Birne Binegar: Okay, I guess we're ready to start so yeah we have.
Birne Binegar: Another vacation last week I guess we've lost a week so far in this course due to snow.
Birne Binegar: i'm not changing the schedule i'm just gonna applaud long through the course and if we end up covering only 36 lectures out of the 41 i've written now that's okay we'll just plod along what I want to do today is i'm going to start off reviewing.
Birne Binegar: The notion of Sub spaces bases and linear independence that we covered a week ago, and then i'll.
Birne Binegar: point out the substances that you can attach to any n by n matrix and, in particular i'll show you how you actually find a basis for the role states of a matrix that calculation.
Birne Binegar: Although i'm going to introduce it as a way of finding a basis for a particular subspace attached to a matrix that same calculation is going to be a basis for finding.
Birne Binegar: basis for arbitrary substations as well anytime that faces sub spaces, presented as a span of a set of vectors will have a means to find a basis for.
Birne Binegar: So let me.
Birne Binegar: Let me start.
Birne Binegar: Reviewing the concepts are going into this first of all.
Birne Binegar: And linear algebra we're always dealing with.
Birne Binegar: This particular vector space of order and list of numbers and.
Birne Binegar: What we'll see and you'll certainly see your applications after you leave this course.
Birne Binegar: it's usually not individual vectors that are most important in linear algebra understanding what linear algebra is good for, but rather the sets that they that linear algebra axon.
Birne Binegar: And so we introduced this notion of a subspace as a particular subs a particular set of factors that are living inside a larger vector space, but what we want for these.
Birne Binegar: Particular subset is that they behave well with respect to the fundamental vector space operations scale multiplication index tradition.
Birne Binegar: So, as long as we have a subset that behaves well with respect to that meaning if I do these operations I stay inside the said I don't go somewhere else, as long as we have that, then we have what's called the subspace.
Birne Binegar: And there's two typical ways that well, there are many ways that sub spaces arise, particularly in applications one.
Birne Binegar: One way that if you've had differential equations you've already seen is that when you solve a linear differential equation.
Birne Binegar: The solution set this a subspace of the set of functions well with never stated that way in linear algebra walk but we'll get to that point, a little bit later in the course.
Birne Binegar: Typical way you'll see some spaces arises, they usually can be recast and one of the following two forms one way, you can create a safe space.
Birne Binegar: To take a set of factors living inside your vector space and just consider all the factors, you can create from them like taking linear combinations well that will produce the subspace.
Birne Binegar: What you need to be a subspace is a set that to close under scale multiplication in depth tradition.
Birne Binegar: The way you such like this, or be constructed if we're taking all possible scalable applications in depth traditions, to create the set so it's automatically closed under those operations, because coast and operations were made to construct the set.
Birne Binegar: But the second way that a subspace can arise is so not as explicitly as a span of a set of vectors, but rather as the solution set of a homogeneous linear system.
Birne Binegar: Yet some coefficient matrix times vector of variables equal to the zero vector.
Birne Binegar: Now here you don't see this the subspace directly right you only know that it all the vectors that live in that tub they satisfy a particular equation.
Birne Binegar: that's kind of different than if we're at least here, you know how to write down the vectors that live inside the subspace but it turns out that these are the two basic ways that sub spaces arise and what we're going to concentrate on at least in the first half of this lecture is.
Birne Binegar: How we deal with substations of this actually more simple form, where you can actually see how you're going to make the vectors inside that subspace.
Birne Binegar: let's look at a geometric texture of the situation.
Birne Binegar: If you have a set of vectors thanks to the blue arrows as being the vectors V1 to bk.
Birne Binegar: Then what you can do is you can start taking all possible linear combinations of those guys.
Birne Binegar: And what's going to happen is that yeah they'll begin to fill out the space in which the vectors live but they're all be confined.
Birne Binegar: Because you only have this many directions to work with they'll they'll be confined to a smaller set of directions and therefore sort of confined to a hyper plane that's embedded inside.
Birne Binegar: A larger vector space, so the picture to keep in mind here is, you have an overall vector space in which everything lives.
Birne Binegar: You have a set of generators that are just individual vectors and then you have the subspace is thing that made this hyperplasia made like looking at all possible linear combinations of you've given list vectors.
