Birne Binegar: Okay, I guess, we get underway.
Birne Binegar: On so.
Birne Binegar: I just.
Birne Binegar: A sign.
Birne Binegar: Over two or.
Birne Binegar: Three days a week, from today class.
Birne Binegar: Well, we may have opportunity later this week to go through some problem in the world.
Birne Binegar: But.
Birne Binegar: Maybe i'll start with my first time seeing you guys face to face ask if there any questions for.
Birne Binegar: You so much OK, so now, so what we're gonna do today is, we spent the first two weeks, you don't take the calculation that will be using throughout the course.
Birne Binegar: And now we're now making a transition things we can actually do with these tools first thing we're going to do is learn how this whole equation, but before I start back.
Birne Binegar: up very quickly what went on in lecture last week where.
Birne Binegar: The second week of class what we did is first week of class, we talked about vectors and vector spaces to the second week of class was largely developed to.
Birne Binegar: understanding what matrices are and how to how to make you like them yeah what i'm going to review start and then we'll turn to a really perfect application.
Birne Binegar: Okay, so.
Birne Binegar: Firstly, all remind you about.
Birne Binegar: Oh, what are they gonna.
Birne Binegar: Be is.
Birne Binegar: Some rectangle around.
Birne Binegar: To.
Birne Binegar: That.
Birne Binegar: Hello first row of this matrix second row a to a.
Birne Binegar: Particular term to.
Birne Binegar: devil rose till we get to the last row.
Birne Binegar: And this is what we would call an m matrix because it has enrolled and comb and notice.
Birne Binegar: As is typical this of course.
Birne Binegar: About mathematics, is the first index here on the actually the first row it, and the second entry always records which call that is in progress so that's going to be important, because when we find matrix multiplication to pay attention to roll call to the very back.
Birne Binegar: Alright, so.
Birne Binegar: The first matrix.
Birne Binegar: number is called value.
Birne Binegar: Okay well.
Birne Binegar: well.
Yaqoub Alkandari: Okay.
Birne Binegar: Well, the other than.
Birne Binegar: All people.
Birne Binegar: are going to have this thing called that they are going to have exactly the same.
Birne Binegar: rosedale can have that is he mentioned.
Birne Binegar: Oh.
Yaqoub Alkandari: Well, we can.
Birne Binegar: We can tend to.com.
Birne Binegar: running down the.
Birne Binegar: Road rather be in a particular role there's going to be fixed number call number call and edited with call me now you go down the road, the number of these families is equal to the number.
Birne Binegar: Number of entries that didn't call the number of rows of so you need those dimensions to that job so this dark profit is fine, but the longer that.com find you get an entry for each choice of a road in a in each choice of a row column B, and in fact that's how we compute young.
Birne Binegar: matrix let me do a quick example on the board.
Birne Binegar: The formula, not very hard at all.
Birne Binegar: What you should be good at, maybe, by the end of this week, or, if you do the homework to do the calculation.
Birne Binegar: pros we have a 1101 that you and I, I have recall your goal need something with at least three rows here for that product fine let's take 110 Okay, now the quick way to do this.
Birne Binegar: And it is a fine, because this is a matrix with.
Birne Binegar: Two rows of three calls multiplying a matrix with three rows him called, this is a necessary back up.
Birne Binegar: Because of enemy mentioned matt job the product is you find and what's going to get kicked out well that's going to be determined by the outer game, this is kind of go through a tube by two matrix this number.
Birne Binegar: There, and that number goes to jail so just looking at the dimensions of the vcs I can be cat whether the product is be fine and in fact I can even predict the size of the matrix is going to be produced.
Birne Binegar: let's calculate it.
Birne Binegar: I need to take that up to get this entry, the first slot first row and first column of pride when i'm going to do is i'm going to take the first row first factor and take it.com with the first column of the second factor, so I have two.
Birne Binegar: plus two.
Birne Binegar: Plus first one.
