Birne Binegar: not working, maybe the ball.
Birne Binegar: technical difficulties here gotta.
Birne Binegar: Get the projector to work.
Birne Binegar: let's see if we could reboot the system and see if that helps.
Birne Binegar: Okay, so i'm going to wing it i'm just going to improvise instead of working from prepared slides I did record my 930 lecture and i'll i'll post that but.
Birne Binegar: yeah unfortunately For those of you online.
Birne Binegar: yeah i'll try to keep track with a slight, even though in class i'm not unable to display the slides but i'll try to.
Birne Binegar: adhere to them.
Birne Binegar: Just that i'll be writing more on the blackboard than the normal Okay, so let me.
Birne Binegar: remind you, where we're at we're trying to solve linear system large systems of linear equations and.
Birne Binegar: What I showed you last time that it's possible to actually solve these systems by manipulating the equations rather than the variables.
Birne Binegar: That you can systematically transforming the equations from whatever they started off it to the equations of the solution, using only operations that preserve the solution set preserve the points in space it happily satisfy the equations and so.
Birne Binegar: I did that one example of that, but what I would began to describe on Wednesday, with a more efficient way of doing that, where instead of constantly rewriting the equations we manipulate a single matrix that matrix it's called the augmented matrix.
Birne Binegar: And it has a form of where you take the say we have a system of equations anyone one X one plus dot dot dot plus a one m X m equals B one and more equations a say equations.
Birne Binegar: and variable and m X m equals and the system of equations can be encapsulated in a single matrix take the coefficient of the variables.
Birne Binegar: arrange them in an n by n matrix.
Birne Binegar: And then you append to that this would be the coefficient matrix when we write this system is a matrix equation, but so the coefficients there are captured there we ignore the variables, but we do capture the numbers that appear on the right hand side as the last call.
Birne Binegar: Of this augmented matrix and point is, is that you can go from equations to augmented matrix and augmented matrix see factor equations.
Birne Binegar: Going the opposite direction.
Birne Binegar: You just have to remember that each column on the left hand side corresponds to a particular variable.
Birne Binegar: Where you recover the first equation, you should look at the first row, and you go a one one times X one plus a one two times X two etc a one in time to exam and then this line, the interview now convert to.
Birne Binegar: Think of this as an equal sign equals B one and so i've got the data that goes into specifying the system of equations all encapsulated in the single matrix.
Birne Binegar: itself, instead of manipulating the equations will manipulate this matrix until we reach a matrix in.
Birne Binegar: what's called reduced row Echelon.
Birne Binegar: That will be an augmented matrix again, but it will be the augmented matrix at the solution, and so you all right, this symbolically as a prime.
Birne Binegar: The prime.
Birne Binegar: If this was the original augmented matrix coefficient matrix with the last column, I think the idea is to take this matrix and.
Birne Binegar: Do operations that mimic the operations, we perform on equations directly until we get it into reduce relational on form, and then we get convert back to equations and we get the equations of the solution.
Birne Binegar: Okay, so let me.
Birne Binegar: review the nomenclature, we developed so far let's see the augmented matrix than the scribe ah OK, so the question would then be well.
Birne Binegar: How do I actually go from here to the augmented matrix of the solution we use operations that are not going to change the solution set and if those are three types.
Birne Binegar: First of all, we can take two rows.
Birne Binegar: And simply reverse their order in the in the matrix if this is the first slot and the last slot I would move the last roll off the top of the POPs down to the bottom, for example, so that's one thing we're allowed to do.
Birne Binegar: second thing we could do we could replace the I throw.
Birne Binegar: By a scale or multiple itself, but because we don't want to lose information we don't want to multiply a row by zero.
Birne Binegar: Because that would just make the whole row zero and and the equation that would correspond to a line at the augmented matrix where you only had zeros would be.
Birne Binegar: zero times X one plus zero times two plus zero times X is equal to zero, which would just be the equation zero is equal to zero, so we will have lost information if we.
