Birne Binegar: Okay, so I guess, we should get started Good morning, so today.
Birne Binegar: we'll get into the nitty gritty of how to solve them to your Cisco Systems.
Birne Binegar: First i'll show you how.
Birne Binegar: The idea of solving am which is not going to involve matrices directly, but when we actually.
Birne Binegar: Make the calculation more efficiently matrix easy going to play a key role until now let's see.
Birne Binegar: here.
Birne Binegar: Alright, so now i've got what we doing.
Birne Binegar: What are your systems and what their solution sets look like then i'll show you how you can actually solve equations not by.
Birne Binegar: Eliminating variables substituting into subsequent equations but, rather, instead of focusing on variables will focus on the entire set of equations and will systematically manipulate the equations until we get the equations of the solution.
Birne Binegar: And so that's that's sort of new so i'll spend some time doing that and then in the third step what we'll do is we'll transport our basic calculation into a matrix setting where we can do it even quicker.
Birne Binegar: Alright, so let's start.
Birne Binegar: It n by n linear system that needs any equations in em unknown and linear means that the variables are occurring it works to the first power if they occur at all, but they're always occurring in separate terms you never have an x one X two in the same term.
Birne Binegar: And what we found I guess.
Birne Binegar: On Monday, and also last week was that their solution sets are always going to be a fighter plane these hyper planes are generalizations of single point.
Birne Binegar: mind three dimensional planes or a higher geometrical objects basically your copy of a little space sitting inside of a larger vector space.
Birne Binegar: So points are just.
Birne Binegar: Take a points when we have one direction to depart from that point we get align when you have to we get a plane, etc, so what i'm stretching is from the start, because in the end, when we sold linear system, our goal is going to be, to express a solution like that.
Birne Binegar: Okay, so what we've seen is that if you have any equations in Mr known then the possible solutions are either no solutions at all, which case we would say the.
Birne Binegar: solution set is empty or we always get a hyper planes some solution set to hash exactly this form that just we'll have to figure out what the p zero is and what these vectors be one poopy me.
Birne Binegar: Okay, and then we also know the dimensions well at least an estimate of the dimension of the solution.
Birne Binegar: Generally speaking, every time you add in the equation you reduce the number of free variables by one So if you start off with no equations and so empty variables and you impose a series of equations what you should get is N minus in.
Birne Binegar: dimensional solution states.
Birne Binegar: You can understand this just from your high school technique every new equation gives you another variable you can eliminate So if you start off with em variables and you have any equations and you can affect you eliminate and variables and you're left with a minus in.
Birne Binegar: Three variables in your solution, however, sometimes you can add an equation, and it doesn't really add any new restrictions on your variables.
Birne Binegar: And so it's not always the number of equations if I said N minus the number of independent equations then this would be.
Birne Binegar: A strictly quality, however, we can always know if our equations are dependent or not, so we have to be prepared that are solutions that actually might be larger than we might expect.
Birne Binegar: All right, and then i'll end up proving that every when you solve a linear system, you always get a solution set like this, this is kind of come out from our technique of solution will show you how you sold it and then i'll show you how you write the solution in that form.
Birne Binegar: Okay, so i'm in high school and probably either to at this point in your mathematical career is that you solve the equations by.
Birne Binegar: Focusing on one variable usually an equation, to eliminate it substituting result in the subsequent equations and eliminate more variable but that's got to get really complicated because.
Birne Binegar: If you.
Birne Binegar: Use okay for two equations, but if you had 10 equations so you solve for X one in the first equation now you have to substitute X one into nine other equations.
Birne Binegar: Until you then you can eliminate to but then you have eight other equations to put that into it it's it becomes a mess and it's very error prone so the technique we're going to pursue but we're not going to worry about.
Birne Binegar: variables per se rather what we're gonna do is we're going to manipulate the entire set of equations.
Birne Binegar: Rather than the variables until we sort of massage the equations until they're actually the equation at the solution that seems sort of far fetched and i'll show you how easy, that is.
Birne Binegar: Alright, so.
Birne Binegar: The first thing he did explain is, how can I modify equations without changing their solution.
Birne Binegar: And these these operations or should be familiar to you.
Birne Binegar: If I have.
Birne Binegar: Two equations and do unknowns.
Birne Binegar: That, then, I can manipulate these equations and you want frequently does, for example, you can change the order of the equations that's not going to change the solution it's just.
Birne Binegar: Changing what you regardless, the first equation, and the second equation, so they have that like this, it seems like it's sort of a trivial operation but i'll still take time to.
