Birne Binegar: Okay.
Birne Binegar: Just let me know i'll mask up again.
But.
Birne Binegar: Now that we're into March.
Birne Binegar: Okay, but again, if anyone has any concerns, let me know i'm happy to mask up you don't have to let other people know.
Birne Binegar: i'm happy to abide by you wishes, but.
Birne Binegar: I thought we test this out anyway, so what we'll do today.
Birne Binegar: On Monday I can do research phase one can catch.
Birne Binegar: Up take the road space which spanned all the road after.
Birne Binegar: That all face that similarly that's all calm.
Birne Binegar: And the last phase of a matrix well if you have a matrix you can create a linear system from it and the north face of a matrix is the solution set.
Birne Binegar: solution set up.
Birne Binegar: A system of equations by using the matrix as a coefficient matrix setting a time factor variables equal to zero vector so.
Birne Binegar: For example, the row space.
Birne Binegar: spam.
Birne Binegar: Why the span of course we mean taking all possible linear combination.
Birne Binegar: Of the Rebecca factors.
Birne Binegar: That they're easily introduced the focus today is going to be how funny face.
Birne Binegar: And then, for you, then, once we do that i'll talk a little bit about mentioned such an easy one.
Birne Binegar: And then in a special case of these sub spaces attached to a matrix.
Birne Binegar: i'll introduce the rain no these are basically Council imagine the break is going to be the dimension of the road space or the columns face, while the know he is going to be the dimension of the subspace solutions of the.
Birne Binegar: homogeneous linear system that wanting to captivate.
Birne Binegar: Okay, so what are some things you have to factor is going to be in stuff face up a well respected in the button after State after scale multiplication and division.
Birne Binegar: We have to Beijing prototype for ways of making today.
Birne Binegar: Firstly, it just take a list of factors living in time you've got your status to her all possible combinations all both ways both find them on numbers and adding them together we do that when you actually create the space, by doing these operations to.
Birne Binegar: Effectively it's going to be closed if.
Birne Binegar: You find anything that's not they might feel a tradition that you will grow in fact still being funny.
Birne Binegar: But how will you make.
Birne Binegar: per second point is a little bit, the solution of a.
Birne Binegar: System of equations come about by coefficient.
Birne Binegar: Maybe the variable.
Birne Binegar: solution sets illusion of obedience equation that they have to buy the property sorry, etc.
Birne Binegar: For all the major applications and distributed over of vectors that all the fire So if you had two solutions have the added them together, it would still get a new solution you distribute.
Birne Binegar: matrix.
Birne Binegar: So what i'm saying here is X.
Birne Binegar: y.
Birne Binegar: equals zero then.
Birne Binegar: Just.
Birne Binegar: A why because that matrix multiplication trivia of the factors and then you'll get euro euro euro, and similarly.
Birne Binegar: You scalable by a solution that equation line number it's still a quick to get it resolved is still solution So these are the two principal ways that subspace arrived and what we've been doing is trying to play bass each.
Birne Binegar: bases from.
Birne Binegar: particular set of factors.
Birne Binegar: That are good at providing port for the back of the other factors of liberalism.
Birne Binegar: or precisely.
Birne Binegar: So basis is going to be a special set of factors that is large enough so that he can reach every point in a linear combination.
Birne Binegar: But also not so big that you have some simplicity of what you want ours, instead of independent directions, or which you have enough directions to reach every point in yourself.
Birne Binegar: And then you want to make sure, because the idea is that once you can expand every vectors like not state the linear combination give a list of that then don't expansion cool it should see one day, they can now use this coordinate for the rest of the factors such as.
Birne Binegar: So you have this picture here is three dimensional box to imagine that representing an arbitrary vector states are in maybe our camp and then sitting inside that vector space to have a sub sub state lower dimensional that he that he thought the larger than.
Birne Binegar: The blue line represents our retreat Center city.
Birne Binegar: But if refined represent these special factors now they're they're special in the sense of the rectangle grid that used to detect that stuff they sort of indicate if they provide a good set of coordinates of the point I know.
Birne Binegar: Like you're a great point.
Birne Binegar: That be like go out this far along direction, and then I go this far along the direction to do not have then.
Birne Binegar: He coordinates for this point in terms of the coefficient of an expansion of the vector in terms of the basic.