Birne Binegar: We have lost a condition if we multiply by zero but as long as we multiply by a non serial number, then we don't lose any information, put another way, if I multiply by I want to.
Birne Binegar: multiply by one half I can get back to where I started by multiplying by two, but if I multiply by zero I can't divide out by zero until I end up losing information, so this is not a problem, I just want you to be aware that you don't want to be multiplying by zero.
Birne Binegar: And then the third thing that we can do, and this is probably the most common thing you do and in the process of reduction is you take a row, and you replace it.
Birne Binegar: With it, some in general, with a multiple of another realm.
Birne Binegar: So the I throw gets replaced it's somewhat of a multiple in the other room now each of these operations is performed on a particular row of the augmented matrix until we call them elementary row operations.
Birne Binegar: What they correspond to that the level of equations or operations that don't change yeah.
Change.
Birne Binegar: yeah yeah I mean.
Birne Binegar: I think, so I mean maybe you're looking a little bit ahead of your calculation, you see that i'm going to have to do this, eventually, am I just go do it now.
Birne Binegar: You don't always I mean you, you can be true, you can be more strategic but generally speaking you're going to use these three kinds of operations to.
Birne Binegar: The point is to go from the original matrix down to the matrix of your solution, and these are the allowed operations, they won't change your solution set if you do something else that your party.
Birne Binegar: Exactly exactly so i'm going to explain that today okay so.
Birne Binegar: So let's Let me explain, so far as I remind you, an augmented matrix is what our elementary row operation is now we're getting to your point.
Birne Binegar: What is a matrix in reduce real Echelon form that's going to be something will be able to spot visually will look at a matrix and we can tell immediately if it's in the right form and not.
Birne Binegar: and, moreover, having that visual idea of what we're shooting for will tell us what we need to do next at any given step so first of all row Echelon for.
Birne Binegar: i'll describe it verbally but it's really visually that you want to see this, so a matrix is in real Echelon for.
Birne Binegar: row.
Birne Binegar: Echelon form all usually be abbreviating so it's R E, F and i'm.
Birne Binegar: If.
Birne Binegar: well.
Birne Binegar: pivot.
Birne Binegar: And upper rows.
Birne Binegar: up here.
Birne Binegar: To the left.
Birne Binegar: and lower rows.
Birne Binegar: pivot pivot means the first non zero entry in a row so that's what you're focusing on.
Birne Binegar: We have a matrix 120001000 i'm underlining you the pivot, then you said, the guy if you watch during this calculation and the other other entries.
Birne Binegar: Well, you made it be like them, but your focus is always on what what what's happening with the pits and this matrix is enroll Echelon form, I should put one more restriction here this row is all zeros it has no pivot you want those at the bottom of the matrix and so with zeros.
Birne Binegar: At the bottom.
Birne Binegar: Alright, so this is a matrix in row Echelon form.
Birne Binegar: Is it reduce yeah it's also in reduce your show on porn why because in reduced.
Birne Binegar: row Echelon form.
Birne Binegar: Here you want row Echelon for.
Birne Binegar: All right, that's okay here's your pivot as you move down pivots move off to the right a pivot.
Birne Binegar: Always equal one for this matrix and thirdly above.
Birne Binegar: And below.
Birne Binegar: pivot.
Birne Binegar: Only zeros.
Birne Binegar: So, if you look at the pivot in this column that contains the pivot I have only zeros besides the pivot itself same thing here only zeros.
Birne Binegar: This is the kind of matrix we're shooting for because it's going to rip when we have an augmented matrix in reduce your actual form is going to correspond to the augmented matrix of the solution.
Birne Binegar: What I want to describe to you today is how do you go from an arbitrary matrix.
Birne Binegar: To the standard form for the.
Birne Binegar: augmented matrix of the solution.
Birne Binegar: and
Birne Binegar: i'm going to do this in several steps.
Birne Binegar: The main thing I want to show which is going to be the meat at the problem in general is to how to get a matrix in row Echelon form.
