the number of vectors you have in the basis is exactly exactly he finds the dimension of the subspace.
Birne Binegar: OK, so now we've got the concepts down let's look at an example where you have to compute things.
Birne Binegar: So i'm going to look.
Birne Binegar: First of all, I want to introduce some substations that are attached to a matrix.
Birne Binegar: And then, and then i'll have to wonder little sub spaces, no show you directly how you find it spaces.
Birne Binegar: Alright, so.
Birne Binegar: There are two ways of making sub spaces, that you gave you at the beginning of the lecture either you take a bunch of vectors and you take all possible linear combinations that will give you a subspace that's that spam construction or.
Birne Binegar: You take the role vectors out well here the row space of the matrix is going to be the span of the roadmap, which means you'll get to take.
Birne Binegar: The rows of the matrix reinterpret them as ordered list of numbers and then take all possible linear combinations of the ordered list and you'll get from the span construction you'll get a certain subspace.
Birne Binegar: And lets me describe a little bit more explicitly where that subspace lives you're a is a matrix say what am calm.
Birne Binegar: And rows.
Birne Binegar: So i'm just putting in the corner entries of this matrix.
Birne Binegar: But, of course, the whole thing is false, I just want to point out that when I.
Birne Binegar: Look at these rogue actors, I grabbed them as what a list of numbers, because I have em columns that each row vectors going to be an n dimensional vector.
Birne Binegar: And so, each of the factors it's going to be an m dimensional vector when I take linear combinations of them i'm going to get some subset of the set of ordered list of em numbers.
Birne Binegar: So the row space.
Birne Binegar: it's going to be a sub spaces each row vector can be interpreted.
Birne Binegar: n dimensional vector so the road space of this matrix.
Birne Binegar: it's going to be a subspace of our em I could also similarly view the columns it's ordered list of numbers i'll call that the column space.
Birne Binegar: i'll be the span.
Birne Binegar: Of the columns of a.
Birne Binegar: And because each column it's going to contain any entries, this is going to be a subspace.
Birne Binegar: of a general a different vector space, so the row space lipson one vector space determined by how many columns you have, and they call space it's gonna it's gonna live in a better state, depending on the number of rows people.
Birne Binegar: So those are two vectors spaces like create as soon as I write down invite him matrix.
Birne Binegar: Of this row space to span the Rose calm faces the span of the columns and there's a third one, because I had a different way of making a subspace.
Birne Binegar: If I have a homogeneous linear system, I can look at the solution set and that will also be a sub space, and so I can take a matrix.
Birne Binegar: reinterpreted as the coefficient matrix of homogeneous linear system and then look at the solution set of that that subspace is called the no space of a matrix.
Birne Binegar: And it turns out.
Birne Binegar: Well, maybe if we have time we'll see at the end of lecture today that we already know how to find a basis for this, we just don't have the nomenclature for it.
Birne Binegar: But um what else concentrate on first, though, is looking at that row space of the matrix and describing how we find a basis for the road speaks of the matrix.
Birne Binegar: So we got pretty close to that last Monday.
Birne Binegar: I showed you several things last Monday.
Birne Binegar: First of all, I pointed out that if a matrix is in row Echelon form then it's non to rose automatically form the basis for the space that they generate.
Birne Binegar: that's.
Birne Binegar: me put us matrix in.
Birne Binegar: In row Echelon for what does that mean it means.
Birne Binegar: It may have a non zero entry up in this corner, but below it only going to have zeros and maybe we have.
Birne Binegar: No pivot in the second row until we reach the third column, we could have even have a non zero entry here, maybe we have a three there well if it's a relational on forums.
Birne Binegar: here's a pivot here's the next pivot and as we proceed over maybe 1000 the pivots are Korean this downward staircase.
Birne Binegar: that's what i'm assuming when I see a matrix is in real Echelon forms now what I want to show you is that the non zero rows are going to linearly independent.