Birne Binegar: I hope you're seeing what i'm doing i'm putting my fingers on the two numbers i'm looking at multiplying together to find one, and then I add to that one to two and then I add to that the last component last component what.
Birne Binegar: All right, let's see, then, I have another column, I could dump is first.
Birne Binegar: First row with like you started with the second poll and that's going to give me the entry in this corner, so I have two times 001 time to to at one time to.
Birne Binegar: Now I haven't second row, for the first matrix that I can use to compete to your doctor first i'll take it off of the second level first call that's going to give me a second row, for me, the second row the product and also be the first call.
Birne Binegar: I 02.
Birne Binegar: plus two.
Birne Binegar: And then, I have one last call to work with.
Birne Binegar: 012 to.
Birne Binegar: tell it looks like this matrix Tony valuation just work out 544.
Birne Binegar: Okay, so there's a complication.
Birne Binegar: A finished product of a two by three matrix read it.
Birne Binegar: Okay, so need understand that.
Birne Binegar: The rule for multiplication but is that.
Birne Binegar: we're not going to spend too much time will make use of this course nevertheless it's really important to understand how to.
Birne Binegar: carry out the population, it is T because you end up taking a whole bunch of different products that completely fill a matrix so, for example, if I had.
Birne Binegar: If I wanted to hold it together two three by three matrix season there's nine different products i'm going to have to compute just to fill in the entries of the result.
Birne Binegar: But also always designed problems were both, even though these can be any real numbers are usually designed problem so that you work with entertainers and so let's read the ticket not going to be hard at all you just have to understand the rule you just have.
Birne Binegar: To okay so that's.
Birne Binegar: matrix multiplication Another thing we brought this particular matrix.
Birne Binegar: So.
Birne Binegar: that's the square matrix going to have a statement for growth is it at all and it's for us it's all you have one from all this diagonal for the upper left little right but everywhere else happy.
Birne Binegar: You.
Birne Binegar: Know it's called the identity matrix has a nice profit, if you multiply anything.
Birne Binegar: By.
Birne Binegar: A lot of a product sells five what is that well this product be fine, the number of calls with the first factor has to equal the number.
Birne Binegar: Of number of growth.
Birne Binegar: it's a lot of Apps true.
Birne Binegar: I have as many people as the Austin overtaken in the US, in order to get.
Birne Binegar: A flat the number one.
Birne Binegar: And then.
Birne Binegar: let's see there's two other operators.
Birne Binegar: That are now more scalable vacation in depth tradition and.
Birne Binegar: Applying.
Birne Binegar: The rules.
Birne Binegar: So, for example, I have to make.
Birne Binegar: As long as they have the same song.
Birne Binegar: Just like the adventures together as long as they have the same dimension.
Birne Binegar: And the rule is you simply have all the components in it, because they.
Birne Binegar: gave.
Birne Binegar: You just have the.
Birne Binegar: same.
Birne Binegar: Time.
Birne Binegar: And we have feels old page where you have a matrix and you both both.
Birne Binegar: And all that means is that every entry of the patient is modified by take back Jesus land up they both funny every entry of the original by that number yet Giro.
Birne Binegar: A lap.
Birne Binegar: pretty easy.
Birne Binegar: This last one I don't think I actually covered very well you know what a true but.
Birne Binegar: there's another operation, do you.
Birne Binegar: See, this will be useful down the road, and that is well we formed a to Steve either ready gamma list rose or right across on this call, but.
Birne Binegar: So.
Birne Binegar: We can always convert.
Birne Binegar: The road from the bakery is called as an attorney and if that operation is called as.
Birne Binegar: They give you a quick example.
Birne Binegar: To 111.
Birne Binegar: well.
Birne Binegar: individually.
Birne Binegar: But more simply, what you do is you look at the road and you convert the call.
Yaqoub Alkandari: um so, can you please turn it to the list I can't see you and.
Birne Binegar: Dr say.
Birne Binegar: No, but.
Birne Binegar: Now i'm not going to be doing it right.
Birne Binegar: Now.