Birne Binegar: Right, the last picture in general, our problem, we have a an arbitrary subspace to be specified is taking all possible when you're combination that factor, but if that were the case against situation more like this, where.
Birne Binegar: You represent any particular vector well you, you really only two directions to get every point in time, if you have multiple directions which direction you choose the more defined a whole okay.
Birne Binegar: So that's the idea, the bases will give us good word for the point system side of stuff.
Birne Binegar: Right and so.
Birne Binegar: We talked about how we find.
Birne Binegar: The first example the subspace remembers taking his list of factors and then maybe every possible linear combination.
Birne Binegar: But that gives us some space, but it doesn't tell us how to coordinate.
Birne Binegar: Your device it, though technique we developed on Monday was you take the bankers and write them as the roads and the matrix.
Birne Binegar: And then, what we do is we produce that matrix to grow up on the purpose of that is well it's kind of like what we did production or world you're solving equations we will grow reduce.
Birne Binegar: Your matrix corresponding who's listening equation so don't you know eliminate all the redundancy.
Birne Binegar: and effectively what we're doing here we have a bunch of directions, all in the light of day one, a limited interaction, but we serve those factors into a matrix growing into a road or.
Birne Binegar: copy producer zero wrote at the bottom of made up doing.
Birne Binegar: Both of their roles are going to represent some redundancy among the original direction after we've done the road will be left with.
Birne Binegar: A new set of roses generate the same rose things but are now linearly edit and so.
Birne Binegar: don't represent independent direction and we'll use those corrections as the basis for the corresponding substance.
Birne Binegar: So.
Birne Binegar: you're given.
Birne Binegar: A sub stage, presented as a little off linear combination of Mr that you're on the basis for that update right these guys that wrote a trick reviews that you read one and then just grab a non zero, and they will be, they will be your face.
Birne Binegar: Okay now there's a certain type of stuff face.
Birne Binegar: Which when we produce a sub state by the nanny has a factor in it satisfies the equation.
Birne Binegar: turned out that we already know how to find the basis just we don't have don't lecture.
Birne Binegar: But what i'll do is i'll give you a linear system all solid and then i'll just point out where the basic.
Birne Binegar: let's look at this guy here's a system of equations for them, obviously going to be.
Birne Binegar: redundant directions, because we have basically for variable to equation, we can eliminate the leaf probably do the variables, but let's apply our method resolving the years this scotty graphical they should put them on the left hand side of Oregon naked one minus 111 maybe first row.
Birne Binegar: Here that equation seven bro.
Birne Binegar: it's all the system fine fine young men were using all the way usual.
Birne Binegar: And then we use the augmented matrix in the usual Echelon on the right go back to the equation, but we do sell in a special way we interpret.
Birne Binegar: The variable a corresponding home with it as the variable and the third variable the corresponding home to outdated you think of those as providing three variables.
Birne Binegar: But with this example X one X two corresponding to the first two columns they both been painted but i'm going to regard those it's free because fixed variable meanwhile X three x four.
Birne Binegar: minutes this regard X three x force regrettable.
Birne Binegar: Why then use the equations for finding the augmented intelligence, you want to give me the equations that expressed experiments in terms of the premier.
Birne Binegar: premier X one plus x for half that you zero which tells me that next one variable can be replaced by minus X four X four is one of the free variable and suddenly the second equation rose giving me a second equation that expresses its variable.
Birne Binegar: Good yeah once we.
Birne Binegar: can replace the fixed variables in terms of their freshman.
Birne Binegar: In terms of variable.
Birne Binegar: solution back you're in the last step if you want to realize these solution backers it's living on pipeline, so we expand in terms of the free variable.
Birne Binegar: and your support these two concepts factors that are never mentioned so explicitly before are now open.
Birne Binegar: These two constant factors are actually going to be a basis for the solution set So what do you think it here and the calculation that you could do already perfect them on now tell you that confidence factors that come out of your answer are actually the basis that one.
Birne Binegar: Right so yeah they are really basis that.
Birne Binegar: First of all, what do you need for a set of that used to be a basis.
Birne Binegar: So you have a certain factors, the one okay.
Birne Binegar: This is going to be a basis.
Birne Binegar: Word stuff basic.
Birne Binegar: yeah.
Birne Binegar: Every factor.
Birne Binegar: can be written can be.
Birne Binegar: uniquely.