Birne Binegar: Why is that well suppose I take the first row and I multiply it by see one, and then I take the second row and or to scale or multiplication by in the second number see two and then.
Birne Binegar: I add everything up, so this should be seeing Okay, so I want you to imagine this number.
Birne Binegar: multiplying each entry net first row see to multiply each entry in the second row is better and then adding everything up well here, we would get to see one in that first slot.
Birne Binegar: Just multiplying to buy see one and then adding the contribution from the other rose well they're all being multiplied by zero so they're not going to contribute now if this linear combination of the robe actors is supposed to equal.
Birne Binegar: The zero vector i'm going to need the first component, the equal to zero well I can't cancel it the see one I get here with a see two or three or ck other terms here, so the only way I can get this leading entry to vanish is like settings see one equal to zero.
Birne Binegar: And so.
Birne Binegar: If the matrix is enroll Echelon for.
Birne Binegar: The only way I can get a linear combination of my rose to equal to zero well I can't cancel the pivot entry of that first row with the other rose until I have to take see one is equal to zero.
Birne Binegar: Well, what that does to me, is it makes the first row irrelevant like it's like I have to sit see one is equal to zero now, I have a simpler equation and back, but I have the same problem.
Birne Binegar: Now, are tooth you are okay, are those rows are still in real Echelon form, I just figured out a way of eliminating that row I have the same problem here.
Birne Binegar: I have a C two times a second row I can't cancel out this leading term with the entries from the lower rose, so I have to set see to is equal to zero, in order for the total sum to equal the zero vector.
Birne Binegar: And then, once I do that, then the sun gets even smaller now I have see three, but it still enroll Echelon form in one by one, you systematically can prove that.
Birne Binegar: You need see one is equal to zero, but the see one is equal to zero, then UDC to is equal.
Birne Binegar: But then you need see theory console and recursive Lee You conclude that is that all these coefficients have to separately equal to zero.
Birne Binegar: If this linear combination of the road actors is going to equal to zero vectors well, that means that the only way I could get linear combination two equals zero by sending all the coefficients equal to zero, it means like rows are linearly independence now.
Birne Binegar: What does it mean according to my second my rice definition, if I have a set of vectors that's large enough to generate an entire subspace yet.
Birne Binegar: there's also linearly independent.
Birne Binegar: that's what it means to have a basis, and so what other effectively showing you here is that a matrix in row Echelon form is non zero rows are not only going to be enough to reach every point in the row space, but they're also maybe linearly independent to there'll be a basis.
Birne Binegar: Right so.
Birne Binegar: So that's the first argument it's kind of specialized because the argument so far only worked for major season real Echelon for.
Birne Binegar: So next thing is to wells.
Birne Binegar: we're used to dealing with major seasons, that are not in Russia on for usually what we do is we try to put them in real Echelon Florence, and so what I want to show you next is that when you.
Birne Binegar: Okay, so here's this the argument that the non heroes of the matrix should relate to one form or necessarily linearly independence, but what I want to show you next is that if you do elementary row operations.
Birne Binegar: A space, the roads are going to change, but the space that's generated by the roads is not going to change.
Birne Binegar: Now.
Birne Binegar: One way to understand this is remember where these elementary row operations came from.
Birne Binegar: We develop elementary row operations originally when we were trying to solve linear systems.
Birne Binegar: We correspond to each elementary row operation was something we could do to a set of equations that will not change the solution.
Birne Binegar: Well, those elementary real operations actually act on the matrix that corresponds to the original set of equations.
Birne Binegar: So elementary row operations correspond the operations that don't change solution set and, in fact, what they correspond to our operations that you can perform on sets of vectors that are not going to change the space that they generate.
Birne Binegar: So.
Birne Binegar: So let me show you quickly why elementary roll operations did not change the row space here's just a matrix with two rows so this should be enough for me to transmit the idea.
Birne Binegar: Elementary roll operation really only excellent one row, so I can represent any elementary row operation, acting on any matrix by just focusing.