Birne Binegar: So when I didn't throw row I wrote it the call, I think the second row and I like that at all, and then this matrix which is now a three by two matrix is the transport original two by three.
Birne Binegar: Okay, so transpose.
Birne Binegar: Though.
Birne Binegar: actually get back with you start off with what you do is you take your take a tricky one, but if I do that operation again.
Birne Binegar: What happened well This call is going to get converted into the same as first call me comes the first row.
Birne Binegar: Until we get to the why and the South pole because the second.
Birne Binegar: That I applied to transpose offering quite get back to where I started.
Birne Binegar: LSE we have.
Birne Binegar: station or really happy.
Birne Binegar: To see you for the kind of same with all the.
Birne Binegar: first day.
Birne Binegar: And then, lastly, this is.
Birne Binegar: About.
Birne Binegar: Transport, you have a product that you say just.
Birne Binegar: let that happen that you get the product of transport, but the trip in reverse for that's important.
Birne Binegar: Okay.
Birne Binegar: applications of the transport operationally.
Birne Binegar: we're not going to see those for a bit later.
Birne Binegar: Now people with this with all the poverty right sure if that will start fresh with a new.
Birne Binegar: kid.
Birne Binegar: Oh so.
Birne Binegar: We call it.
Birne Binegar: The rational way I do things.
Birne Binegar: equation.
Birne Binegar: extract the data that used to establish.
Birne Binegar: The binders is sort of a laundry.
Birne Binegar: In fact, and thought I take all these quotation collected together in one room.
Birne Binegar: And that i'm pretty much able to get this right, I simply racing.
Birne Binegar: Everything on this side of the equation racing hold the variable.
Birne Binegar: Like up correctly, the matrix of patients, for five.
Birne Binegar: variables excellent Madame.
Birne Binegar: me.
Birne Binegar: Another matrix this is how many three blue arrow in a single call.
Birne Binegar: Every year, I take it all the.
Birne Binegar: Time and have a great.
Birne Binegar: Day baker know, essentially, it is.
Birne Binegar: A fact.
Birne Binegar: The left hand side here is to be interpreted as a product of this whole may increase by this.
Birne Binegar: By this rectangular.
Birne Binegar: Like a show you real quickly.
Birne Binegar: The translation.
Birne Binegar: We had.
Birne Binegar: one.
Birne Binegar: To one is.
Birne Binegar: equal to.
Birne Binegar: Do equations.
To.
Birne Binegar: call a two by two your system.
Birne Binegar: Reflecting the language of matrix you've been in be progress linear system, I think, right now, associate.
Birne Binegar: The coefficient matrix I get that like grabbing the multiplies and each of the variables each equation.
Birne Binegar: First equation two times first variable one.
Birne Binegar: coefficient a tree and then the second equation with old client X, Y minus one was like heck to begin.
Birne Binegar: So there's a coefficient matrix I tried to variables three x two X one X two and let's put the numbers that appear on the right hand side on to.
Birne Binegar: Our thought.
Birne Binegar: Is a matrix equal.
Birne Binegar: Right.
Birne Binegar: To these patients should equal this picture, what does that have to do with the original system of equations well when you compute that product two times X one.
Birne Binegar: X, to the product of this first row this first call it can be to add on to.
Birne Binegar: The second row with that call this one minus X one.
Birne Binegar: and two and we're not doing anything for the right hand side.
Birne Binegar: But now notice the only compare both sides of the equation component by component these bases are equal each component has equal and we just recover the original linear system.
Birne Binegar: Talk what you should understand this, is that what i'm doing is i'm taking large linear system and are able to encapsulate it in a simple just.
Birne Binegar: Pre pre lunch right Kwame doesn't always mean, although.
Birne Binegar: Each of these old days of.
Birne Binegar: Letters actually represents a major.
Birne Binegar: law or you know those are.
Birne Binegar: The matrix equation, they replicate the original linear system and what we're going to do and.
Birne Binegar: Also, your system that you would have placed an ad solving one variable.
Birne Binegar: equation very.