Birne Binegar: Alright, so our problem that I introduced last week was Okay, given the subspace, how do we describe the individual vectors that occur well.
Birne Binegar: We can write them down, but what we are really seeking our good coordinates so that we can actually specify individual vectors inside of subspace uniquely.
Birne Binegar: So that led me to this.
Birne Binegar: Talk a bit about bases and how we coordinate I sub spaces Oh, again, the picture is.
Birne Binegar: We have the factors that yeah which question so are some spaces.
Birne Binegar: I guess.
Birne Binegar: Are these three dimensional from the hydroplane or are they always.
Birne Binegar: Well, so so hyper planes are everywhere yeah so, in fact, what else you see this picture.
Birne Binegar: Here I like our end is actually a three dimensional States represented by that cube and then my subspace this is flat plane it's embedded.
Birne Binegar: So what i'm just trying to pick yours, that some of the vectors inside your larger vector space are confined to a smaller vector space, you can see that vector space sitting in.
Birne Binegar: There on the hyper play yeah so maybe not clear, if I could rotate around you would see that these arrows always lived in this flat plane.
Birne Binegar: And the problem you have, though, is that when I have a bunch of actors, creating a vector space there's not a good natural set of coordinates here.
Birne Binegar: If you look at this grid here you kind of can see how you might coordinate ISIS you know, maybe you go this direction, a certain distance and you go over this to.
Birne Binegar: A certain direction then you've got a unique point but, unfortunately, those directions that are coming from you this.
Birne Binegar: graphical picture are not really there you erased all this grid points here you just see a bunch of lines range arbitrarily in a plane that's embedded in a three dimensional space.
Birne Binegar: For the problem we're facing is if this is the way we're making a subspace, how do we actually identify the points that live in here.
Birne Binegar: uniquely rather than there'll be multiple ways that I could, for example, reach this point here, using these factors, I could maybe take these two vectors and put this guy.
Birne Binegar: On former parallelogram using these two directions, I can reach this point or I can pull a parallelogram using these two directions and reach the same point i'll be able to reach.
Birne Binegar: Any two of the factors I think will be sufficient to reach any point on the plane, but when I use all them generate it I don't get natural for this, because I have.
Birne Binegar: Too many different ways of writing down a given vectors.
Birne Binegar: Alright, so.
Birne Binegar: that's four bases come in.
Birne Binegar: The idea of a basis well if you know, a subspace with generated by given listed vectors.
Birne Binegar: That means that every vector in the subspace has to be capable of being written as a linear combination of your given list.
Birne Binegar: But the problem is unless we put some restrictions on these vectors here there's going to be multiple ways of representing a single vector using these coordinates and we want unique coordinates for every point.
Birne Binegar: So.
Birne Binegar: So actually yeah Let me move past this picture, except for the main point is that you can make a subspace using a bunch of actors.
Birne Binegar: But the vectors us here may not are not going to give you good coordinates because there's too many different ways of writing of.
Birne Binegar: Particular vector as a linear combination of these 123455 vectors there'll be many, many ways of writing a given point as a linear combination of those five vectors.
Birne Binegar: So what we need is, we need to put an additional condition on our generators, to make sure that.
Birne Binegar: That are the coordinates when we write them down that they're going to be unique and it turns out that the condition that necessary is called linear independence, it means that if the vectors do you want to be K that generate your subspace can only.
Birne Binegar: Have a have a linear combination that correspond to the zero vector, the only way, the only way that can happen is by taking all the coefficients individually and separately equal to zero.
Birne Binegar: I can only satisfy this equation by these constraints, then we'll say that the vectors are linearly independent what this effectively means is.
Birne Binegar: let's look at the counter example of this suppose, this was not true suppose I had a way of a relationship between the vectors V1 through bk it didn't require me to put all these coefficients X one X case separately equal to zero suppose I had three be one plus to be two plus.
Birne Binegar: The three is equal to the zero vector well if I had that what I would be able to do that and it's right v3 as minus three V one minus two the two and.
Birne Binegar: This coming from this equation, I would be able to replace one of the vectors in terms of the other two, and so the three would not be independent of V1 and V2.
Birne Binegar: What we're denying like this condition is the ability to write one vector in terms of the other vectors and that's why we call them linearly independent.