Birne Binegar: Yes, he is equal to see one.
Birne Binegar: Well that's yeah.
Birne Binegar: Just being able to write every doctor is a linear combination to get a list of that does not make it a basis.
Birne Binegar: This uniquely is the key key point, you need because when it uniquely that's what we need in order to interpret these coefficients see one tk as the unique coordinates of the vector be with inside that subspace.
Birne Binegar: But this unique like cause it's just so that we can use these guides and scored.
Birne Binegar: Ah yeah indirectly.
Birne Binegar: done a more general criteria would be set them.
Birne Binegar: up one UK is a basis.
Birne Binegar: yeah.
Birne Binegar: that's a basis.
Birne Binegar: For so if, first of all, yes, has to be capable, every point, and so you have to be able to reach just using these factors.
Birne Binegar: But so not just making this true but it's not not ensuring that their unique so The second thing you need is that the vectors V1 bk.
Birne Binegar: are literally independent.
Birne Binegar: So.
Birne Binegar: So we actually.
Birne Binegar: This part of the utility of basis factors is guaranteed by this, except for the uniqueness and the need is guaranteed by this condition of linear independence.
Birne Binegar: A step back with linear independence.
Birne Binegar: That be the only solution.
Birne Binegar: See One the one.
Birne Binegar: is equal to zero is.
Birne Binegar: That all the code.
Birne Binegar: Zero don't In fact the way you guarantee that a given set of factors.
Birne Binegar: you're going to be able to represent every factors like yourself state, be a linear combination of the sower when you guarantee that you have to get uniqueness.
Birne Binegar: And she just have to check and make sure the zero vector issue if the zero vector he then then you'll know that this is a workable set okay top.
Birne Binegar: Oh, I know that he's he said attackers are clearly maybe every solution here's a solution you're always we do it.
Birne Binegar: Well that's I thought this month every solution is lily lily inside the span of those these two constant.
Birne Binegar: How do I know that they're literally ended well.
Birne Binegar: Was we try to make the solution factor equal to the directors were looking for linear combination or two solutions vector that athlete zero.
Birne Binegar: will notice that the only way to do that Well he does extra time in the third component you can't be cancel anything here Oh, if the some of these two factors producer director i'm going to be extremely thorough otherwise at.
Birne Binegar: home and certainly.
Birne Binegar: In this world component there's nothing in the first factor I used to cancel that contribution of the general done it's all i'm also guilty X 40.
Birne Binegar: But that's always going to happen.
Birne Binegar: When we're solving linear systems see this third component which you can cancel that the other vector that's because.
Birne Binegar: it's free parameter to that one, this is a different plan, but what's going to happen is when you write down the solution that form you're never going to be able to produce a linear combination that will produce zero unless you pay all the coefficients.
Birne Binegar: So back or method of finding a solution to a linear system automatically produces solution as a linear expansion of basis.
Birne Binegar: So there was nothing really this calculation that we've done the world important perfect now, you know how to do this calculation, now we know a little bit more how the poll results, not only giving us the solution set advocating as a basis for.
Birne Binegar: Okay well the Schumann explicitly i'm here i'm here at that hole here are my two concepts that are the whole point of factors, the solution, and those are exactly can be a basis factors.
Birne Binegar: to generate the entire solution.
Birne Binegar: Okay.
Birne Binegar: um.
Birne Binegar: I had.
Birne Binegar: Two different types of spaces, I make yourself face either by taking so all possible linear combinations of them.
Birne Binegar: And I know how to find the patience for that right, the back roads in the matrix reviews grabbed it on through.
Birne Binegar: I also have a way of find a basis for subspace coming from this linear equation.
Birne Binegar: Now let me describe how like basis, but something attached to it, and by invasion.
Birne Binegar: well.
Birne Binegar: Each of their through some space that I attached to a given matrix but it's going to turn out the two of them are just the span of that factor, and the third one is going to be a solution to a linear system.
Birne Binegar: First row stage you take the road back yourself the matrix and then.
Birne Binegar: consider their spam.
Birne Binegar: And that will produce the sub base, or you can take the call of the metric at all possible, and you have combinations of column that you get another stuff.
Birne Binegar: And then, finally, you can use your matrix you find your audience.
Birne Binegar: And then you get a self states, by demanding.
Birne Binegar: Restricting your attention to the vector X and after stratify this linear system.