Birne Binegar: proud of the whole process of self up solving for a variable itself to to you.
Birne Binegar: Know general.
Birne Binegar: Thank you.
Birne Binegar: pretty much ignore the variables throughout the calculation all said done.
Birne Binegar: Okay, so.
Birne Binegar: that's where we're headed this formulation of the solution of Mars this equation, the matrix formulation.
Birne Binegar: But.
Birne Binegar: Before we actually implement that what I want to be the star is simply.
Birne Binegar: build some intuition, as what exactly beat the soul, a system of linear equations and here i'm going to focus on a few basic interpretation, so you can rather than dealing with order brain, you can actually visualize what's going on.
Birne Binegar: OK so.
Birne Binegar: again.
Birne Binegar: An equation is linear.
Birne Binegar: The variables that appear in it all separated into separate term at.
Birne Binegar: Any individual term of the equation is either number times a variable or number on a single variable or just a pure number include all of your right hand side the variable term on on blessings.
Birne Binegar: That we have multiple languages because compiler.
Birne Binegar: till we generally assistant and your system system, and then you require you to follow him.
Birne Binegar: So.
Birne Binegar: What I want to talk about is, what are the solutions that looks like.
Birne Binegar: and
Birne Binegar: Space target date we don't have any equation on after.
Birne Binegar: Well, for the edification both are completely free so.
Birne Binegar: solutions that have no equation in Mr Townsend just going to be the variables, of which there are free to rank over the entire real life.
Birne Binegar: And so what you get as a solution that is simply every word list of numbers, these numbers are arbitrary and surprise them as a column if you're effectively we've got an ordered list of.
Birne Binegar: Words listed, but we.
Birne Binegar: just get that.
Birne Binegar: Effectively vector space very nice addition.
Birne Binegar: let's go to the first page post we impose a civilization.
Birne Binegar: right there.
Birne Binegar: And i'm not really don't have to tell the quotation this, because this is only one equation.
Birne Binegar: But we do.
Birne Binegar: say.
Birne Binegar: This is a non trivial equation something other than zero is equal 01 of these coefficients has to be knowledgeable so we had a non zero turn on the right hand side.
Birne Binegar: let's just assume that that non zero term is well at least the last term is not zero okay and 93% is assumption is that as long as.
Birne Binegar: I can solve this equation for the variable.
Birne Binegar: take all these terms, all the terms, but the last one, and you move into the office.
Birne Binegar: And then the isolate the variable itself you divide.
Birne Binegar: divide that that song efficiency and so solving this equation for this last variable at amp lt the class variable is be fine I am fine just a one divided by a phonics.
Birne Binegar: So that's how you sold.
Birne Binegar: One equation, you know it on Amazon America.
Birne Binegar: will show you is how to interpret that solution you met.
Birne Binegar: So.
Birne Binegar: First of all right that solution is.
Birne Binegar: Now, the solution you're looking at this equation really implies is the last element can be expressed in terms of the preceding export the receiving X one N minus.
Birne Binegar: One i'll do the right solution that's.
Birne Binegar: The variable except for the laughter evil or unrestricted by the equation you use the equation, just to determine the last variable X in Seoul substitute into the general vector.
Birne Binegar: This particular for my only have to change the lap and I take the last entries, but when exactly the solution of the way.
Birne Binegar: And then.
Birne Binegar: You find out.
Birne Binegar: So.
Birne Binegar: Look at this actual.
Birne Binegar: terms are proportional to variable filters are independent variable and in fact filter proportional X one.
Birne Binegar: X two.
Birne Binegar: X what what i'm doing this between these two on this i'm taking a song splitting apart.
Birne Binegar: Because i'm really good all in one day because their differences there until I read this last 10 years or.
Birne Binegar: So let me just grab it and put it in a single after zero everywhere, because there's no question about.
Birne Binegar: that I have.
Birne Binegar: A lot.
Birne Binegar: Of these.
Birne Binegar: Little line here there's No one to get nothing but there are some contribution of X one note entry, but I do have an entry up here and I have an entry down here that both X one well I write it down that I received a little bit well as to occur and over here, no.