Birne Binegar: tell you what i'm doing to a set of equations if i'm simply changing your order.
Birne Binegar: That just means flipping.
Birne Binegar: Alright, so that should be easily understood it's not changing the solution another thing you can do to an equation, if you can, if you see equation has a common factor on both side.
Birne Binegar: Common thing to do when solving equations is to simplify the equations like pulling out common factors that appear on both sides that doesn't change the solution either.
Birne Binegar: Right and so i'll note that by saying, an equation goes to a certain number of times that equation this case and multiply the second equation by one half to get that equation, but notice.
Birne Binegar: What i'm doing is in these operations.
Birne Binegar: I think of it is acting on the set of vectors, even though the exact operational only ox on one of them.
Birne Binegar: All right, and then, finally, what you can do is you can.
Birne Binegar: replace an equation with in general, you can add it to another equation and use that new equation as as a replacement but, more generally, what you can do is you can replace an equation like to multi.
Birne Binegar: Let me, let me just do the.
Birne Binegar: Easy method first demonstrate that don't let me work with these two equations X plus y is equal to two and X minus y is equal to two.
Birne Binegar: yeah So if I add these two equations X plus x is equal to X, Y plus minus y equals zero and two plus two is equal for.
Birne Binegar: This equation is implied by these equations and in fact I can replace either one of these two equations with this equation and i'll get exactly the same thing, because, if I can add these guys to get this equation, I can take any two and subtract them to get that this guy so.
Birne Binegar: Another operation that will not change the.
Birne Binegar: solution set is to replace an equation by it somewhat on another equation, or even a multiple of another equation, because in this step, you see, that you can also replace equations by their by multiple of that equation.
Birne Binegar: Alright, so all right, that is, for example here if i'm going to.
Birne Binegar: i'll replace the second equation like it's some with two times the first equation, let me do that slowly.
Birne Binegar: Maybe here.
Birne Binegar: Yes, it will show up best here, so what i'm gonna do is i'm going to take equation to.
Birne Binegar: Two X minus two y and then let me figure out what.
Birne Binegar: equation to minus R plus two times.
Birne Binegar: equation what is well that's going to be.
Birne Binegar: Two X minus two, why are you here.
Birne Binegar: To X minus two y.
alright.
Birne Binegar: Two X minus y equals four.
Birne Binegar: And then two times the first equation is going to be two X.
Birne Binegar: plus two.
Birne Binegar: equals two times two is four.
Birne Binegar: And so what I can do if I add these two equations together i'll get for X, these will cancel equal to eight.
Birne Binegar: And so.
Birne Binegar: This that's that's how you would actually.
Birne Binegar: Oh, so I didn't hear correctly well on either on the slide.
Birne Binegar: I guess, this should have been.
A.
Birne Binegar: slight typo there a top you see how that operation is working, and if that you shouldn't have to be writing these things down on the blackboard.
Birne Binegar: Ideally, what you can do is sort of in your head at two times this equation to this equation show, I have two X plus two X is equal to four X.
Birne Binegar: To y minus two is equal to zero, and four plus four is equal to eight, you should should just be able to read it off the equation what the result of this operation is.
Birne Binegar: OK so.
Birne Binegar: Just so three basic operations so i'm going to be sufficient for me to convert a set of equations into their solution.
Birne Binegar: But.
Birne Binegar: Let me introduce the vocational use, so what I like this, I have this double arrow between the two equation labels that means i'm going to just switch their order.
Birne Binegar: And if I write it this way, this means that the equation is getting replaced by a multiple of itself.
Birne Binegar: And this last operation is going to be we replaced an equation by at some with a multiple of another equation Those are the three operations i'm going to allow myself and now i'll show you an example that I can use those operations to solve a system of equations.
Birne Binegar: So here's the original SUP first thing i'm going to do is i'm going to add the two equations and I use that some to replace the second equation, so you add and then you use that the the result replace one of the.
Birne Binegar: One of the equations that you started off with so what i'm doing is i'm going to replace the second equation, like some of its first, and the result is you add these two equations you get X plus x is equal to two X, Y minus y is equal to zero and one plus two equals four.
Birne Binegar: All right.
Birne Binegar: Well, here we have a common factor on both sides, so i'll simplify that system by dividing the second equation by like to that converts it to that.
Birne Binegar: And now, what i'll do is i'll take the first equation X plus y Z equals one and i'll subtract from it, the second equation X is equal to two.
Birne Binegar: A double I want to do yeah X minus X is going to produce zero looks I did do this correctly good I looks like.