Birne Binegar: I really only need to look at what happens to one row but don't include the other row, because sometimes that comes into play, but if we have a matrix consisting of two rows here's a tro space, instead of all possible linear combination of it drove actors.
Birne Binegar: What happens when we interchange two rows well row one in row to get switched but the set of linear combinations will sing just reliable these variable see one and see to just interchange them and you see that this is exactly the same set of vectors.
Birne Binegar: What happens if we replace a row by a scale or multiple of itself well there you have it i'm replacing row to buy the scale of multiple so the new row space is going to be a setup all possible linear combinations of real one, and the new road to.
Birne Binegar: Look at this constructions.
Birne Binegar: i'm not allowed to bury see one and see to over the entire real line I can make this number any number, I want, and although see two is being multiplied by Lambda by simply readjusting definition of CO2, I can make this second coefficient any number I want.
Birne Binegar: And so again i'm reproducing the same the same set of vectors.
Birne Binegar: What if we do, the more complicated elementary row operation, where we place a row buy it somewhere for multiple another else right so here's you see the effect.
Birne Binegar: The roast face at the new matrix is going to be the set of all possible in your combinations the original first row and the new second walls.
Birne Binegar: But just rearranging that terms here a little bit, you see, I still have some number times are one plus some number times are two.
Birne Binegar: And i'm taking all possible linear combinations by letting those variable see one and see to range over the entire line, so the roads are changing his ice carry out elementary row operations, but the space that they generate is staying the same.
Birne Binegar: Right so.
Birne Binegar: That tells us that if he has any invite him matrix and we calculate any rational and form of that matrix than the non zero rose up the real excellent form are actually going to be a basis for the roof space at the original matrix.
Birne Binegar: Why because.
Birne Binegar: Elementary real operations don't change the road space, and so the road space at the relational on form it's going to be exactly the same subspace.
Birne Binegar: As the row space of the original matrix.
Birne Binegar: But then we know how to find a basis for the row Echelon form a matrix and rely solely on form it's just the non to row so we're going to be automatically linearly dependent and he will generate the entire row space so.
Birne Binegar: The Non heroes are the worst one form of going to be a basis for this, but it's the same as the original aerospace and so that's how we find a basis for not the roof space up an arbitrary matrix.
Birne Binegar: We write it down we computed try to inform and we grabbed a nonzero rose.
Birne Binegar: All right now that calculations.
Birne Binegar: i'm going to his bed in a different problems supposedly now forget about matrices for for just a couple seconds suppose I have a list of factors sitting inside an n dimensional space w one w K and I want to find a basis for the span of vectors.
Birne Binegar: Well here's how we can do it, like the factors as heroes of a matrix.
Birne Binegar: Then, when I take the span of these vectors it's going to be the same as the span of the roads of the matrix I create from him.
Birne Binegar: Then I can reduce that matrix to which rationalized form by grabbing the non to rotor the rotation was born I get a basis for the role space of this matrix, which is the same as the basis as a basis for the original matrix which is same as the span of the.
Birne Binegar: The original set of vectors because they were exactly the rows of a.
Birne Binegar: itself, the way the calculation proceeds, and let me write it down more systematically.
Birne Binegar: defined a basis.
Birne Binegar: For.
Birne Binegar: span.
Birne Binegar: of a given list of actors who.
Birne Binegar: Have UK and let's say they're all n dimensional vectors.
Birne Binegar: The first step is to form.
Birne Binegar: A matrix and in fact it's going to be a matrix was a road.
Birne Binegar: And exactly and cold.
Birne Binegar: Why.
Birne Binegar: Am I doing this right.
Birne Binegar: Maybe i'll put this inside our and.
Birne Binegar: that's usually the way I.
Birne Binegar: label columns.
Birne Binegar: yeah so each of the.
Birne Binegar: Each of these factors is going to be an n dimensional vector so I can write them in a horizontal list with em entries, and I can write them on top of one another's to create that I have K of them have paid vectors with em components or K by a matrix holiday.
