Birne Binegar: Okay we're a little early today, but I can at least start the most people walk in there any difficulties with a web assign a problem set.
Birne Binegar: On some of those you have to enter a matrix.
Birne Binegar: there's a trick for doing so, you may know this, by now, but what the sign is going to give you, you will need to do, like a two by three matrix but you'll see on web assign.
Birne Binegar: it's typically three entry ball blocks to fill in.
Birne Binegar: A two by three matrix what subtle here is what you do is you fill in the first row because you'll have sorry three columns to work with.
Birne Binegar: you'll fill in the first row and then there's a little I forget what the symbol is there's a little symbol down here like maybe a little error or something like if you click on that and then another.
Birne Binegar: set of entry box for the next row of the matrix will occur you just put single numbers, or whatever entries are in those boxes and.
Birne Binegar: Then it's easy to fill out, but the first time you do this, you see you've got to enter a matrix that you only have three entry boxes it's confusing as to what you're supposed to do.
Birne Binegar: But you so you enter row at a time and then you use the little down arrow whatever to bring up the next row.
Birne Binegar: Okay anyway i'm excited assignment to is going to be technically do on Monday i'm I am accepting late homework, so if you don't any given week.
Birne Binegar: You don't get your homework in on time, you should still do your homework because it's good preparation for the exams i'm not going to take any points off for like homework what.
Birne Binegar: What the sign is going to annotate this file title with turned in late, so I can track if people are doing homework or not, and that may become most relevant.
Birne Binegar: At the end of the Semester someone comes into my office with a D and say look I did all the homework yeah but you did all the homework night before the exam you can expect to do do well in the course so.
Birne Binegar: I do keep track of it just so I have some.
Birne Binegar: Some way of understanding, your performance but um but other net is the key thing is always do the homework and particularly this section now what we're doing reduction.
Birne Binegar: it's an easy calculation tends to be tedious but, step by step it's really easy and.
Birne Binegar: You really have to go through a number of these problems before it begins to flow and you just see I got a row, reduce and matrix you just go ahead and do it so that's what we're going to talk about today.
Birne Binegar: Essentially way i've set things up so far.
Birne Binegar: Is this going here, here we go.
Birne Binegar: In the last lecture I introduced augmented matrices it's a way of Encapsulating the data for linear your system into a single matrix and not a matrix equation now what we do is we capture the coefficients and the numbers that appear on the right hand side in a single matrix.
Birne Binegar: From that matrix.
Birne Binegar: It effectively represents the system of equations and rather than manipulate equations themselves what we do is we manipulate that.
Birne Binegar: matrix representation of the equations and we manipulate them using operations on the roads of that matrix that correspond to operations on the original set of equations that did not affect the solution set, and so what we do is we take.
Birne Binegar: Not we take a linear system convert it to an augmented matrix and then we start doing these country row operations, to put it in a special form so called we do throw Echelon form that will be.
Birne Binegar: The form of the augmented matrix when you've actually reached the solution of the up the linear system and, once you get there, then the the idea suggest to convert back to equations and write down the solution.
Birne Binegar: So what are we going to be focusing on today is.
Birne Binegar: How we take it.
Birne Binegar: A matrix and we convert it into reduce relational and forth.
Birne Binegar: Right so actually have several slides that we're going to replicate what I guess said, you have a linear system you grab all the coefficient of the variables you put them in the.
Birne Binegar: left hand side of this augmented matrix this vertical line is just a separate the coefficient data from the numbers that appear on the right hand side.
Birne Binegar: Oh, I should also point out.
Birne Binegar: Because this will be important, later on, you think of this as a coefficient matrix and you take, think that the individual columns is being tied to the variable so the second index is going to tell you which variable its associated with this call it should be X M and the last call.
Birne Binegar: that's relevant as soon as you try to.
Birne Binegar: convert an augmented matrix back into equations this first row is going to correspond to the equation.
Birne Binegar: A one one now we use the association of the first column with diverse variable a one one plus a one two X two plus dot dot dot plus a one m X.
Birne Binegar: equal to one, so we can go back and forth between augmented matrix ease and equation what's nice about this is we don't have to write down all the variables we're all the plus signs and other rotational apparatus if we simply you're trying to track how our equations are changing.
Birne Binegar: Just represent them by augmented matrix sees.
Birne Binegar: All right, and then.
Birne Binegar: What we can do.
Birne Binegar: To those augmented matrix cases we can manipulate them.
Birne Binegar: In a way that will lead us at the end back to the equations of our solution.