Birne Binegar: that's turned down slot you keep doing that for you reach the last variable.
Birne Binegar: Is appeared in the elements are the laughter and also if you're laughing I get.
Birne Binegar: Those you can.
Birne Binegar: tell what i've done is i've taken my solution Center rip it apart, and now the next step.
Birne Binegar: If you realize it each of the first half first one, this one every component of the factors or.
Birne Binegar: Even zero is very contextual and several several factors that.
Birne Binegar: will only return with do.
Birne Binegar: multiply by X, to the last factor all through my life.
Birne Binegar: So I can pull out of that here's.
Birne Binegar: What you see it's effectively I have.
Birne Binegar: hi we're playing is a set of factors that before like 50 starting point and that allow herself to factor that.
Birne Binegar: In a fixed set of direction by arbitrary good.
Birne Binegar: luck here's here's a constant.
Birne Binegar: And is it just sitting here, and I would.
Birne Binegar: prefer after taking the best and all the time.
Birne Binegar: We prouder N minus one on it.
Birne Binegar: So what are displayed here, Chris presentation of the solution of that linear equation by.
Birne Binegar: holy for variable rather choosing to display my solution, as we want.
Birne Binegar: And that this is the way we'll proceed in this course, when we solve linear equation you won't be so much for the value of individual variable rather will try to capture all the solution as living on people.
Birne Binegar: are no that.
Birne Binegar: I will deal with two sets of linear.
Birne Binegar: What I want to do now is actually do you give you some examples of what this looks like in three dimensional space where we should all be able to traditional it fairly easily what's going on.
Birne Binegar: So the docket bird the.
Birne Binegar: Bird fly.
Birne Binegar: Let me show you what happens if you have a simple equation three there.
Birne Binegar: see this into.
Birne Binegar: Software programming graph solutions of the equation Well, this is what you get now don't be.
Birne Binegar: A music label that you produce this thing people does it chooses the box and the treatment phase, and then it shows you the point inside that box it satisfies the equation so here, you can see the box, I hope you can see from this just the.
Birne Binegar: infinite plane this pain well be part of the infinite plane before this illusion that this is where exactly did we fail.
Birne Binegar: well.
Birne Binegar: Maybe not so clear in the.
Birne Binegar: screen here what I needed.
Birne Binegar: Protocol black is better yeah.
Birne Binegar: My mom.
Birne Binegar: And pop only captured with farm it out.
Birne Binegar: But hopefully it's depicted in such a way that you can visualize this last few minutes of service is.
Birne Binegar: That what is part of your exactly because.
Birne Binegar: Of this equation.
Birne Binegar: Okay, so that's single variable what have.
Birne Binegar: We had a single equation imposed on three variables are solutions that, as you mentioned, to that's going to be typical.
Birne Binegar: Every time you open.
Birne Binegar: or restrictions on your variable lies the solution set it small.
Birne Binegar: Those jumps a dimension when you started posting.
Birne Binegar: More equations that's what i'm thinking to talk about today, but.
Birne Binegar: There will be.
Birne Binegar: A buck, generally speaking, if you have.
Birne Binegar: Three equations and for.
Birne Binegar: Your solution set is going to be the number of variables.
Birne Binegar: minus the number of equations you've been posed, and that will be the dimension of your solution, usually.
Birne Binegar: What part of the point here is to show you when that's not.
Birne Binegar: But let's let's try again what you should expect if you saw.
Birne Binegar: say a system of equations that Mr down issue should expect the size of your solutions that be equal to number of variables you started off with minus the number of quake two.
Birne Binegar: So here's an example, but now we have to eat.
Birne Binegar: So.
Birne Binegar: let's do Kathy mentioned three other two equations three minus two is one I should expect a one dimensional solution.
Birne Binegar: And that, for this picture show rafi is how that one event.
Birne Binegar: That arises geometric.
Birne Binegar: Each equation corresponds to.