Birne Binegar: What I should have had.
Birne Binegar: Is.
Birne Binegar: equation why.
Birne Binegar: Well let's see what I got yeah so i'm replacing equation one and i'm subtracting equation to from it so X minus X should have been equal to 01 minus y would equal, why not sorry.
Birne Binegar: yeah I did I did did do it correctly.
Birne Binegar: X plus y minus X is simply why and then.
Birne Binegar: One minus two is minus one, so I did do that correctly.
Birne Binegar: What threw me off the side with upper i'm operating on the first equation so i'm replacing the first equation I this difference of equations.
Birne Binegar: Alright, so here I got two equations all do now let's flip them to get the variables in your canonical order and you have to you have the solution of the original system, the only way, you can solve this equation is type city X is equal to two in y is equal to minus one.
Birne Binegar: Alright, so that's that that's basically the idea.
Birne Binegar: The idea is to not work with variables, but rather entire set of equations and systematically manipulate them until you reach the equations of the solution.
Birne Binegar: Once we get the equations up solution, then they'll still more work to be done, we have to express that solution has a hyper plane, but we basically sold the system at this point what would remain is how do we express it in a tidy form.
Birne Binegar: Okay, so that's that's what's going to be going on in the background, but the calculation that that one uses sort of thing.
Birne Binegar: doesn't rely on looking at equations at all, rather, what we do is we introduced matron she's in the game, so that these manipulations of the equations.
Birne Binegar: can be performed really rapid city, in fact, that the first thing we're going to do is just throw away all the variables and work with matrices that contain the coefficient of the equation, and the numbers that appear on the right hand side.
Birne Binegar: Let me remind you what the coefficient matrix is very matrix of variables and the numbers be on the right hand side so called in a homogeneous turns here's a system of equations in in an unknown.
Birne Binegar: We grab all the coefficient on the left hand side put them in bed by m matrix if we started off with any equations and.
Birne Binegar: Then we put the variables in a column vector and we put the numbers that appear on the right hand side in the column vector.
Birne Binegar: And the point of this and thus far has been to now realize that this entire linear system is encapsulated by this single matrix equation.
Birne Binegar: Where you understand the connection between these two is, you have to perform the matrix multiplication you have to multiply these two factors together.
Birne Binegar: And then you compare the result component by component what you'll find is when you multiply these matrices together you'll get a column matrix that whose first entry is going to coincide with the first entry here second entry is going to coincide there.
Birne Binegar: However.
Birne Binegar: Even though this idea of matrices is important, and if that will use equations like this later on our initial technique for solving linear systems it's not going to rely on solving this matrix equation directly yet will still use matrices as follows what we're going to do is grab from.
Birne Binegar: This matrix equation.
Birne Binegar: will get to create a new matrix so called augmented matrix we're in we take the coefficient matrix for coming from our equation, we take the in homogeneous terms.
Birne Binegar: And we just append those in homogeneous terms it's the last column of a larger matrix so here, you see if I ignore this guy.
Birne Binegar: Here that just the regional coefficient matrix what i'm adding on to it in a separate call i'm going to separate by align because the left hand side plays a different role than the right hand side.
Birne Binegar: But we we separate them, but we write this down this way and that's will be.
Birne Binegar: known as the augmented matrix corresponding to a given linear system.
Birne Binegar: Alright, so let me show you how this is done here's the system, a system of three equations and three are nouns and here's it's augmented matrix.
Birne Binegar: Really easy right you go coefficient of X one in the first equation it's going to give you that first coefficient to there's going there minus one is going there to is going there there's the first row that augmented matrix.
Birne Binegar: so effectively you get it by erasing the plus signs in the variables putting a line for the equality and just in closing the whole thing is square brackets.
Birne Binegar: And not only can we go from equations to augmented matrix it's simple to go from an augmented matrix back to equations right, I know what equations this first row corresponds to think about.
Birne Binegar: augmented matrix ease.
Birne Binegar: The variables are not appearing explicitly, but this first column always corresponds to the coefficient of X one, the second call always correspond to the second coefficient of X two and the third call.
Birne Binegar: Always contains a coefficient of X three so what's happening here is, although i'm not no longer writing down the variables if we need to recover them, we can get them from the augmented matrix.
Birne Binegar: Okay, so.
Birne Binegar: here's the here's the plan and how we're going to solve linear systems.
Birne Binegar: Make sure this appears on screen, the first step will be as i've done last slide i'll take a linear system and i'll express it as an augmented matrix and I can this should be a two way arrow because I can go both ways.