Birne Binegar: Okay, so what we're going to do them is we're going to put a restriction on our generators if we require them to be literally independent then in fact we get a.
Birne Binegar: Work let's see.
Birne Binegar: If we restrict our generators.
Birne Binegar: till that they're linearly independent well, it turns out that that's going to be enough to make your set of generators have basis.
Birne Binegar: way I want you to understand basis is that capable of giving unique coordinates.
Birne Binegar: For every point inside the subspace so as long as you have generators are literally independence.
Birne Binegar: cannot take a linear combination them equal zero without taking the coefficient of zero.
Birne Binegar: As long as that condition is true, then what's going to happen is that each vectors are automatically going to give you a good set of coordinate.
Birne Binegar: What this what this condition effectively does is it throws out the vector sure you don't need and it keeps only the vectors that are sufficient to reach every point in your subspace.
Birne Binegar: And, but also under the condition that you can only reach a given point you're subspace by one linear combination of the given list of vectors.
Birne Binegar: Alright, so my original definition of what a basis was is.
Birne Binegar: It was practical, I just wanted to set a set of directions, from which I can generate the entire subspace and which would give me a unique coordinate.
Birne Binegar: The problem I had was is that an arbitrary set of vectors would have too many directions until I wouldn't get unique coordinates So what we did was we introduced the condition of linear independence and.
Birne Binegar: From the theorem on the previous slide if I start off with generators that are linearly independent then.
Birne Binegar: I will get a basis, and I will be able to coordinate all my vectors.
Birne Binegar: Until let's change our definition.
Birne Binegar: This one.
Birne Binegar: The original definition was more utilitarian I introduced the basis because I wanted to get coordinates until I introduce this notion precisely silica get good coordinate, but now we have a simpler condition to impose requiring our vectors are generators to be it literally dependence.
Birne Binegar: Why don't we use that as a definition that the basis, so will now say that a set of factors is a basis for subspace.
Birne Binegar: If you can reach every point in the subspace by taking linear combinations of your backers alright well that's practical, we want to be able to reach a point in our subspace but Moreover, if we demand that they're linearly independent then.
Birne Binegar: what's going to happen is because they're linearly dependent, the only way to write a particular vector is by choosing a particular set of numbers one through ck.
Birne Binegar: And then we can use those numbers as a way of coordinate sizing the vectors inside the space, now the picture that that you should have now is that OK, so these factors are living in a higher dimensional space and some are.
Birne Binegar: They don't have a maybe a list of 410 dimensional vectors that are used to create what one would hope would be a four dimensional subspace there'll be.
Birne Binegar: Think of K is the for here, and then we would have for parameters that we would need to specify every point inside my subspace and in fact I can arrange those numbers in an order list, and I can actually think of that as a four dimensional vector but.
Birne Binegar: These objects here each member.
Birne Binegar: Each w here listen attend dimensional space i'm just using.
Birne Binegar: Four dimensions again considered i'm suggesting for to be a concrete example if case equal to four, then these coordinates are going to be four dimensional, even though the objects that they're trying to coordinate eyes are living in attend dimensional space.
Birne Binegar: But so.
Birne Binegar: If we have a set of actors that are large enough haven't enough independent direction, so we can reach every point in our subspace and these guys are linearly independent.
Birne Binegar: Then, this is precisely how we can coordinate ice vectors living inside the subspace we can then just grab these coefficients and use them as coordinates if the vectors are not linearly independent, then this way of specifying a vector is not going to be unique yeah.
Birne Binegar: Are.
Birne Binegar: Only four dimensional only going to use the first four basis.
Birne Binegar: i'm somewhat somewhat, let me, let me give you a picture.
Birne Binegar: i'm not sure if you can see it from here, but i've got in read what I would like to regard is basis vectors they they consider to live in the subspace but what i've got are two directions.
Birne Binegar: And just using I don't have to use all all of these five directions here to reach every point it turns out to reach every point on this flat plane, I just have to go a certain direction along.
Birne Binegar: Each coordinate directions and a certain distance long beach core nutrition to just specify a component in this direction and a component in this direction to enough to specify any point in the plane, so this is, this is the strange thing.
Birne Binegar: All the if you think of n is equal in 10 every one of the vectors are drawn here so 10 dimensional vector is going to have 10 components, but to specify the vectors that live on this.