Birne Binegar: And they want you do issues, reduce.
Birne Binegar: A to.
Birne Binegar: A row Echelon forms a prime and then what we have is.
Birne Binegar: The basis for aerospace.
Birne Binegar: A prime well that's going to be the non zero road.
Birne Binegar: Well, this is a row Echelon form until it's not zero rows.
Birne Binegar: are going to provide a basis for the fourth row space, but this row space it's the same thing as a row space of the original matrix, which is the same thing as the span of the original set of vectors because, after all this matrix was created by using the given list of vectors.
Birne Binegar: So, to find a basis for this subspace create the matrix reduced it it's real life long form and then just grab the non zero rose and you're done.
Birne Binegar: Right so that's the procedures and, in fact.
Birne Binegar: This is the basis for lots of calculations we're trying to athlete coordinate I sub spaces.
Birne Binegar: first thing I have to do is generally find good coordinates that's going to mean finding a basis, and this is the procedure by what you find the basis you list your backers of the Rosa matrix revenue Sharon us on form and grab the non zero rose.
Birne Binegar: Alright, so let's do an explicit examples here's a set of three four dimensional vectors I take all possible linear combinations of them I get a subspace of our four and it's going to look like it's three dimensional right because I have three different directions, I can hit out.
Birne Binegar: But what will see that it turns out not to be three dimensional.
Birne Binegar: What we'll do is we'll write these factors as heroes of the matrix here, you see that first vector written this the first rally.
Birne Binegar: And then, what i'll do.
Birne Binegar: Is i'll note that a span of these for these three robe actors, this is exactly the roast days of a but it's also equal to the set i'm trying to describe.
Birne Binegar: next step is I wrote reduce make the matrix I constructed from the original vectors I reduce it to a row Echelon forums and stuff here, I went all the way to reduce your actual on floor but that's not critical, the main thing is to get it into row excellent.
Birne Binegar: Now, what happens is somehow I got a zero row during this row reduction process that meant that that there was a way of combining these rows so that I got the zero.
Birne Binegar: That means that the original set of doctors were not linearly independent, there was a way of taking linear combinations of them and somehow producing zeros.
Birne Binegar: Alright, so how about that means now is that the row space of a it's the same thing as the row space of this matrix but I don't need that third vectors to generate that space.
Birne Binegar: And, in fact, these two non two rows of the road epsilon form are going to be a basis, not only for the rope face of the row Echelon for but also the roast base of the original matrix, which is the subspace we were trying to coordinate eyes to begin.
Birne Binegar: With exactly all possible in your combinations of those three rows, as is the aerospace of this matrix, and so there, it is the answer to the original question is what is a basis for this subspace of our for that generated by these three robe actors it's these two actors, they will suffice.
Birne Binegar: Okay, so.
Birne Binegar: that's.
Birne Binegar: One calculation one type of subspace that we can now handle in general.
Birne Binegar: Wherever we're at now.
Birne Binegar: two ways of making subs basis.
Birne Binegar: Was the SEC creative subspace why considering the span of a given list of actors.
Birne Binegar: And the way we found a basis for this, yes, we write the vectors.
Birne Binegar: rose up a matrix.
Birne Binegar: Reduce.
Birne Binegar: form.
Birne Binegar: You matrix a prime and grab the non zero rose.
Birne Binegar: And then these non heroes they're going to be your basis.
Birne Binegar: basis for this original subspace.
Birne Binegar: The second way of making a subspace what is the solution set.
Birne Binegar: Up homogeneous linear system.
Birne Binegar: So what I want to show you next is that for sub spaces of this type, how we find a basis and the good news is you already know how to do it, you just don't know the nomenclature.
Birne Binegar: So let me.
Birne Binegar: Show you remind you what you should know about this situation, the way you find the solution set.
Birne Binegar: To take the matrix before the augmented matrix and you reduce to some new matrix that.