Birne Binegar: The next step.
Birne Binegar: will be instead of manipulating the equations until we get the solution, the idea is going to be take this augmented matrix and manipulate it until we get the augmented matrix of the solution.
Birne Binegar: Then we'll take that augmented matrix of the solution and we'll see what the corresponding equations.
Birne Binegar: And that will that will furnish us with the equations of the solution, and finally, the last step will be to take the equations of the solution and reformulate that as a set of vectors that actually hyperplasia.
Birne Binegar: All right now there's lots and lots to explaining or with this algorithm you though it's pretty simple, first of all.
Birne Binegar: i'm going to manipulate matrix to another matrix how am I how am I going to recognize what the matrix augmented matrix of the solutions, well, I need to explain that yeah there'll be a.
Birne Binegar: target that we shoot for as we perform operations here, so one of the first thing i'm going to explain is what does the augmented matrix of the solution, look like, so we know which direction we had our operations to lead us.
Birne Binegar: And second of all.
Birne Binegar: So that that's the key step in the course the other, the other part of that and step two is OK i'm going to perform a sequence of operations to simplify my augmented matrix.
Birne Binegar: What operations, do I use it what time we have three different choices and that will have more than three choices we can enter choice interchange various pairs arose, we could.
Birne Binegar: replace roads license with multiple another equation it's not clear that if you just start doing that randomly you get kind of go around in circles so.
Birne Binegar: what's going to be developed primarily is a systematic procedure to take a given augmented matrix.
Birne Binegar: And do a sequence of well defined operations where each step, you know what to do next, and it will lead you to a particular form of an augmented matrix that will know is actually the augmented matrix of the solution.
Birne Binegar: This is the hard part step two once we get those augmented matrix of the solution it's easy to convert that to equations and then writing the equations that's a high profile that's also pretty easy as well, so it's actually step to it's going to i'm going to spend the most time on.
Birne Binegar: I should also point out that what i'm into discussing last week I talked about how, when you solve the equations you can have.
Birne Binegar: Every equation corresponds to a hype reply and the solution sets can be understood is intersections of hyper planes and so.
Birne Binegar: they're giving us their instance, when you think about solution sets if the hyper planes were there parallel and so there's no solution there's instances where the intersect and there's instances where they coincide.
Birne Binegar: What i'm going to do is I developed this step two is i'll point out how you identify what each of those phenomena is happen okay.
Birne Binegar: And that will help you better understand the nature of the solution, even before you reach the solution.
Birne Binegar: Okay, so.
Birne Binegar: here's the point.
Birne Binegar: I hopefully convinced you that there are three operations three families of operation that you can perform on a set of equations it's not going to change the solution set.
Birne Binegar: What i'm going to describe now is well.
Birne Binegar: I can take a set of equations and when they play and get an equivalent set or.
Birne Binegar: I can take a set of equations.
Birne Binegar: and convert it to a equivalent equation.
Birne Binegar: On the other hand, these guys correspond to augmented matrix.
Birne Binegar: And so to these guys.
Birne Binegar: But what I was going to try to avoid to make the calculations proceed is not due to figure out what the augmented matrix.
Birne Binegar: According to this changes, rather than taking equation manipulate it get a new equation, to get a new augmented matrix I want to see directly what an operation up here does to the answer here.
Birne Binegar: But it's it's really, really simple i'll show you the next slide how you transcribe.
Birne Binegar: operations on equations the operations on these augmented matrix these.
Birne Binegar: Alright, so, first of all, if you interchange the equations you just.
Birne Binegar: reverse the order, but then, when you write down the augmented matrix your equations are occurring a different order the coefficient coefficients in what equation, are going to rows of the augmented matrix.
Birne Binegar: So interchanging equations is just going to have the effect of switching the corresponding roads and the augmented matrix.
Birne Binegar: If I replace an equation, if I multiply equation by a number then each coefficient and the number on the right hand side of the equation is also going to be multiplied by Lambda.
Birne Binegar: When I convert to augmented matrix that needs every element in the row corresponding to that particular equation is going to be multiplied by Lambda.
Birne Binegar: don't want that, if you think of this is a concatenation of robot years when I multiply one of those equations by Lambda, what are the roles of this augmented matrix is going to be scalable time by Lambda because every entry in a given road which can be both by by the same number.
Birne Binegar: Then.
Birne Binegar: The last off basic operation for every replacing equation by some with a multiple of another equation, if I take an equation I replace it by itself for multiple of another equations what that's going to correspond to.