Birne Binegar: hyper planes it to bedded in the 10 dimensional space turns out that they only need these two vectors.
Birne Binegar: And that, by using these two vectors there's going to be a unique linear combinations of what i'm calling V1 and V2 to reach any point in this hyper plane this hyper planes it's embedded in your larger space.
Birne Binegar: The previous slide.
Birne Binegar: Is.
Birne Binegar: Better.
Birne Binegar: unique coordinate vectors.
Birne Binegar: Oh here okay all right.
Birne Binegar: Oh uppercase w.
Birne Binegar: O this w alright so in the picture here yeah think of think of w's goodness so s here, so I changed location, unfortunately, between the picture and and what I wrote before think of.
Birne Binegar: Think of w is being this so this this this plane here.
Birne Binegar: yeah it's a subspace and.
Birne Binegar: So so yeah I had w w I could have had yeah who inside of us, but so, who is the subspace or amnesty is the overall vector space in which everything lives, and this is just.
Birne Binegar: A hyper plane that sitting inside here, but the thing is what we're trying to imagine it's a lower dimensional hyper plane, so we don't have to use all 10 or whatever coordinates.
Birne Binegar: Of the original vector space to specify points that live on that lower dimensional surface.
Birne Binegar: What the basis allows us to do is then specify points that live inside that is bedded surface using a smaller set of coordinates and these are good quarters, because now for each choice of see one two ck i'll get a unique Dr rather than some ambiguity.
Birne Binegar: Okay, so, so this is a picture to keep in mind, because it shows all the players that are coming into this discussion it's showing the overall vector space.
Birne Binegar: Also, the subspace which all the vectors were interested in living, it shows a bunch of examples of vectors that live entire desktop face and, in fact, could be used to generate the subspace.
Birne Binegar: But it's also showing the special vectors the basis factors that are capable of still reaching every point inside that subspace.
Birne Binegar: But doing so in such a way that we can actually coordinate is a point on this up spaces.
Birne Binegar: I certainly figuring out the coefficients well how much it's going in this direction and how much it's going in that direction, we know that information, then we know exactly the point we're talking about okay so.
Birne Binegar: So we'd have an overall back your space subspace inside the subspace We have lots of vectors what we're concentrating on our even smaller set of vectors that capable of reaching every point in the subspace in a unique ways.
Birne Binegar: We have.
Birne Binegar: If I use more than just these two vectors, then I would have multiple ways of reaching any particular point.
Birne Binegar: And that's what we're trying to avoid.
Birne Binegar: Okay, so what I want to do next now is.
Birne Binegar: sort of pass the intuition is going on, what I want to do is pick a country some subspace and show you how you find a basis for it.
Birne Binegar: So let's start.
Birne Binegar: With me describing some substations that we can attach to a any any matrix any matrix with in rows and m calls.
Birne Binegar: One thing we could do this is the subspace is called the row space of a matrix it's going to consist of all possible linear combinations of the factors that make up the road of the matrix.
Birne Binegar: So if you had a.
Birne Binegar: let's say a is equal to the matrix 1110311 minus one zero.
Birne Binegar: Then row space of a.
Birne Binegar: it's gonna.
Birne Binegar: it's actually defined as the setup all possible linear combination of these row vector so you have some multiple the first 1110 plus constant multiple second row vector zero to one plus a constant multiple say the third row vector.
Birne Binegar: And then you let those coefficient range over the entire rely.
Birne Binegar: That generate a certain subspace up well since it for all three dimensional vectors it's going to be a certain subspace of our three taking three three dimensional vectors and I consider all possible linear combinations of them that creative subspace.
Birne Binegar: And that's the first one i've named here, but you can do a similar thing instead of taking all possible linear combinations of this robe actors.
Birne Binegar: You can concentrate instead on the column vectors and take all possible linear combination of these three column vectors that will produce another stuff space.
Birne Binegar: And let me do this a little bit more generally.
Birne Binegar: If I had a was a n by n matrix.
Birne Binegar: So I have a one one a one m a n one a N, am I just put in the corner entries, but you can imagine the rest of the matrix well.
Birne Binegar: Each row vector here it's going to have as many entries in it, as we have columns so the row space.
Birne Binegar: Of this matrix a.