Birne Binegar: So there are three things that we can do to an augmented matrix one is.
Birne Binegar: Switching the orders of the row just flipping road and that that's kind of innocuous if you think of it in terms of the equation that just needs changing the orders of equations.
Birne Binegar: We can also replace a row by scale or multiple of itself so think of these as real vectors so this each row is going to have lots of components and Lambda times a row meantime multiplied every element in that row, by the same number Lambda.
Birne Binegar: and
Birne Binegar: it's important that you don't choose Lambda equal to zero, because what would that do if you multiply anything by zero it's going to produce zero so what happens is you take a row multiply by zero you.
Birne Binegar: destroyed all the information that was in that row, because everything that was non zero there now must be zero so when we do the scale of multiplication we do restrict.
Birne Binegar: Lambda tonight equal to zero, so we don't lose information via these operations and then the last elementary row operation corresponds to taking two equations.
Birne Binegar: Adding them together and replacing one of the equations with that the new son or, more generally, taking the equation and replacing it with a sum up the original equation, plus a multiple for a second question alright So these are the three operations were allowed to perform in.
Birne Binegar: manipulating an augmented matrix.
Birne Binegar: Until we got the.
Alex: augmented matrix of the solution.
Birne Binegar: Right so here's the Program.
Birne Binegar: right down the augmented matrix will try to convert the augmented matrix into reduce row Echelon form i'll remind you that just on the next slide.
Birne Binegar: But the important point at this stage is just remember that the reduced row Echelon form of the argument and matrix is the augmented matrix that we're trying to get to the one that's going to correspond to the solution.
Birne Binegar: And then, once we get the augmented matrix of the solution we convert back to equations and we'll get the equations of the solution.
Birne Binegar: And then the last step is last two steps from maybe maybe I won't even get to these today, but the last step is because our solution to a linear equations are always nice linear surfaces hyper planes more exactly.
Birne Binegar: The last step in our scheme for solving linear systems and linear algebra is to get to that hyper plane that athlete consistent all vectors that satisfy the original set of equations.
Birne Binegar: Okay, but today we're going to work on role reaction that's step two.
Birne Binegar: Alright, so.
Birne Binegar: Let me remind you.
Birne Binegar: What row Echelon foreigners in general, and you have a matrix you can have non zero entries anywhere 601231 okay so.
Birne Binegar: These are just a random matrix.
Birne Binegar: What we're trying to do here is identify a particular form of an augmented matrix and get to come in two steps first you have a notion of pit it.
Birne Binegar: These are the first non zero entries in any given row, so I underlined, a pivot in the three by three matrix the first row here's the first non zero.
Birne Binegar: second row here's the first non zero entry and that's the first non zero entry in the third row What we require is that these pivots they occur in a downward staircase if it's the matrix is in relational on porn.
Birne Binegar: So this is not getting colored.
Birne Binegar: Look at the first row in the second row the pivot appears to have moved off to the left, but that's not allowed.
Birne Binegar: not rational inform what happens interviewing real Echelon form is, as you move down the matrix pivots have to be moving off to the right, so this is not in one form.
Birne Binegar: But a matrix like 10101001 well you're the pivot and that's required of rotational on form the pivots are occurring in this downward staircase from the upper left to the lower right.
Birne Binegar: Right so that's all you're looking for visual.
Birne Binegar: way of identifying and a matrix in row excellent.
Birne Binegar: This is halfway to the arm augmented matrix system, reduce rashwan form.
Birne Binegar: To get to reduce your national form we have two other requirements.
Birne Binegar: What we want.
Birne Binegar: Is a matrix to be in real Echelon form but also you want all the pivots to equal one so anytime the first non zero entry any given row has to be the number one rather than any other.
Birne Binegar: real number, and then the other thing we require is that when we look above the pivot or below a pivot in the same column, you only have zeros.
Birne Binegar: So give you an example that let's see 1132.
Birne Binegar: Let me put a zero here 01210000.
Birne Binegar: All right, so this matrix is in reduced row Echelon for because.
Birne Binegar: First of all, it's unreal I salon for there's only two pivots in the entire matrix is 00 you'll always put them down at the bottom of the matrix if you do a real real entertainment get the zero down at the bottom if you're trying to reach reduce relational on on and then.
Birne Binegar: The pivots here, you do have that downward staircases like descent through the matrix the pivot in the Rose are moving off to the right.
Birne Binegar: And then we could have non zero entries in the matrix but they're not the pivot entry pivot entries have to equal one that's this condition here and then.