Birne Binegar: Okay, so let's see.
Birne Binegar: By the way, i'm wondering.
Birne Binegar: Is my computer so on.
Birne Binegar: Well, even even if.
Birne Binegar: Even if I do that okay i'm i'm still trying to think it maybe I could use the computer here to display but I can't alright so that's a.
Birne Binegar: problem i'm working.
Birne Binegar: On unfortunately on this side the blackboard because that's where the cameras pointing it.
Birne Binegar: Alright, so let me show you.
Birne Binegar: An example.
Birne Binegar: Where I explicitly.
Birne Binegar: converted matrix to row Echelon form.
Birne Binegar: So the matrix i'm going to start off with.
Birne Binegar: it's going to be.
Birne Binegar: 01131227 row I have 00000 third rolling him 10100 and the fourth I have 1101.
Birne Binegar: Okay, show it's a rather large matrix yeah i'm not putting the line here because i'm just looking at the matrix itself and not yet interpreting as the augmented matrix.
Birne Binegar: Of a linear system, because these row operations don't require you to put a line here so i'm just going to ignore that for now alright, so this matrix is not in row Echelon form.
Birne Binegar: Look at where the pivots live.
Birne Binegar: And you see they're not following this.
Birne Binegar: staircase pattern from the upper left to the lower right.
Birne Binegar: So what we'll do is we'll.
Birne Binegar: rearrange this matrix using the elementary row operations first thing i'll do seeing this zero row is i'm going to put it right at the bottom of the matrix.
Birne Binegar: So the operational uses all take row two.
Birne Binegar: And i'll interchange it with a fourth row.
Birne Binegar: Right, so that will produce a new matrix.
Birne Binegar: So first row is not changing.
Birne Binegar: 1312.
Birne Binegar: second row it's all us not last row 11011 last second row is moved to the bottom, and the third row with has been left unchanged.
Birne Binegar: All right, good, so now we got the zero at the bottom.
Birne Binegar: But there's no pivot in this upper corner, and this is where you want to start when you start the process of production.
Birne Binegar: You want to work start initially at this upper corner and you can work your way down towards this last row in this last corner, so what i'll do is i'll somehow get a pivot up here a tip that I can work with and and Nice one to grab it just interchange two more rows.
Birne Binegar: So here all now switch row one in round two.
Birne Binegar: And what i'll end up with is not 11011013120131 sorry I interchange to roshan i'm not expecting.
Birne Binegar: them to 00000000 good look what we got so far at least the first two rows or in row Echelon form, but this guy.
Birne Binegar: So I moved from the second row to the third the pivot is moving off to the left, it should be moving off to the right what we'll do is i'll use this entry in the first row to effectively cancel this guy.
Birne Binegar: Now I don't simply just scratch the sky out and replace it with zero, I have to use an elementary row operation, so the next step would be.
Birne Binegar: Now we'll go we'll send.
Birne Binegar: The third row.
Birne Binegar: i'm going to replace it with a third row.
Birne Binegar: minus the first row because one minus one it's going to equal zero now get a zero in that slot.
Birne Binegar: First rose not changing 11011 second row not changing 01312 the third row is changing one minus.
Birne Binegar: The third row minus one minus one is zero good.
Birne Binegar: that's the thing we were trying to do, and then.
Birne Binegar: Zero minus one is negative one one minus 010 minus one is negative one zero minus one is negative one.
Birne Binegar: notice how fast, I can do these manipulations it's not like, if I had to add equations together and get a new equation, be much more tedious i'd have to keep track of the variables and the coefficients.
Birne Binegar: But here I just have to keep track of the position of the entries of the various roads and make sure entries in the same column get combined.
Birne Binegar: Okay, so, and then the last row stays zero.
Birne Binegar: Alright, so what I did here yeah there's the operation I performed and now oh good.
Birne Binegar: Which is still not in real Echelon form because well I got rid of this problem.