Birne Binegar: Every one of its members is an m dimensional vector so it's going to be living inside the vector space our EMS, while the column space that matrix.
Birne Binegar: Well, we look at the columns the columns the number of entries, we have in a particular column is equal to the number of rows.
Birne Binegar: So the color space, in a sense, it's made out of n dimensional vectors it's going to be subspace are in.
Birne Binegar: So, given a arbitrary matrix, we have to substations we can immediately create from it, we can take the Rose use after generate the row space or we can take the columns and use that to generate.
Birne Binegar: Another vector space.
Birne Binegar: Another subspace living in a different vector space.
Birne Binegar: And then we have one more way, we can create a sub space.
Birne Binegar: And attach it to a given matrix what we can look at is use the matrix to form a set of equations and then.
Birne Binegar: Regardless are subspace a set of solutions of a X is equal to zero, and that will be called the no space of a matrix alright, so I wanted to give you, these three of ways of producing subs basis but today's lecture.
Birne Binegar: least the first.
Birne Binegar: Three quarters of it we're constantly we're going to concentrate on the roof space here.
Birne Binegar: If we create a sub space, by using the roads of the matrix to generate it, how do we find a basis for that.
Birne Binegar: Alright, so.
Birne Binegar: that's the topic that I actually began last Monday, let me now complete that.
Birne Binegar: What we're going to our goal at this point now is take a matrix create it drove space and now, to find a basis for that road space to find a way of actually coordinate icy aerospace of the matrix.
Birne Binegar: Alright, so here's what we do.
Birne Binegar: Is.
Birne Binegar: We first all deal with a simple case where a matrix is in relational on form.
Birne Binegar: What does that do for us well it's going to be very useful because we'll be able to actually visually see the linearly independent of the generators, if you have a matrix that's in relational on form.
Birne Binegar: What that means is that you might have a two in this upper corner here, but if it's in real life long form below a pivot in the first row, you can have only zeros and then maybe you have a second pivot in the second row maybe it's three well.
Birne Binegar: Because it's in a row Echelon form you're going to have nothing but zeros you can even have something up here, if you want, but if it's in Russia long form, you can have nothing but zeros below alright so keep this in mind.
Birne Binegar: This is the form of a matrix.
Birne Binegar: In row Echelon form.
Birne Binegar: And up my statement is is that if a matrix is in real Echelon form then necessarily the non zero vectors are going to be literally independence.
Birne Binegar: Why is that literally independence means that there's no way to get the vectors to something to no way to produce a linear combination of vectors that equals zero vectors without taking all the coefficient separately equal to zero.
Birne Binegar: Alright, so let's look at this matrix and think about what it would take to get a linear combination of the trope actors to equals zero.
Birne Binegar: Well, look at this guy you're going to multiply this first row like something number see one all right, you do that, let me just move that those factors in see one times to zero time to you not see one there and so.
Birne Binegar: Now I want to add to it a multiple of see to do ck and have it produced zero well, you see.
Birne Binegar: If I multiply the second row by see to well first time i'll get a non zero contribution is way out here, and similarly for the road below so there's no way i'm going to be able to get rid of this first component like adding to.
Paxton Krafft: The first row.
Birne Binegar: Some linear combination of the remaining rose because there's nothing here to cancel that guy.
Birne Binegar: So that shows you that see one is definitely going to have to equal zero if the sum is equal to zero, because otherwise there'll be no way to cancel this leading component, the pivot component of the first row.
Birne Binegar: Alright, so if this equation is true and my rose Horn Rochelle on form i'm going to have to take see one is equal to zero.
Birne Binegar: Again, because otherwise I will not be able to get rid of this, the contribution of this particular component when I do the total sons.
Birne Binegar: But then, once I said see one is equal to zero i'm left with a similar situations.
Birne Binegar: get rid of this row.
Birne Binegar: The matrix is still in relational on form, so you might have a.
Birne Binegar: pivot here in the third row, but in any rate when I take a linear combination of rows two, three and all the way up to the last row.
Birne Binegar: I won't be able to cancel out this guy by taking linear combinations of the second row with remaining rows so that would mean I would have to take see two is equal to zero, just to get rid of this component.
Birne Binegar: But then I get a smaller linear combination with the same problem and step by step, you just work it out that, in order to get a some like this to equal zero.