Birne Binegar: And we don't have to find a seat.
Birne Binegar: We have a span of factors well yeah yes right that the record is amazing you guys have already.
Birne Binegar: So we just wrote us a few in a row, so one of the graph and on your road there's are based.
Birne Binegar: On the basis for those but as well.
Birne Binegar: We know how to do that, too, we simply fold is linear system and grab the constant factors that are being multiplied by three parameters when we present the solution is a hybrid.
Birne Binegar: And then probably called face.
Birne Binegar: Well, we can do this i'm just using techniques at our disposal if we have a is equal to.
Birne Binegar: Any.
Birne Binegar: basis for the common space.
Birne Binegar: If you had a is equal to 1101 on.
Birne Binegar: One and suppose the thing rhodium says two.
Birne Binegar: matrix or maybe not do that yep.
Birne Binegar: So you get a basis for the road face of this.
Birne Binegar: What i'm doing it for maybe a moment.
Birne Binegar: me.
Birne Binegar: It may get one.
Birne Binegar: Okay, so this this matrix get a road.
Birne Binegar: That this metric and then a basis for the road space.
Birne Binegar: yeah that's going to be.
Birne Binegar: You just grabbing onto your route 112 and 000 that's not based for the roast day basis for the column space Well now, what I want.
Birne Binegar: For the column space.
Birne Binegar: it's going to equal the span.
Birne Binegar: The column backers 101.
Birne Binegar: Once you're a one i'm afraid and and to one drupal well.
Birne Binegar: Now that even though on the road to the matrix that I know I know have any way find a basis for what I do is, I take the factors, they have been as the road on the road, for us, and then grab the non zero so what I would do for this case is we write.
Birne Binegar: A column columns.
Birne Binegar: As the rows of the major.
Birne Binegar: So I.
Birne Binegar: guess i'd get 101101213.
Birne Binegar: And now, what I could do is reduce this matrix in a row H1 or grab the non to arose i'll have a basis for the row space at this matrix which is identical to the column face of the original me so.
Birne Binegar: What i'm suggesting here, and this is not the way we'll do it in general, but one way, we could find a basis for the columns stages of a matrix is effectively take the trend photos of the matrix by converting rose in the column then reducing.
Birne Binegar: transpose matrix until you reach a trough on for grab the non through a road, and that will be a basis for the most days of the transpose, which is the same thing as the basis for the columns base of the original matrix, but if we do that, then we have to do two step.
Birne Binegar: process.
Birne Binegar: To get bases for all heat by reducing a to reduce reliance on.
Birne Binegar: A special case which one one or databases, or the road space, but also get a basis for the call that time.
Birne Binegar: But entwine the basis for the columns face I could use, production, again, but I have to reduce a different matrix.
Birne Binegar: But what i'm going to show you on the next slide is a theorem that allows us to use the same reduction process that we use to.
Birne Binegar: get a row state and normally, we can also use that a reduction, you get a basis for the columns face it's just that the algorithm or what we grabbed from that calculation is different.
Birne Binegar: Oh gracious.
Birne Binegar: statement.
Birne Binegar: Oh here's an alternative way of finding a basis for Tom.
Birne Binegar: What you do is you reduce your matrix to a row so on for now i'm not reducing the transpose a.
Birne Binegar: greeting a itself.
Birne Binegar: But then what I do is identify the columns of the row excellent warm they have it, I don't use those column I just need to know their position in the row H1 or but I did you like go back to the original name hey and I grabbed all of the row H1 or a to F.
Birne Binegar: Those those.
Birne Binegar: Those columns are actually going to be a face.
Birne Binegar: But let me show you visually what's going on.
Birne Binegar: Oh, I.
Birne Binegar: basis work.
Birne Binegar: Now, a call.
Birne Binegar: A.
Birne Binegar: Few choices.
Birne Binegar: First, one is to.
Birne Binegar: mouth.
Birne Binegar: Which rows and columns.
Birne Binegar: produce.
Birne Binegar: The matrix.
Birne Binegar: Reduce.
Birne Binegar: transpose matrix.
Birne Binegar: To.
Birne Binegar: grab the Rose.
Birne Binegar: And then take those non to arose and convert them.
Birne Binegar: convert them to columns.
Birne Binegar: And then the column that you get by converting the growth factors in the columns are going to be your basis, but this is the way i'm.