Birne Binegar: Well, the it equation corresponds to the I throw of the augmented matrix.
Birne Binegar: And then it's going to be replaced.
Birne Binegar: By it some somebody I throw out the augmented matrix with Lambda times the J through the augmented matrix alright so i'm only saying here is to.
Birne Binegar: implement this operation at equations on the augmented matrix we look at the corresponding row, this would be real I, and then we replace that row well I similar operation this this this is going to be exactly that the row corresponding to this side is exactly that.
Birne Binegar: Okay, so what I got out.
Birne Binegar: or three operations.
Birne Binegar: Did I can perform directly on the augmented matrix ease and, in fact.
Birne Binegar: i'll do note that this way this corresponds to interchanging two equations that now and interchanging two rows of the augmented matrix.
Birne Binegar: This corresponding to multiply equation by constant number, and then I can just do a scale or multiplication if the corresponding row.
Birne Binegar: And this alteration i'm taking corresponds to replacing it an equation, the equation by some with multiple with the jt equation, but on the augmented matrix that's implemented.
Birne Binegar: via this vector some right that's it just the linear combination of the two robe actors that are sitting inside the augmented matrix.
Birne Binegar: Alright, so.
Birne Binegar: Now let me show you.
Birne Binegar: demonstrating.
Birne Binegar: These these operations and action, how we can use it to solve a linear system.
Birne Binegar: Okay, so here's X plus y is equal to three x minus y is equal one.
Birne Binegar: First Step will convert it to an augmented matrix alright, so this is nicer notice the variables have disappeared and all i'm going to be doing subsequently is manipulating these these this matrix.
Birne Binegar: So the first thing i'll do is let's see i'm going to replace the second row with it's different with the first row, and so we can do this in place, so the first row 113 it's not changing because the operation is only acting on the second row.
Birne Binegar: And what's going to happen to the second row well i'm the subtract the first real from let's do this in place one minus one is zero negative one minus one is negative two one minus three is negative to don't notice I didn't have to carry any variables around and the.
Birne Binegar: The step which would be more tedious if I actually was working with the equations it's done really rapidly.
Birne Binegar: Alright, so now we ended up with this matrix let's let it be late it here we have common factor of two on both side, so what i'll do is i'll replace the second row by it multiple by minus one half that will convert this minus two to one and, similarly, here and now.
Birne Binegar: We have a new equation or new augmented matrix and the last thing i'm going to do is i'm going to take that first row and i'm going to.
Birne Binegar: subtract the second row from it, then i'll get one minus zero is one one minus one is 03 minus one is too.
Birne Binegar: All right now i'm stopping here but.
Birne Binegar: My claim is that this is actually the augmented matrix of the solution.
Birne Binegar: What you might observe here is the identity matrix you see here that's a good clue that.
Birne Binegar: What we'll see in as soon as I write down the equations why regard this has the equation of the augmented matrix of the solution.
Birne Binegar: Because when I take that matrix that augmented matrix and I convert back to equations i'll get one times X one is equal to two or simply X or sorry one times X is E plus zero times y is equal to two were simply X is equal to, and then we had zero times X, let me.
Birne Binegar: remind you that when you go back to the equations.
Birne Binegar: You think of the columns it's representing veritas.
Birne Binegar: And so I get one times X one plus zero times y is equal to two and I have zero times X times y is equal to one, and now there are you see your solution.
Birne Binegar: In this case.
Birne Binegar: i'm not i'm not going to.
Birne Binegar: Well, I could write down that solution as a hyper plane.
Birne Binegar: But it's so simple anyway there's only one value of X and one value why will permit me to solve the original set of equations, and so my solution set.
Birne Binegar: Is the hydroplane but it's sort of trivial one it's just a single point that corresponds to something with coordinates X X has to equal to, and why has to equal one.
Birne Binegar: So there, I actually finished the entire program for this.
Birne Binegar: For this linear system i've solved it and i'd like to express that solution as a hyper play.
Birne Binegar: But the hyper planes the trivial one is just a single point.
Birne Binegar: Okay, so let me pause here any questions so far yeah.
So.
Birne Binegar: um ideally, so what the identity matrix it tells you, is it that you have a unique solution and.
Birne Binegar: Remember, we don't always get solutions, and so we don't not always going to get the end of the identity matrix.
Birne Binegar: The next thing i'm going to do in today's lecture is actually tell you what that augmented matrix of the solution must look like.