Birne Binegar: Using the rows of a matrix and relational on form, the only way you could do it is by separately setting all these coefficients equal to zero, therefore, the rows of a matrix and real Echelon form are always linearly independent.
Birne Binegar: Good That means, if you have a set of vectors that are linearly independent and they generate a subspace they are a basis for the subspace so.
Birne Binegar: i've actively prove this lemma if a matrix is enroll isilon form than the non zero vectors of that matrix are going to be a basis for the.
Birne Binegar: For the subspace that you generate by using those robe actors.
Birne Binegar: All right now yeah so.
Birne Binegar: The next thing I wanted to point out theorem I just described only worked for matrix ease in raw children or.
Birne Binegar: We see, generally speaking, we start off with the matrix is not in real actual unborn and free modifying its inverse wonderful is find a.
Birne Binegar: solution set to an equation that solves you want to be able to.
Birne Binegar: recast the matrix into row Echelon form or reduce show Echelon forms this next lemme tells us that well when we do row operations on it on a matrix it does not change the space that the road vectors generate it's analogous to the way.
Birne Binegar: Elementary real operations came up in the first place, we introduced elementary operations, acting on equations because they did not change the solution set.
Birne Binegar: Then we transported that idea into a matrix setting by replacing those operations did not change the solution set of equations with operations, we can perform on the augmented matrix that represented that linear systems and then with.
Birne Binegar: The elementary real operations were things that would not change the solution set up the corresponding equations but they're actually operations that you perform directly on a matrix what I want to show you on this slide.
Birne Binegar: Is that those operations don't change the row space either.
Birne Binegar: So here i'm not going to prove it in general i'm just going to show you a matrix with two rows one and two and i'll do some real operations on these and you'll see that the road spacing even though.
Birne Binegar: The way we write down the road space it's going to change the actual set of vectors that produces not going to change.
Birne Binegar: Alright, so here's a matrix with to row vectors one row vector one row back to when I write down the road space, I take all possible linear combination, so those two Rebecca.
Birne Binegar: All right, that's just the definition of what the road spaces now let's create a new matrix by interchanging the rows of a.
Birne Binegar: What does that do well, when you switch these two rows these guys effectively just switch places.
Birne Binegar: But since see one and see to in both cases range over the entire real line it's pretty clear that this set is identical to this set all you have to do is reliable, see one a see to see to see one.
Birne Binegar: So you can get exactly the same set of to interchange Rosa the matrix What if you change this matrix a by.
Birne Binegar: multiplying the second row by a non zero multiples Lambda right, then what happens is when I take the look at the road space of the new matrix well.
Birne Binegar: I still have an arbitrary number times the first row, but the second row change the Lambda to do, I have an arbitrary number of times the second row of the new matrix but here.
Birne Binegar: I varying see to over the entire real line I can make this coefficient to be anything I want and so again i'm reproducing this set it's basically any possible number times are one plus any possible number times are too it's just that.
Birne Binegar: The possibilities are sort of tabulated differently, but they still no matter what Lambda is I let see to range over the until entire real line this product can be any number I want.
Birne Binegar: Alright, so again we reproduce the original the original set of vectors that we create I taking the span at the original rows of the matrix.
Birne Binegar: Alright, and then let's see, let me pull this up a little bit.
Birne Binegar: The third element tree real operations where you take a rough Oh, and you replace it with us some it somewhere from multiple have another else well if you do that.
Birne Binegar: Alright, so here's your original first row it doesn't change them to this operation, but your second row it's changed well, you can rewrite this just taking the terms, involving r1 and collecting him together.
Birne Binegar: Now again you have a linear combination of the two original rows are wanting to buy very see one and see to this can be any number one and by Bering sea to itself, this can be any number, I want, so I again I reproduce the same sets.
Birne Binegar: The morals of the of the slide is that well, although not every.
Birne Binegar: matrix is in real life line, for we can put a matrix in row Echelon form and it rotates is not changing.
Birne Binegar: Alright, so.
Birne Binegar: Now we can draw the conclusion suppose we have an arbitrary n by n matrix and let a prime be any of it traditional on forums.
Birne Binegar: Well then, what we can do the non zero Rosa the matrix general action one form provide a basis for the road space of that matrix so the room from the first lemme.