Birne Binegar: going to avoid because this is a separate reduction process.
Birne Binegar: we reduce the transpose of a rather than eight so so the new way.
Birne Binegar: i'm suggesting.
Birne Binegar: Yes to.
Birne Binegar: Reduce.
Birne Binegar: yourself.
Birne Binegar: Let me say 121011211 let's suppose that this role reduced.
Birne Binegar: To a matrix 100 or 10000 all right talk, this is not true, but suppose this were true.
Birne Binegar: Then, my direction here is.
Birne Binegar: I figure out where the pivots are in this row Echelon warm up my original matrix I don't use each COPs.
Birne Binegar: Although to get a basis for the road space I would use these two rounds, but here i'm just figuring out where these columns that have the pivots live I go back to the original matrix.
Birne Binegar: And I grab a column of the original matrix that corresponds to a column of the rational form that has a pivot and so when I grabbed those guys i'll get 122211 in my time.
Birne Binegar: These two vectors are going to be a basis for the column.
Birne Binegar: And so that's not because.
Birne Binegar: they're so it's.
Birne Binegar: True it's not nothing.
Birne Binegar: But it's absolutely true that if you got any major to enroll in home.
Birne Binegar: And you reduce it to our global use it to a restaurant or, then you get a basis for the homepage that matrix simply by figuring out where in a row at one on the page.
Birne Binegar: yeah credits in column 123 you go back to the original language and grab a cold 123 and that will be your face.
Oh.
Birne Binegar: yeah so, in fact, on the slide i'll show you their civil production process, we take a matrix a produce one for Kelly so funny with bases for both of those days, all day in addition to know.
Birne Binegar: When you wrote us a to a restaurant or a non zero rose again give you as that most things.
Birne Binegar: When you produce a matrix do a video on board or even rowan.
Birne Binegar: The halls of the row H1 want to have it, are going to correspond all of the original matrix or activity.
Birne Binegar: And so.
Birne Binegar: You still reduce the matrix but you go back to the original matrix that grab your calm.
Birne Binegar: The roadmap, you have to still involved with the risk of injury.
Birne Binegar: And then finally together a basis for solutions that.
Birne Binegar: You system that corresponds to a matrix you saw all the users and then you just grab the common factors be multiplied by the arrogance and so what you see from this is that a single role reaction process you answers.
Birne Binegar: require all is the basis for growth day, what is the basis for the homepage and what is the basis for the whole spectrum.
Birne Binegar: But most to an explicitly then here we have a matrix we wrote on.
Birne Binegar: I will show you the population, but this matrix after fine sequence elementary row operations and you probably do it is a reduced our natural born.
Birne Binegar: or pretty close to actually folding the linear system associated major.
Birne Binegar: First let's notice that this is a row excellent on a and so it's non zero road or can be a basis for the roasting.
Birne Binegar: Now I i've got a basis for the roasting.
Birne Binegar: You get a basis for the column space well the pivots in this role Echelon form or living in columns one into I go back home wanting to have the original matrix and I use those and maybe supply me with a basis for the columns.
Birne Binegar: And then.
Birne Binegar: yeah a basis for the solutions that have a.
Birne Binegar: Zero well, I thought that by Mary baker and write down at solution Center.
Birne Binegar: Here we go so.
Birne Binegar: Here we have a matrix yes, the Roi on a the reduce row Echelon form of the identity matrix okay augmented with the zero vector it just going to equal the augmented matrix you can buy a new flashlight warm okay and joining the zero vector because the row operations that we do.
Birne Binegar: Is augmented matrix to reduce electron or it's going to be the same row operations that act on this on this part so.
Birne Binegar: genius linear system nothing ever happened in the last call sometimes I put their thumb remember that we're solving a linear system, but other time time could be in a row, reduce the matrix any.
Birne Binegar: At any rate, what we do in this case, one REACH, we do select one or form we convert back equation women that are patient.
Birne Binegar: We looked at the call that contain it goes are going to be the variables that we handle who are explicitly in terms of referring the Greek parameters, on the other hand they come from calls that are not contained in this example next three.
Birne Binegar: Get drawn this screen rather than blackboard but I was unwilling to pivot columns at least the second the third and fourth calm and we're finding the free variable.