Birne Binegar: The satisfies both equations you have to live on both.
Birne Binegar: And so what we're actually experience is not the individual to get the solution set up very quickly you want the Arabic.
Birne Binegar: What happened.
Birne Binegar: The intersection of two.
Birne Binegar: decimal places in general.
Birne Binegar: So that's another.
Birne Binegar: point is also the geometric.
Birne Binegar: Right.
Birne Binegar: Now.
Birne Binegar: Free variable.
Birne Binegar: More.
Birne Binegar: direction.
Birne Binegar: Be.
Birne Binegar: hydroplane illusion, this was brought out.
Birne Binegar: By this granted.
Birne Binegar: I know.
Birne Binegar: Generally speaking.
Birne Binegar: The situation you picked it on a five year old German every time you will post.
Birne Binegar: If I only had a single event date imposing additional information charge down a bit further definitely drop one dimension solutions that for each equation.
Birne Binegar: thirsty.
Birne Binegar: But that's always.
Birne Binegar: An important part is like your show you the caviar.
Birne Binegar: Why.
Birne Binegar: is both the solution set of X plus y plus easy to yeah the solution that.
Birne Binegar: You live with two z's equal to every question yeah but you actually it is quite popular program graph solution, you only get one play.
Birne Binegar: I have free variable equation.
Birne Binegar: At a one dimensional solution.
Birne Binegar: Remind one well what's happening here is that these these two equations have the same solution, so if this is true and I posted i'm not preview restriction on my period already.
Birne Binegar: it's already listed this does this sentence, please just two times the first people to.
Birne Binegar: fly equation by another.
Birne Binegar: Number you don't change.
Birne Binegar: Okay, well, it is not true, if you have waited for you always have zero plus or minus work we do know what could happen is that something.
Birne Binegar: could be done doesn't have the equation, because the implied by up because it says proceed, in which case the imposition of the newbie question without further ado restriction on the variables, and so it will not be the solution.
Birne Binegar: Okay, what am I worried now in this particular example is easy to see that this equation as essentially says this after all, I just have to take this equate to cover the equation what.
Birne Binegar: This course is that what I have.
Birne Binegar: Can equation 18 unknown there may very well be redundancy boxers equation, but I saw.
Birne Binegar: yeah you may want to know what the size of the solution, set, and so one of the things that we try to do in this course is telling you how to take a set of equations and through our redundancies, we really work with any any questions.
Birne Binegar: let's do.
Birne Binegar: one more example.
Birne Binegar: Of that there's nothing.
Birne Binegar: Here I have.
Birne Binegar: To equation, for years now.
Birne Binegar: And in this case.
Birne Binegar: The solution time to not care sector all.
Birne Binegar: The data that's the case that we go to our solution to go crazy.
Birne Binegar: Every solution to this.
Birne Binegar: list this plane every solution of this life on this planet, these two planes don't comment.
Birne Binegar: Well, that happened.
Birne Binegar: But you cannot have.
Birne Binegar: He to simultaneously.
Birne Binegar: One because this equation, combined with this equation.
Birne Binegar: Equal one transitive property on call.
Birne Binegar: is equal to vc stayed at see well, you will have is one, so we get it.
Birne Binegar: Okay, so what i'm worried about here is that look at our equation it's not necessarily true that you get a solution where you get the size of the solution space tracking.
Birne Binegar: number, though, you could have done that equations that are going to destroy that or or, even worse, is your set of equations or maybe in contradiction that will prevent you from finding a solution.
Birne Binegar: we're going to be concerned with data.
Birne Binegar: For a.
Birne Binegar: System of equation, are the, these are the contradiction and as long as we can satisfy ourselves.
Birne Binegar: Either contradictions.
Birne Binegar: That basic rules in the fall you get the size of the solution, set you just subtract the number requirement.
Birne Binegar: In addition.
Birne Binegar: The solutions that.
Birne Binegar: well.
Birne Binegar: they're all going to be.
Birne Binegar: Jordan legal point your your solution set for starting points going arbitrary arbitrary distances all.