Birne Binegar: I still have a pivot below appendix until i'm going to need to get rid of that guy so what i'll do is i'll send the third row operated on again this time sending to it, some with the second row i'm trying to get this get rid of this term, so I add this row to it and i'll get exact cancellation.
Birne Binegar: So I get 1101101312 i'm just rewriting the first two rows the operation it's going to be act on the third row zero Plus 001 plus minus one is 03 plus one is for.
Birne Binegar: One minus one is zero and two minus one is one last row unchanged.
Birne Binegar: know there are many you'll the statement I guess I should make it when you go to row Echelon form many infinitely many different ways to write down.
Birne Binegar: But when you get it to reduce your actual on performance unique.
Birne Binegar: Luckily i'm not going to prove that in this course it's hard to prove, but in fact it's true that once in some sense, you can see it from the algorithm.
Birne Binegar: is, as I described it, you see there's you're going to end up in one final destination, then the question would be what if you took a different route to get to that.
Birne Binegar: yeah you'll you'll get a slightly different answer, but we haven't yet got to reduce real Echelon force.
Birne Binegar: Until but now you can look where the pivots live you see at least we've got to row excellent forms.
Birne Binegar: Alright, so this is what happens in the first stage of this process, we call row reduction we take a arbitrary matrix and the first task is to get it into row Echelon, for the way you do, that is, first of all, you start up in this upper corner if there's nothing there, then you may.
Birne Binegar: may move some rows around so you get a pivot up in this upper corner, but then you get your starting point, and you can I guess starting here, you can begin to remove.
Birne Binegar: entries below pivots and higher rose your startup which for the first round Clara all the entries below its pivot.
Birne Binegar: And that led to this and then we stare at them result we see Okay, we got rid of this guy but now we have a pivot directly below the pivot in the second row.
Birne Binegar: tickets in the higher road have priority use them to eliminate the entries below entries non zero entries below them.
Birne Binegar: And so the next step was to clear out this entry again I did another elementary row operations and in the end it's like descent through this matrix i'm naturally.
Birne Binegar: clearing out all the entries below the pivot and that will, when you do, that all the zeros are going to be concentrated toward in this lower left hand corner and the non zero members are could get kind of pushed up into the upper right hand corner.
Birne Binegar: And so, sometimes we call these such matrices upper triangular.
Birne Binegar: You see how the non zero entries tend to be up in this corner all right, but this is just a.
Birne Binegar: row Echelon form.
Birne Binegar: i've only got the pivots.
Birne Binegar: arranged correctly, but my pivots are not all equal to one yet nor.
Birne Binegar: Nor do I have I cleared out the entries that lie above tickets, so in this pivot and the second rule rule, I have a nonzero number and to get it in reduce your last one for might have to get rid of that guy.
Birne Binegar: alright.
Birne Binegar: So in my slides.
Birne Binegar: Right yeah flashing through this.
Birne Binegar: But um.
Birne Binegar: let's let's move on, I think I have a different example yeah one requires so much writing.
Birne Binegar: The next example I want to give you is OK, if you've reached.
Birne Binegar: Real Echelon form.
Birne Binegar: What do you do next.
Birne Binegar: So in this example like think that matrix is.
Birne Binegar: Two for 1036 and 00 minus one so that's an online form.
Birne Binegar: But not.
Birne Binegar: Reduce relation one for.
Birne Binegar: What do I do, to put it in reducing.
Birne Binegar: The first thing you should try to do is get those pivots equal to one easy to do.
Birne Binegar: This pivot is too so let's divide that first row by two.
Birne Binegar: And this pivot is three so let's divide the second row.
Birne Binegar: By three.
Birne Binegar: And we'll get a new matrix one to one half 012 and zero all a send up the third row also the two minute minus 0001 Okay, so I just using scale or factors I can.
Birne Binegar: easily convert these pivot to the value one.
Birne Binegar: All right now what remains is to get rid of.
Birne Binegar: These entries that lie above pivot you've got zeros below the pivots but now you want to get rid of the entries above pivot and.