Birne Binegar: So what you do is you take convert back to the equations putting the breeder mode on the right hand side, so this first row helped me to X one there are plenty of X three two times X four equals zero or that one is equal to minus X three minus two X war, so I.
Birne Binegar: I switched assign it to turn because the idea was to take them over the opposite side of the equation internally the second row rose tell me if X is equal to two X three.
Birne Binegar: X four.
Birne Binegar: Now i've got.
Birne Binegar: six.
Birne Binegar: We are able to.
Birne Binegar: express the terms of preparing.
Birne Binegar: The next step is to write down exclusion factor of replacing where I can be expression in Bowling free variable i'll use those to replace the.
Birne Binegar: The perfection so here, I have my evil temptress 3.242 X three.
Birne Binegar: X or X three three so I can substitute there at 4370 there right so then just just a typical solution vector.
Birne Binegar: And in fact what we do, yes, is then taking a solution vector and expanded in terms of the free parameter so X three others that have.
Birne Binegar: occurred in the first article coefficient minus one.
Birne Binegar: component of coating to the third component and coefficient one it doesn't appear in the moment.
Birne Binegar: And then we have turns involving H4.
Birne Binegar: First proposed it encouraged to come to us to the second doesn't occur third and coefficient one last.
Birne Binegar: look good so now we've got two vectors that generate.
Birne Binegar: Every vector that's in the solution set of the linear system, a contact here, and these are basic.
Birne Binegar: Things being multiplied by the three parameters.
Birne Binegar: notice again that they're literally dependent, because the third component that through the third the variable X three can't be Council with about the other factor this fourth component naturally Todd for can be cancelled.
Birne Binegar: Apart that's coming from.
Birne Binegar: Variable X.
Birne Binegar: All right, alright so.
Birne Binegar: Now.
Birne Binegar: We now have display these two doctors as a basis for the solutions that a xp zero another name, for that is Lol.
Birne Binegar: Okay, so that's how we proceed now.
Birne Binegar: What I want to do next is.
Birne Binegar: introduced a little bit more nomenclature.
Birne Binegar: But halfway they'll be pretty simple ideas.
Birne Binegar: know.
Birne Binegar: What dimension of her.
Birne Binegar: Face might be so I mean that's a problem.
Birne Binegar: Yes.
Birne Binegar: w is equal to the span.
Birne Binegar: Of one okay.
Birne Binegar: First, that need.
Birne Binegar: If you take linear combinations of.
Birne Binegar: Those coefficients range over the entire Milan and then gather everything up into a set so.
Birne Binegar: we're used to thinking of this as a way of making a tea mentioned ourselves right I pre K parameter talking Barry and so like during the.
Birne Binegar: Day parameters I should be generating K eventual state, but in fact in general.
Birne Binegar: The dimension of w.
Birne Binegar: it's not.
Birne Binegar: Because it could be that some of these directions are not independent of another direction and, if I have two vectors that are parallel to another pointing in the same direction i'm not increasing dimensional space by adding in additional pieces that are already in that states.
Birne Binegar: But what.
Birne Binegar: What does happen, though, and the reason why I cannot immediately say that.
Birne Binegar: The dimension of Sub faith created this way and be okay it's because these guys are not.
Birne Binegar: So there are two basic facts about something, first of all, every cell face except.
Birne Binegar: The space interesting only the zero vector as a base.
Birne Binegar: So let me just point out why the zero vector there's.
Birne Binegar: A reason why does it have a basis well.
Birne Binegar: To be a basis you're the vector she used to create this update by picking their spam, they have to be literally independent if I only have one factor.
Birne Binegar: So.
Birne Binegar: Then the span of the it's going to be a set of all.
Birne Binegar: well.
Birne Binegar: it's just going to be.
Birne Binegar: Just this right there at all possible combinations of that single factor, however, that factor exactly equal to zero you go you actually done.
Birne Binegar: This sector.
Birne Binegar: It we're not zero, it would be automatically literally dependent on it was non zero, the only linear combination zero will be will be forced to take this code fishy.
Birne Binegar: But this guy is already done through a factor I can multiply it by a non zero number and still produce zero so zero is not.
Birne Binegar: linearly independent.
Birne Binegar: of itself.
Birne Binegar: Because two times zero factor still equal to zero, so I have a non trivial linear combination this factor that produces the zero vector and since that the basis of the vector to generate such things not literally tended not going to be introducing our storage.