Birne Binegar: Right so.
Birne Binegar: here's what we got.
Birne Binegar: Three boys is.
Birne Binegar: What usually happens.
Birne Binegar: Quite contradictory not regarded as the addition of.
Birne Binegar: The social said after going to be the intersection to play which will be along.
Birne Binegar: But it's also possible that the second equation is already.
Birne Binegar: Up get rid of your this case, your intersection is still playing, but the intersection of two solution says see you happy.
Birne Binegar: Or you can be this situation for victory equation and there's one plane that intersect.
Birne Binegar: yeah so that's that's what happens when you.
Birne Binegar: Are you one more attainable now.
Birne Binegar: three.
Birne Binegar: here's the general typically what's going to happen in July.
Birne Binegar: Read equation each is going to be two dimensional flat surface embedded you're reading books book.
Birne Binegar: That is, that is all we.
Birne Binegar: live at all.
Birne Binegar: And it usually happens hard to tell from that graphic.
Birne Binegar: And maybe.
Birne Binegar: On the Blackberry well they don't really have any flat surface but generally speaking.
Birne Binegar: The blackboard he wants to eat vegetables.
Birne Binegar: And I intersected with another latitude to intersect along, if I have a third one well.
Birne Binegar: it's going to intersect single phone and so that rule every penny of equations trapped out there dimension one dimension is a time that's still true, but now.
Birne Binegar: three independent increase.
Birne Binegar: The size of the solution that would space of dimension three minus three and zero.
Birne Binegar: Which is not empty when we talk about zero dimensional spaces what we're really talking about single points because we don't we just have no degrees of freedom to depart from that point but it's still a single better.
Birne Binegar: What happened in the generic place you are give you an arbitrary said equation in three years out confused that the solution is probably.
Birne Binegar: Your three equations and.
Birne Binegar: Actually.
Birne Binegar: performing store.
Birne Binegar: i'll try to add this.
Birne Binegar: One platform to another place another button, they can intersect along.
Birne Binegar: know.
Birne Binegar: All of that power curve well.
Birne Binegar: It was actually a pig redundancy amongst people equate it's not.
Birne Binegar: But it is.
Birne Binegar: If you, yes, you simply add the first equation F1 people aren't as good as.
Birne Binegar: Your career.
Birne Binegar: Wonderful, why are you on the line the zero to 10 zero N minus one.
Birne Binegar: or two people find it too, so that these two equations implies a certain way.
Birne Binegar: But that we got to see is sort of him and that's what i'm trying to get a handle.
Birne Binegar: On, then you can have three pages here the contradiction, though, I guess, it is because.
Birne Binegar: In general, you can have three equations in which.
Birne Binegar: Is a better example, I have a question correspondent for the other equations.
Birne Binegar: that you have a triangle.
Birne Binegar: yeah.
Birne Binegar: Three play the intercept all one played it from there.
Birne Binegar: playing with their own now think of the blackboard out that laid out this way that three step or plane, but there's no point in time to all three you get points in a comment to do with the surfaces all three.
Birne Binegar: Okay, so.
Birne Binegar: we're gonna pay attention to.
Birne Binegar: Whether or not they're redundant equation, or whether or not they're.
Birne Binegar: All.
Birne Binegar: Right.
Birne Binegar: You have.
Birne Binegar: A linear equation.
Birne Binegar: that the solution is general a high profile if you create by going to be some.
Birne Binegar: Vertical dimension to be number various.
Birne Binegar: ways.
Birne Binegar: That is all the equation or any candidate and not on.
Birne Binegar: However, if there's some way.
Birne Binegar: that's going to be that your solution that always chop yeah like what dimension by the imposition of position.
Birne Binegar: So your solution.
Birne Binegar: or.
Birne Binegar: If your equation is a vision anything.
Birne Binegar: at all.
Birne Binegar: Okay well that's a good place to start at least hopefully establish.
Birne Binegar: intuition, for what the solutions are gonna look like what will you do on Wednesday actually begin, so the place.