Birne Binegar: or athlete a second ella was see know exactly the second lemme in which we show that the row space above.
Birne Binegar: matrix it's obtained from another matrix via elementary row operations has exactly the same row space.
Birne Binegar: But from the first one must sorry this ordering is mixed up the first lemme was that if you have the non to rows of the matrix general Echelon form is a basis so.
Birne Binegar: The Non zero rose up this role Echelon form it's going to be a basis for the road, the road space of the row Echelon form.
Birne Binegar: But because the row space at the rotational on form it's exactly the same subspace as the roof space of the original matrix the non heroes reduce row Echelon or the rotational on form of the original matrix it's going to be exactly a basis for the original matrix.
Birne Binegar: Alright, so that's the idea, let me show you the application and how this how this is used.
Birne Binegar: Forget about matrices for the time being let's just consider a subspace sitting inside, some are in that generated by given list of vectors.
Birne Binegar: i'm not supposing East vectors are linearly independent, so I don't have good coordinates yet for the subspace I just have a way of constructing it and I don't have a way of coordinate time to eat it yes.
Birne Binegar: So what i'll do is i'll take the vectors years and write them down as the roads are the matrix.
Birne Binegar: And then, what i'll do is i'll reduce that matrix to it's relational on form and then what will happen, then, is the non zero rose up this matrix now.
Birne Binegar: I use the original set of factors to create a matrix now I change that matrix to which relational on form and i'll grab the non to Rosa that.
Birne Binegar: The Nadia wrote up a prime are going to be a basis really it's ro space.
Birne Binegar: But the road space up the original up the relational form it's the same as the road space of the matrix we just created, which was exactly.
Birne Binegar: Like construction we use these vectors as the rows of a, and so the roof space of as exactly the span of the original set of vectors.
Birne Binegar: So, since the row space up the row I salon form is identical to the substation we're trying to quiet ordinary life what we'll do is we'll use a basis for this vector space to coordinate ice is equal, that your space w.
Birne Binegar: Right and so.
Birne Binegar: In the end, the non heroes of the matrix a prime are going to be exactly what we need to get a basis for the original vector space w's here there's a concrete examples here, I give you three four dimensional vectors.
Birne Binegar: And I watch we're going to find a basis for the subspace are for that generated by these three factors, so I take.
Birne Binegar: The vectors and I write them as the rows of the matrix.
Birne Binegar: And what i'll do, then, is well first, we note that.
Birne Binegar: The span of vectors that we're looking for is exactly the row space of this matrix.
Birne Binegar: And so what we'll do find a basis for the space of that matrix we're rubber do some matrix to row I salon form and in fact here, I went further I reduced it all the way to reduce your show on forums.
Birne Binegar: But what happened here, so I got zero row and that zero well we're gonna end up ignoring all we need are these two factors there, the non zero rows of a rational on form of the original matrix so the row space of a prime.
Birne Binegar: It has is a basis, these two non zero vectors but since the row space of a prime is the same as the roast facing a these two factors will also be a basis for the space is generated from these three vectors.
Birne Binegar: Which is exactly the vector space substation we're trying to coordinate ties so yeah well we thought it's a suitable basis for the original span of the first of the original three vectors if these two vectors.
Birne Binegar: To non zero rows of the real Echelon form of the matrix that we formed by writing the road.
Birne Binegar: Like writing the rows of the original list of vectors right ready, the original list of vectors is Rosa the matrix so let me.
Birne Binegar: Let me step through this again.
Birne Binegar: So w equals span.
Birne Binegar: say.
Birne Binegar: w.
Birne Binegar: wk.
Birne Binegar: You find a basis.
Birne Binegar: For you here's the procedure.
Birne Binegar: first thing you do, and you form a matrix.
Birne Binegar: be the span.
Birne Binegar: Well it'll be the matrix we get by writing these are.
Birne Binegar: Real these factors as the road to the matrix will have w one written as a row.
Birne Binegar: The second.
Birne Binegar: Second factor in this list written as a row, to the matrix and then we'll have wk at the bottom all right, then the next step is to rubber use.
Birne Binegar: To a row Echelon form.
Birne Binegar: A prime also will do elementary real operations on this matrix until we get it in row Echelon form once we get that.
Birne Binegar: The Non zero rows.