Birne Binegar: A quarter cup logic why this guy doesn't have a basis, but on the other hand, it also customize my terminology, I said, you should think of points is being zero dimensional objects and might be one dimensional hockey playing the.
Birne Binegar: server.
Birne Binegar: well.
Birne Binegar: If every third phase of the roles of faith has a basis and i'm sorry remarkable fact is that every basis for a beautiful place has the same number.
Birne Binegar: And that combining.
Birne Binegar: We reduce.
Birne Binegar: What will use us a definition of what dimension.
Birne Binegar: So every sub base every non zero sub base going to have a vision, and second, of all if you have a sub base may have lots of different choices with a sport so they're always going to have the same number of.
Birne Binegar: situation, I have a two dimensional subspace.
Birne Binegar: I mean two directions and that to reach every direction inside that today's but I don't have to use these directions, I could use this direction in this direction and I still have a property that these two factors generally the mistake so i'll have general philosophy of choices for.
Birne Binegar: bases for a 500 fly but.
Birne Binegar: The number of factors that you need to generate something that's going to be completed and that's what we call dimensions.
Birne Binegar: And it's kind of mash all your earlier intuition mentioned he just needs an isolated number three parameters need reach every point is like that.
Birne Binegar: Okay.
Birne Binegar: For those definition let's return to the subspace that we can attach two matrix and see if we can get a better description as to what their side, whether dimension.
Birne Binegar: So for the row space where we find the basis that versus we reduce.
Birne Binegar: What we produce the matrix to arrive on more and we grabbed the number on the road, and so the number of the dimension of a rose is going to equal, the number of non 001.
Birne Binegar: And that, in turn, is going to equal, the number of roles.
Birne Binegar: Because.
Birne Binegar: Once you reach row Echelon for.
Birne Binegar: Their.
Birne Binegar: care.
Birne Binegar: In three down here once I have a non through row.
Birne Binegar: I have a pen because of the words were a zero again by not having.
Birne Binegar: So what counting the number of visits in raw form of a I guess you mentioned.
Birne Binegar: But on your.
property.
Birne Binegar: dimension of the calls that as well we're counting the number of basis factor for calls, how do we find these guys.
Birne Binegar: We use a toothbrush born and then we grab the call of the original matrix they correspond called the lowest form but notice the same number.
Birne Binegar: That number of visits you found when you kind of find the most in here is exactly the same numbers you do want to count the number of phone or basic factors with columns.
Birne Binegar: And so we have this.
Birne Binegar: Again, that the dimension of the road stage is always equal cannon dimension, about the tone as.
Birne Binegar: Well, so you want that surprising is because when you take a matrix.
Birne Binegar: One a one in a in.
Birne Binegar: A in.
Birne Binegar: A corner and to.
Birne Binegar: show you that.
Birne Binegar: row space.
Birne Binegar: Okay, well, I created the roast beef by taking a linear combination of the road after you've wrote vectors of exactly entries in this is going to be a sustained.
Birne Binegar: or yeah.
Birne Binegar: Meanwhile, the columns how many components as a column, has it has end components so every column vector it's going to have in common, I take linear combinations of them.
Birne Binegar: Since I have any component i'm going to get some some space of our in.
Birne Binegar: General number roads and number of columns are different, so these two sub spaces live in time there are factors.
Birne Binegar: that's why I had an amazing at their Convention is the same.
Birne Binegar: What it all boils down to is that once you reach the.
Birne Binegar: form of.
Birne Binegar: You you can use that information either regret web based editor rose or you can figure out where the pivots go back to us to get paid on.
Birne Binegar: All different dimensional vectors but the overall size of the side is still be, at the same.
Birne Binegar: Time well.
Birne Binegar: Now one last thing I can hold on for one.
Birne Binegar: Well, no, I won't.
Birne Binegar: play, but the back of a verbal real quick.
Birne Binegar: When you wrote this matrix to grow actual form, you have on.
Birne Binegar: On.
Birne Binegar: The call that number pitch you have equal to mention of rosedale or the call because eventually part of the state, we just call it the right.
Birne Binegar: But that leads us to discuss the call without a call about it for free parameters that's going to give you mentioned solutions going back, looking at the parents have brought on board is going to give you the dimension of results.
Birne Binegar: Okay, so i'll stop today and we'll pick up with that discussion on Wednesday or Friday.