Birne Binegar: At this stage.
Birne Binegar: Like what we did in the stage one where we were trying to clear out entries below pivots only instead of starting at this upper corner you start at the bottom one.
Birne Binegar: You work your way up there, and then you move up and sort of in exact opposite direction you a movie before, so what we'll do here is maybe we'll send row one.
Birne Binegar: Two row one minus one half rho three when i'm arranging is for this pivot to cancel that entry in the first row when we can simultaneously do the second row is not going to confuse us I don't think we want to get rid of that too so we'll subtract.
Birne Binegar: Two times the third row from the second row.
Birne Binegar: And the matrix that will result will be let's see one, two, so one minus one half done zero which one two minus one half and zero is to.
Birne Binegar: One half minus one half times, one is zero.
Birne Binegar: And what this operation will do will be 2010001 okay.
Birne Binegar: So I changed two rows in that matrix just using effectively.
Birne Binegar: This operation of adding a multiple one row to another, and now.
Birne Binegar: Last problem we have is, we have a pivot for non zero entry about it and so again i'll use the.
Birne Binegar: Summer to roast it cancel that i'll send row one two row one minus two Times wrote to.
Birne Binegar: write i'm trying to get rid of that guy and the resulting matrix will be let's see one minus two times zero is one two minus two times, one is 00 minus zero is zero.
Birne Binegar: And then, here we have no.
Birne Binegar: I guess that's all I was doing 010001 all right.
Birne Binegar: Now we're reducing row Echelon form here your pits they're all equal to one and above and below pivots I have nothing but zeros.
Birne Binegar: OK, so now that that's the procedure and.
Birne Binegar: Let me.
Birne Binegar: Next, try to do a full example.
Birne Binegar: Well, the system of equations.
Birne Binegar: And i'll show you the entire process by which we extract.
Birne Binegar: The solution set.
Birne Binegar: Using this role reduction algorithm.
Birne Binegar: Okay, so system, I have in my slide is X one.
Birne Binegar: plus x two.
Birne Binegar: minus X three.
Birne Binegar: is equal to zero.
Birne Binegar: X one minus X two.
Birne Binegar: plus x.
Birne Binegar: is equal, the two and.
Birne Binegar: Two times X one.
Birne Binegar: minus two minus X three is equal to negative three.
Birne Binegar: that's when we start off with.
Birne Binegar: A system of equations in three unknowns first thing we do is we grabbed the coefficient.
Birne Binegar: And the numbers on the right hand side in a separate call.
Birne Binegar: And we'll have one minus 1122 minus one minus one negative three.
Birne Binegar: Okay, so that's.
Birne Binegar: that's the augmented matrix to start off with now Okay, this is a nice example there's No zero row, so we don't have to move any road to the bottom.
Birne Binegar: And there's We already have a pivot up in this corner, so we can just start clearing out entries below the pivot in the first row let's send.
Birne Binegar: row to throw two minus row one unless and row three two row three minus one.
Birne Binegar: Oh sorry minus two times Roman.
Birne Binegar: right that factor of negative to here is, precisely, so I can cancel the to their these other columns we're going to be effective, of course, but what we're focusing on is getting rid of pivots that sit below pivots.
Birne Binegar: So this will produce a matrix let's see one minus 1111 minus one zero first row that's not changing second row I subtract the first row from the second one minus one is zero.
Birne Binegar: minus one minus one is negative to one minus negative one, is actually plus two, when you subtract a negative number, it becomes a plus and then two minus zero is too.
Birne Binegar: And let's see Oh, we can do the third row as well here we'll have to now i'm going to multiply the second row by two so i'll point at these numbers but i'm also going to multiply each of them by two so two minus two is zero.
Birne Binegar: Negative one minus to.
Birne Binegar: look like negative three and then minus one minus.
Birne Binegar: Negative one times two is going to be.
Birne Binegar: yeah I guess yeah but I subtract off the two times this it becomes plus and then we have negative three minus zero that will be negative three.
Birne Binegar: Okay.