Birne Binegar: In reduced row Echelon for now I wrote the zero here because, in fact, for a homogeneous linear system, all the entries in this last column or zero and no matter what row operations before those entries remains zero.
Birne Binegar: All in this production process all the action occurs on this left hand side this right hand deck the vector on the right hand side just stay zero throughout all the row operations, because.
Birne Binegar: By interchange two zeros here i'll get it looks the same if I multiply a zero by a non serial number looks the same it looks zero if I take a multiple one row and I add it to another row i'm adding 00 it stays zeros so.
Birne Binegar: I knew how we saw the things and once we get to the root we do show Echelon form next thing that we try to do is present that solution as a hyper planes so that means we identify.
Birne Binegar: Fixed variables.
Birne Binegar: correspond to the columns of the rope reduce one form that contain pivots.
Birne Binegar: And the columns without pivots.
Birne Binegar: They correspond to the free variable.
Birne Binegar: And in writing gamma hydroplane solution to such a system, the idea is to.
Birne Binegar: Separate your variables in the fiction free very both express the fixed variables in terms of the free variables write down a solution vector.
Birne Binegar: And then expanded in terms of the free variable so let me give you not step through that process, but hopefully you remember the end of the calculations for solving such a linear system is to express your solution vectors.
Birne Binegar: As living on a hyper planes.
Birne Binegar: Right that's the goal to solve this linear system if you want to express the solution is that you're living on a hyper planes now, when we have a homogeneous equations what's going to happen.
Birne Binegar: If that constant vector which basically comes from the entries in this column here that's going to equal zero because.
Birne Binegar: There, everything is that nothing is going to survive in this call it's going to remain zero, but the additional factors that occur here.
Birne Binegar: And these guys are going to be tied to the free variables, which are in turn tied to the columns of The reason we do show Echelon form that do not take pivots these are going to be your free variables.
Birne Binegar: So what this solution in for homogeneous linear system what it shows you is that the solutions are always the span of a given set of vectors.
Birne Binegar: In fact, these factors are always going to be linearly independent you know, and the reason for that is is that.
Birne Binegar: Oh i'm emitted over I think i'll delay.
Birne Binegar: That until Wednesday, but basically what's going to happen is.
Birne Binegar: Essentially spree parameters are going to be like an x four X five it's one of the original variables it's always going to have a component.
Birne Binegar: that's one for it and, but in that X, one may appear and other components, because the other fixed variables may depend on it, but because there's one component that.
Birne Binegar: That is going to be non zero, but the other vectors in this expansion, are going to be different three variables I won't be able to cancel them and the sky, so these factors aren't going to necessarily be linearly independent.
Birne Binegar: Alright, so i'll go through that argument on Wednesday or by the way, I.
Birne Binegar: I posted homework set six it's going to be a do a week from today, but I don't expect you to have the wherewithal to calculate much until after Wednesday lecture.
Birne Binegar: want to hear if I.
Birne Binegar: Give five minutes yeah all your consultation I dia so i'm for my senior design project I.
Birne Binegar: would like you know the real up from Star Wars.
Birne Binegar: Which robot to do that circle, the one that rolls on a sphere oh yeah yeah yeah yeah so.
Birne Binegar: One of the problems we have a test robot that's broken up like isn't tracking and its movement is dictated by to serve those both opposite each other on the XY plane that's controlling radio movement so would that be, would it be beneficial, because I mean, obviously we could.
Birne Binegar: not use any theory and just find the exactly like what motor positions do this, that but I need to to.
Birne Binegar: A PhD controller for it, so what i'm asking.
Birne Binegar: Is would it be feasible and or a good idea to set those up as sub spaces, based on the constants which would be the.
Birne Binegar: Service yeah it's a little bit, let me think about this it's all let's just be better just to translate the linear emotion into radio playing with no reference, because I think I think probably what you have to do.
Birne Binegar: um let's see.