Birne Binegar: And in fact they'll be unique, for that will have a clear goal to head towards, and it will have an algorithm that will lead us to that eventually.
Birne Binegar: Until so that that's what that's what's coming up, but yeah I wanted to point this out because.
Birne Binegar: Here it's sort of easy to read it off Okay, so this this row is saying X, one has to equal to, if no contribution from the other variable and this row is telling me that the secondary will have equal one.
Birne Binegar: So I want you to see that even at this stage, because it will be helpful, later on, but it is not true that you always get the identity matrix yeah you'll get something even more general than.
Birne Binegar: I get into the matrix so that we can include include the possibility that we have three variables in our solution, rather than a unique solution, so in this case.
Birne Binegar: Okay, so let's um.
Birne Binegar: let's let's do explain it now is well.
Birne Binegar: As we reduce well if we use elementary real operations to change an augmented matrix into the.
Birne Binegar: augmented matrix of the solution, the order and choice of those operations, I need to explain how you decide what to do next, as you systematically reduced and augmented matrix to the augmented matrix solution.
Birne Binegar: And then.
Birne Binegar: So that we don't just go around in circles, will have to know how to be able to recognize when we've reached the augmented matrix other solution.
Birne Binegar: So i'm going to concentrate on the second as second question first, what is the solution the augmented matrix of the solution, what should that look like.
Birne Binegar: And then, once we know that there'll be much easier to understand how we get there, we know the destination, then we can.
Birne Binegar: call up Google maps and find out how to get there.
Birne Binegar: Alright, so.
Birne Binegar: Now some nomenclature this this nomenclature.
Birne Binegar: not overly technical but it allows us to talk.
Birne Binegar: use a familiar terms for a calculation that will be doing over and over again, so the first idea to bring out is, if you have a matrix not necessarily an augmented matrix.
Birne Binegar: We have a notion of pivot and what a pivot is the first non zero entry in a row at when you read from right to left alright so here, I have a matrix with three rows.
Birne Binegar: Each row, so why is it has a non zero entry is going to have a pivot, what do you identified as you read from right to left.
Birne Binegar: And so here's a zero that's not can't be a pivot, it has to be on zero entry here's the two it's the first one, I come across so I underlined it I get if I dad is a pivot.
Birne Binegar: In the second equation, I have a non zero entry in that first call so that's my pivot there, and the third equation, I have to zero, so those cannot be pivots but here's the first possibility and that.
Birne Binegar: is exactly the first possibility that we identify is a pivot or he only wants you to do what I say, what are the pivoting is in a in a row, just to figure out what the reading from left to right what the first non zero entry is.
Birne Binegar: What i'm trying to describe to you, of course, is what the augmented matrix of the solution, this looks like figuring out where the pivots live is a key part of that.
Birne Binegar: Alright, so.
Birne Binegar: here's another definition row Echelon floor.
Birne Binegar: What this is going to be, is it the pivots these first non zero entries, we want to have them arranged in in a particular fashion insider augmented matrix we want to have this up.
Birne Binegar: Be sort of concentrated up to the upper upper left hand side more precisely.
Birne Binegar: What we want is the pivots in the rose above are always off to the left of the pivots and the road alone, so this equation is not in line for because here's the pivot here, and this real below it.
Birne Binegar: Is up its pivot is off to the left of the row above it, and so it doesn't satisfy this condition.
Birne Binegar: If you have a pivot and an up arrow to be in Russia on form this guy was the pivot that where the word.
Birne Binegar: pivot is off to the right of pivot and the role of up, but here the pivot is moving left ways up in the Left direction as we descend, so this is not in relational on floor what we're trying to do.
Birne Binegar: Is.
Birne Binegar: get our pinterest sort of form a staircase inside the matrix you want to maybe a pivot up here and then later, a pivot there and a pivot there and a pivot there, so you want to have this sort of downward staircase.
Birne Binegar: If you move off to the right your pitch should be descending order so if you descend through the matrix your pivot should be steadily moving off to the right what you don't want is to be send off to the right to send off to the right to send off and right he sent off to the left this.
Birne Binegar: This sort of phenomena you're trying to avoid.
Birne Binegar: Alright, so let's give some examples here.
Birne Binegar: here's a matrix in row excellent for i've underlined pivots and you see that if I move down the matrix the pivots are steadily moving off to the off to the right.
Birne Binegar: here's a matrix it's not in Russia lot for because this pivot and the first row well the pivot in the second row which current off to the left of the pivot the row about it so that's not rush alone for.