Birne Binegar: So.
Birne Binegar: I do, that correctly.
Birne Binegar: I guess so three minus 313 okay good alright so.
Birne Binegar: Now.
Birne Binegar: Eventually we're shooting we've almost got it in row Echelon form and.
Birne Binegar: let's simplify matters a little bit I mean it's possible for me to get rid of this term here by multiplying the second row by.
Birne Binegar: Three half and then subtracting it, but I don't want to multiply by fractions because then, once you introduce fractions into your arithmetic there you know they tend to hang around and make things more complicated so let's let's um.
Birne Binegar: With the I have sort of simple playing the calculation, let me go ahead and convert that pivot into a one because we're going to need that eventually anyway welcome one john simply sending row two two.
Birne Binegar: minus one half rho 211 minus 1001 minus one.
Birne Binegar: minus one, and then the third row was still zero minus three one minus three.
Birne Binegar: All right now getting rid of this guy is going to be a little bit easier I just take three times the second row add it to the third and i'll get rid of that lady that pivot in the third row now send rope three.
Birne Binegar: row three plus three times row two.
Birne Binegar: And i'll end up with one one minus 1001 minus one minus one nothing changed in the first two rows the last row is going to be zero minus three times zero and zero minus three plus three times, one is 01 plus three times minus one it's going to be negative two.
Birne Binegar: And minus three plus.
Birne Binegar: Three that's look like six maybe.
Birne Binegar: Negative which, let me see.
Birne Binegar: i'm adding the car so three plus three times minus points can be negative six.
Birne Binegar: All right, alright let's see.
Birne Binegar: let's get that last pivot.
Birne Binegar: continue working here.
Birne Binegar: multiply the third row by minus one half to get an augmented matrix one one minus 1001 minus one minus one and now 0012 sorry.
Birne Binegar: let's see oh you're right you're right.
Birne Binegar: Okay, so.
Birne Binegar: rocio on form pivots are all equal to one last obstruction.
Birne Binegar: Or the entries that lie above it.
Birne Binegar: But they're easily remove.
Birne Binegar: The important point, though, is it now to get rid of the cubits up these entries that lie about pivots you start in this corner.
Birne Binegar: clear out the entries in the in the last pivot last row, then you move up and then you'll clear out the entry above the pivots and higher rows of the first thing we'll do is we'll send row one, I want to get rid of this term.
Birne Binegar: I add the third row to.
Birne Binegar: row.
Birne Binegar: I also have a negative one here so i'll cancel that by adding.
Birne Binegar: Two plus rebel free.
Birne Binegar: end up with 110 so what's nice about this is that it's these calculations to become easier because these zero entries here.
Birne Binegar: Even when I add a row they're not gonna have any contribution for most times, most of these entries are not going to have any effect in the rose above, but the entry in question this guy here it's going to be exactly canceled by the corresponding entry in the third row zero and then.
Birne Binegar: i'm adding the third row to the first so three plus zero is three.
Birne Binegar: Man, the second row again, you can see how these add the 01020102 and the third row is unchanged 0013 okay we're getting close.
Birne Binegar: that's The only problem that's left and phil will send.
Birne Binegar: First row to the first row minus the second row.
Birne Binegar: So what we get when he canceled that guy out and we'll end up with 100301020013 now.
Birne Binegar: reached.
Birne Binegar: let's see let's see you're right you're right.
Birne Binegar: I appreciate your good.
Birne Binegar: Okay, so.
Birne Binegar: Okay, so now we've got a matrix fits in.
Birne Binegar: We do row Echelon form and so.
Birne Binegar: Now we'll write down the solution and we just read off remember these columns are attached to variable so I can just write down the solution you'll be.
Birne Binegar: One times X one, let me put the variables here explicitly one times X one no term no contribution is equal to one so that first row corresponds to X one is equal to one.
Birne Binegar: will have zero times X one one times X zero times X three on the left hand side of the equation corresponding the second row but that's simply going to say that X is equal to two.