Birne Binegar: David yellow marker yeah yeah yeah I kind of trying to.
Birne Binegar: kind of see what you're doing so you have a you have something that's going to roll in one direction or another, I guess the Italy yeah so.
Birne Binegar: Basically, this hemisphere, is this, but then we also have another atmosphere rotating and moving on this plan, however, the actual service they're only pulling they pull bike tires or bike breaks.
Birne Binegar: like this, so you know into in perpendicular motion this isn't exactly upwards, but this is like into or out of the page if that makes sense.
Birne Binegar: yeah so um, and these have these little wires that go up to a magnet and then this magnet varies along this point as well, which and that's actually what controls the head, but this needs to tell a processor here where it wants to go and i'm wondering what the mathematical best way to.
Birne Binegar: solve that system of equations because it's a system like yeah and and it may not be linear algebra i'm just.
Birne Binegar: Definitely, because you have so many curves here that it's kind of getting nonlinear and maybe linear algebra is only going to give you an approximation when you can actually see it as a.
Birne Binegar: Not as a curve space, but in a very small region it's practically a plane and then, then you can get those directions.
Birne Binegar: But then the probably what you'd have to do, though, is do some additional programming to know if you roll so far in this direction, how far do you actually move it's not.
Birne Binegar: not exactly the number of terms it's number of terms times the radius times pie, or something or yeah so so it may not, it may not be directly, but there should be a formula that will allow you to go from.
Birne Binegar: directions there which way you want to move, but how much you move that's probably a formula for that yeah.
Birne Binegar: But yeah I think you can do that now, because the issue i'm having is i'm doing controls, but the person who's doing the head is just sending me.
Birne Binegar: How it was like what position it needs to be, and so I need to translate that into voltages to send to my service right Brown and so I was just hoping, I could get away with a little linear.
Birne Binegar: Well, you know I think he'll set it up using linear algebra but then you'll use the formulas for the actual distance travel that that's going to be non linear but at least least.
Birne Binegar: Your controls are going to be linear you know which direction you're going in that's that's gonna be linear but how far moves is going to be nonlinear Okay, but that's Okay, you can you can you can take that into account, the main thing you're going to want to.
Birne Binegar: be able to do is control the direction of movement and then like just do the timing on and off, you can control how far it travels yes well service, you can you can send you can send in a signal, and it will only go to this angle.
Birne Binegar: or only go to this other so I don't exactly need to know like I don't have any timing issues that the biggest issue I have is I don't wanna I don't want to move it so fast that's the, the main problem is I don't want to move it too fast to the point where the magnets disconnect.
Birne Binegar: If and I can't think of a way to tune a PhD so that it slows down at the right spot so that it looks smooth without using linear algebra on top of.
Birne Binegar: You I can translate yeah fear of losing about you may want to keep you have to figure out.
Birne Binegar: What the where does it break down when when is the weather the magnets come disconnected yeah and then you may have to use them for a given movement, you may have to break it up into a bunch of small movement.
Birne Binegar: So that you never cross that boundary will that's what that's what the P, it is useful because it'll allow for the proportional control to get dampened so doesn't just slam it.
Birne Binegar: Like the reason why I was asking for your consultation is I was hoping, maybe I could get away with it just a linear algebra but from what you're saying I am just going to have to bite the bullet and do both yeah you'll have to.
Birne Binegar: But but, but the setting it up properly linear algebra is can be your key step oh yeah and then then once you do that, then you just need to do the math to figure out what happens when you do that yeah and I know i'll set it up as.
Birne Binegar: The maximum amount that's my that's how you know when I can operate it and then everything else is just translating it to a circle, with a given radius.
Birne Binegar: But I was hoping on a limb that maybe maybe I could get away with doing less work well, I think the problem it make it all flat, and then you won't have to deal with yeah.
Birne Binegar: Well that's what mathematicians do that they have a nonlinear probably flatten it.
Birne Binegar: yeah.