Birne Binegar: Okay, so simple enough to identify and back being able to recognize that even better being able to work towards matrices in real life form is going to be the key operation will use to convert a given augmented matrix into the augmented matrix of the solution.
Birne Binegar: But there's a wi fi on this.
Birne Binegar: But I have a matrix in already real restaurant or so it's tickets are occurring downwards and off to the right, then we could put a even stronger condition.
Birne Binegar: Will demanded a matrix will say the matrix you said reduced row Echelon, for if it first of all, is in relational, and for that the pivots all equal one and that above and below the pivots we have all the zeros.
Birne Binegar: So let me give you.
Birne Binegar: Maybe three examples here what we're talking about here one two on.
Birne Binegar: Or maybe like 012311 here's a random matrix.
Birne Binegar: The pivots are here here, here they don't really fall into any particular pattern, then we have a row Echelon for.
Birne Binegar: Be that you might have something like two to 1032001 alright so here the pits.
Birne Binegar: I scattered across roads and i'm underlining the first non zero sure alright, so this matrix is, at least in relational on floor and now.
Birne Binegar: i'm matrix in reduce real Echelon.
Birne Binegar: could look like this 101002000.
Birne Binegar: All right, so a 001 but maybe zero there I can put a to hear it will still be in reduce relational on for all right, let me check to see me verify that that matrix satisfies all three of these conditions.
Birne Binegar: Here the pit.
Birne Binegar: First row first entry second row is the last entry and the third row doesn't contain any pivots at all.
Birne Binegar: So, but it is unreal I salon for because the pivots are occurring in that downward staircase information so it's it real excellent form it's the pivots i've underlined them they're both equal to one.
Birne Binegar: And so there we have.
Birne Binegar: The second edition satisfied and lastly notice that not only below the pivots I had zeros but before we do show Echelon for matrix even above the pivots I have zero.
Birne Binegar: here's why.
Birne Binegar: You like matrices and reduce real Echelon or.
Birne Binegar: Look at the identity matrix 100010001.
Birne Binegar: i've already sort of hinted that identity matrices we've got a correspond to the augmented matrix when you see the identity matrix sitting in the coefficient side.
Birne Binegar: it's gonna you're looking at the augmented matrix solution corresponding solutions, and if you have a unique solution well it's also a matrix and reduce your actual on foreign here's the pits.
Birne Binegar: they're occurring downward in a downward staircase off to the.
Birne Binegar: Lower right the pivots are all equal to one and above and below pivot we have nothing but zeros.
Birne Binegar: These guys.
Birne Binegar: This is the for an augmented matrix of the solution, it generalize is that first example.
Birne Binegar: Where we came up with an augmented matrix hvalur broke out equations that you can see, it was a solution, but more generally.
Birne Binegar: We might get an augmented matrix another solution that has this form and actually what's going to be very important here is that this form, I can stare at it right now, and I know that I have a one dimensional solution set.
Birne Binegar: You don't see that yet, but that's to be explained that there that's the one dimensional nature of the solution, set for this equation like this.
Birne Binegar: You look at the columns that don't contain pivots and that each every call that doesn't contain a pivot is going to be, you have a free variable in your solution that's coming but.
Birne Binegar: For right now, I just want you to understand that this is our target as we.
Birne Binegar: Take a set of equations we convert it to an argument matrix start doing row operations, this is the climate matrix we're trying to reach and then.
Birne Binegar: When we reach a matrix in reducing Echelon for an augmented matrix and reduce cholesterol and four yeah we can immediately convert back to equations and that will be the equations of the solution.
Birne Binegar: Okay.
Birne Binegar: So.
Birne Binegar: Now we know where we're going.
Birne Binegar: next step is to explain how to get there.
Birne Binegar: Okay, so.
Birne Binegar: Let me give you some more examples of this season, reduce your actual on form here notice the pivots sorry we're on the block or.
Birne Binegar: A little bit here.
Birne Binegar: i've underlined the pivots to help you see them but they're readily identify the first non zero entry in any given row they're occurring in the downward staircase.
Birne Binegar: they're all equal to one and above and below pivots yeah above quite, this is not introduce your shoe on for Yet it would be if that was zeros it's yeah it's almost in real life long form but.
Birne Binegar: If I replace those entries above this last tidbit by zeros, then I would.
Birne Binegar: get a matrix should we do celestial on floor, but of course I was trying to solve a system of equations I couldn't say oh this not yet rush along for i'll just put 00.
Birne Binegar: Know get your get your athlete changing the set of equations that correspond to the augmented matrix what you want to do if you want to get rid of these entries here is you're gonna have to.