Birne Binegar: And then the last row simply says X is equal to 30.
Birne Binegar: i've started with a system of equations I performed operations that don't change the solution set and I end up with this set of equations, but this is clearly the solution of the original in your system.
Birne Binegar: Now.
Birne Binegar: Generally speaking in this course linear algebra we're not done yet because we found a solution.
Birne Binegar: But we want to write the solutions as vectors and so all you do to finish this problem is.
Birne Binegar: form the solution vector X and it's going to have components one first component is once happy components, two and three, and if you want to display this solution vector as a hyper planes simply put it inside.
Birne Binegar: curly braces to indicate a higher plane is the set of vectors in this case it consists of a single vector, and so this would be your quote hyper planes solution.
Birne Binegar: But it's kind of a trivial hyper plane consists of a single point.
Birne Binegar: Okay, so.
Birne Binegar: See i've stepped through all this stuff and i'm.
Birne Binegar: The next thing I want to develop, but I think i'm gonna just delay it till Monday, given a technical problem is.
Birne Binegar: i'm now I now have have given you a technique that you can use to solve linear systems, but if you recall, when I first introduced systems of linear equations.
Birne Binegar: I talked about them geometrically first thing I showed you is that if you have one equation in Emma known that the solution set is a hyper plane of dimension one less than the number of variables you started off with.
Birne Binegar: And if you have multiple equations what you're seeing geometrically is the intersection of hyper planes.
Birne Binegar: And what was nice about that visual description of solution set is then you can understand that if you have.
Birne Binegar: Two different equations that had the same hyper plane, it would reduce the dimension of the solution, set because they were redundancy amongst the equation.
Birne Binegar: Or you can have contradictions among equations and those will be hyper planes that don't intersect what i'll show you on Monday is what happens during this role reduction process when you have redundancies, how is that going to get exposed.
Birne Binegar: and also what happens when you have contradiction junior equations look like I don't see any obstructions to this process that would that would fail, even if the equation for contradictory.
Birne Binegar: yeah so so here's here's what will happen if you have a contradiction in your equations you'll get a row at the bottom of the matrix.
Birne Binegar: where you have a non zero number in this last call, and you have all zeros here, because in this case.
Birne Binegar: The reaction will still go through, but what will happen when you try to convert this to an equation they'll say zero times X one plus zero times X, to the zero times X three is equal to one or simplify zero is equal to one that's how you'll spot the contradiction.
Birne Binegar: and
Birne Binegar: What about redundancies way redundancies appears you start off with say a system of equations three unknown, but during your role reduction.
Birne Binegar: You get zero RAL you move it to the bottom of the matrix zero is a vacuous equation is basically say zero is equal to zero.
Birne Binegar: So, instead of having a system of equations and three are nouns one of the equations coming from the augmented matrix is trivial.
Birne Binegar: And so, your system it's really equivalent to a system of two independent equations.
Birne Binegar: And so zeros at the bottom of the augmented matrix indicate that your original system had redundancies and a non zero entry well a pivot in this very last column is going to indicate that your original set of equations had a contradiction.
Birne Binegar: i'll go through that much more slowly with many examples on Monday.
Birne Binegar: Let me remind you that your homework set to is due on Monday and i'll probably assign homework set three to be do the following Monday okay.
Birne Binegar: replace they're either going to be knocking it off with your loving each other or one interception right.
Birne Binegar: Like there's never going to be a place where the angle hits it's where the intersection points like block other than alive.
Birne Binegar: Ah, it will always be in you think of point aligners a generalization or playing that goes off into infinity in two different direction that will always be a hyper plane so.
Birne Binegar: it's hard it's hard to visualize and four dimensions and higher but So what we do is make an analogy was what we can visualize and then.
Birne Binegar: yeah what will always happen i've been playing to intersect don't intersect in a hyper plane of lower dimension so it'll be a line of plane or another hyper plane, but it just will have less degrees of freedom.
Birne Binegar: yeah yeah.