Birne Binegar: Of this row Echelon form.
Birne Binegar: A prime those are going to equal.
Birne Binegar: A basis for the row space.
Birne Binegar: A prime because we've shown that anytime you have a matrix general Echelon form the non payroll provider basis but that's the same thing as a basis.
Birne Binegar: For.
Birne Binegar: Aerospace at the original matrix.
Birne Binegar: Because elementary row operations don't change the row space, but the basis for the rotation of the original matrix well it's exactly equal to the span of these guys because here's your.
Birne Binegar: here's your matrix the span or the Rose is just the span of the original set of vectors so just going to equal.
Birne Binegar: A basis.
Birne Binegar: For w's and we're done, I mean this.
Birne Binegar: This is the theory that's going into it, but the calculations pretty simple you grab the vectors write them as the Rosa matrix reduce grab the non zero roads and that's your basis, so what happened in this example was which started off with the matrix which started off with vectors.
Birne Binegar: We created the corresponding matrix we wrote produced, and then we grabbed the two non to arose and that's our basis we're done.
Birne Binegar: Okay.
Birne Binegar: So um let's see I two minutes left.
Birne Binegar: Let me yeah I think I can say something about how we find a basis for the other way, we can produce this subspace.
Birne Binegar: Remember.
Birne Binegar: two ways of creating a subspace in general.
Birne Binegar: So the first one that we just been dealing with is you take.
Birne Binegar: A given list of vectors and you consider all possible linear combinations with them.
Birne Binegar: Okay, so that's one way, but the second way.
Birne Binegar: of creating subspace is to set w equal to the solution set.
Birne Binegar: up a homogeneous linear system.
Birne Binegar: i've just shown you how you find a basis for it subspace that you produce this way you write these guys's the rows of the matrix row, reduce and grab the non to rose.
Birne Binegar: For this kind of subspace.
Birne Binegar: It turns out, you already know how to find a basis it's just I haven't given you the nomenclature.
Birne Binegar: When you saw the linear system.
Birne Binegar: What you do is you take the matrix.
Birne Binegar: Or you take the coefficient matrix and you put it inside an augmented matrix and you reduce to reduce row Echelon for.
Birne Binegar: Then you write down this is going to be the augmented matrix of the solution.
Birne Binegar: And then, what you do is you write down the equations.
Birne Binegar: You separate.
Birne Binegar: The free variable.
Birne Binegar: From the fixed variable.
Birne Binegar: That requires you to look at the matrix and we do show Echelon for the variables that correspond the columns with pivots are going to be your fixed variables.
Birne Binegar: The variables that correspond the columns without pivots are going to be your three variables, you should hope you remember this, eventually, you get to the point where you.
Birne Binegar: are able to solve for the free bear bricks variables in terms of the free variables and you end up writing Ghana solution that's going to look like some constant vector i'll use the hyper planes location is some constant vector plus some parameters.
Birne Binegar: Times content vectors V1 through DK let's write the the l.
Birne Binegar: Maybe plus dld hopefully you remember that when you solve a linear system we write the solution down as a hyper planes and our goal is to athlete express every point on the solution set.
Birne Binegar: And as an expansion of this type now what happens when you have a homogeneous linear system is this last column always remained zero So in fact that term never occurs and what you're left with is that every vector your solution is a linear combination of a given list of vectors.
Birne Binegar: These constant factors that will be multiplied by the free parameters, these are exactly your basis directors.
Birne Binegar: Those.
Birne Binegar: Your free parameters, maybe it's X one or X for multiplying a constant vector here you ignore the variables you concentrate on these constant vectors because they provide a basis for solution set of linear homogeneous linear equations.
Birne Binegar: Okay, so what i'll show you on Wednesday is that only write down a solution of this form these factors are automatically linearly independent.
Birne Binegar: Since they generate the entire solution set and they're linearly dependent they're going to be a basis so.
Birne Binegar: Are this kind of subspace we already know how to find the basis it's it's wrapped up in our method of solving linear equations.
Birne Binegar: The only thing that's approved is that these vectors V1 to be a a that you uncover via the process of solving a linear system that these guys are actually linearly independent but i'll show you that on Wednesday nights but you'd be easy to see.
Birne Binegar: Okay, so i'll stop here and i'll see you on Wednesday.