Birne Binegar: get those zeros here by doing some sort of elementary row operation because those are the operations that don't change the solution set though it's not simply taking the equation you racing snap so that you get the right answer that only works in Chem labs.
Birne Binegar: Rather, what you're going to do is you're going to somehow get from this matrix to a matrix and we do short form by using elementary row operation.
Birne Binegar: alright.
Birne Binegar: So I think we got the program established.
Birne Binegar: Now let's see how we can implement it.
Birne Binegar: Alright, so.
let's see.
Birne Binegar: here's a matrix in reduce your electronic form, and of course.
Birne Binegar: When you actually have the identity matrix on the left hand side here, you can read the solution directly off this augmented matrix X one times one so remember.
Birne Binegar: Do this explicitly for a little while further first column corresponds to the first variable second column corresponds second variable there to the third variable.
Birne Binegar: You would convert back to equations by going one times X one plus zero times X two plus zero times X three is equal to two or simply X one is equal to zero times X plus one times X two plus zero times X three is equal to one or simply X one X two is equal to one etc alright so.
Birne Binegar: here's an example of a matrix and we do show Echelon for in which it's really easy to see that this is the appropriate augmented matrix for the unique solution of some linear system.
Birne Binegar: right but.
Birne Binegar: here's a matrix that's.
Birne Binegar: Not quite in our producer lashawn form.
Birne Binegar: But.
Birne Binegar: it's in real life, a lot for.
Birne Binegar: But it's not in reduce your platform because first of all, the pits are not all equal to one.
Birne Binegar: here's a matrix that not even a real issue for because.
Birne Binegar: This century here it's like this go down a row, the pivot in the real below it has moved off to the left, rather than the right.
Birne Binegar: And here is a matrix that's in real life real short form.
Birne Binegar: But it's not and we do show Echelon form because the pivots are no not all equal one.
Birne Binegar: let's see i'm just giving more examples of this, or i'm sorry pardon me.
Birne Binegar: If I do run over this time it's because i'm thinking about what i'm going to say next rather than the time just raise your hand and say professors I got a class or something like that okay so i'll let you guys go and we'll.
Birne Binegar: we'll do examples of this on Friday and you'll actually see how you systematically reduce of a trick to it, we do a show on for.
Birne Binegar: yeah.
Birne Binegar: Whenever well.
Birne Binegar: matrix and the solution.
Birne Binegar: Is this is simply a line and another line waiting because yeah I still have a single point intersecting till later will interpret our solution geometrical the first goal is just to get them yeah.
Birne Binegar: How would you.
Birne Binegar: Have a third variable.
Birne Binegar: Ah yeah yeah you would push you watch you.
Birne Binegar: yeah you need a you need.
Birne Binegar: A second free variable in your solution so yeah so you may have any number of variables originally.
Birne Binegar: But just think about solving equations like you do in high school after you've eliminated all as many variables, you can you can have a certain number leftover that's going to be the dimension of your solution.
Birne Binegar: But these are linear equations until that never happens.
Birne Binegar: um if you look at the syllabus like sort of outline the course and it's going to occur after we discussed in verses of matrices.
Birne Binegar: But what I do is I don't I don't set a date on the start, because it varies from section to section year to year exactly.
Birne Binegar: Is it the eighth meeting or the Nice meeting of the course how much anytime we all can be at least a week and if that's what i'll do is i'll pull the class and let them pick the exactly.
I.
Birne Binegar: just wanted to say.
Birne Binegar: In your Home Office oh yeah.
Birne Binegar: And I wish I.
Birne Binegar: built myself yeah so.
Birne Binegar: what's your.
Birne Binegar: I mean, basically, what you do is you, you buy the word Currie cup because it's all done on cnc now anyway, so you just play the word pre cut.
Birne Binegar: You loose with the thing together you lock her up right, but you put your own hardware on I do my own frit jobs and things like that, but uh yeah but it takes the mistakes in it.
Birne Binegar: It takes part of the mystique out a good sense of vendors, when you see how easy they are to build.
Birne Binegar: But.
Birne Binegar: yeah.
example like.
Birne Binegar: X plus y is equal to two plus two.
Birne Binegar: is equal default.
Birne Binegar: OK, so the by this equation by two and you get X plus y is equal to.
Birne Binegar: Five half, but you can't beat both equal to, and I.
Birne Binegar: Just never intersect and then, when you have like three equations we can have individual interceptions but they don't have a common interception gosh.