Birne Binegar: last thing where requires that when we look at a column that contains a pivot not only the pivot one but there's no other non zero entries in the entire column see here's the pivot these entries are zero.
Birne Binegar: here's the pivot these entries are serial in these columns that don't contain tickets it's unrestricted Okay, but it turns out that this form.
Birne Binegar: is actually unique if you start with a random matrix there's only one matrix in reducing Echelon form that you can reach by applying elementary row operations.
Birne Binegar: The task of today's lecture is to actually show you how you do that, how you take an arbitrary matrix and get it in the dis reduce row Echelon form.
Birne Binegar: Now.
Birne Binegar: what's nice about it, for some people, is that the algorithm itself is largely visual you look at the matrix and you then you decide what to do next.
Birne Binegar: But i'm going to show i'm going to step through an example first and then i'll give you some rules to follow, but it's pretty clear from the definitions of reducing rationalize on form you think about it, what to do next kind of becomes obvious.
Birne Binegar: Right and, again, the reason why we're trying to find matrices and reduce relational on form is that those are going to correspond to the solution matrices of a linear system.
Birne Binegar: And this process by which we convert an augmented matrix into a traditional Echelon form that's called row reduction textbook may call it gaussian elimination but we'll just use the terminology row reduction.
Birne Binegar: Okay, so let's do a first example.
Birne Binegar: Here we have a four by a look like a four by five matrix it's not in row Echelon form, so it can't be reduced relational and form.
Birne Binegar: it's not in row Echelon form because this pivot in the first row by descend I see there's a pivot off to the left of it in a lower row well, that means it's not in row Echelon form.
Birne Binegar: The other thing which see if we have a zero row and in a row Echelon form the zeros always have to be at the bottom of the matrix.
Birne Binegar: So the first thing i'll do to this matrix is well i'll move this bottom row that the second row to the bottom, and the way i'll do that as well, simply interchange the second and fourth row.
Birne Binegar: Right that's easy to do you just see what happened here is two rows switched position in the matrix and now.
Birne Binegar: We have a matrix or at least the zeros at the bottom, but we still have the problem that the pivot in the first row is to the right of the pivot in the second row staircase is going the wrong direction, and so what we'll do is we'll.
Birne Binegar: We can do another row interchange to put a pivot up in this corner and then that will mean that you get a pivot up in this upper corner here, that means at least the first row is enroll Echelon for until what i'll do is i'll simply flip these two rows.
Birne Binegar: All right now that's a little bit better because.
Birne Binegar: matrix is not yet in real my salon form because it's like descend from the second row to the third the pivot moved off to the right but.
Birne Binegar: At least the first two rows Horn now in row Echelon polo and the strategy is going to be to start start at the top in fact start in this upper.
Birne Binegar: left hand corner moved it down to the matrix and step by step, ensuring that the pivots are moving off through the off to the right, so the next step is going to be i'm just rewriting this matrix I see that a problem.
Birne Binegar: With this matrix.
Birne Binegar: occurs in the third row this term here, if I can get rid of it replace it by zero I would be.
Birne Binegar: In row Echelon form, but I cannot simply erase it you put her zero there, I have to use one of the elementary operations.
Birne Binegar: So what we'll do in this case is will try to add a multiple of this first row, so that the first component will exactly cancel the first component there and then we'll get a zero here, but we did show using one of the operations that's not change the solution sets.
Birne Binegar: So i'm going to do is i'm going to subtract the first row from the third that's legitimate combination of the third row in the first row, you can think of those as our three plus minus one times, or one, so it does have the form of these.
Birne Binegar: Real operation to where we send a row do it some with multiple up another row that's the operation described here and what's going to happen.
Birne Binegar: let's do this for the most part in our heads i'll have.
Birne Binegar: dealt with the third row and i'm going to subtract the first row from it, so one minus one.
Birne Binegar: Replacing your third row so one minus one is 00 minus minus one is negative one one minus zero is one zero minus one it's negative one zero minus one is negative one, so you can do these.
Birne Binegar: These short little calculations of each step pretty quickly you just have to write things down neatly so you don't make trivial mistakes, but if you write down things.
Birne Binegar: nicely, you can immediately write down the result of a given elementary real operations Okay, well, we were successful you getting rid of that.
Birne Binegar: pivot in the third row we effectively moved it over to the second call, but now we have still another problem now if I descend in rows.
Birne Binegar: here's a pivot city directly below it pivot as you move to a matrix in row Echelon form you can't have tickets are stacked on top of one mobile they always have to be moving off to the right as you descend through the matrix.
Birne Binegar: So how can we get rid of this.
Birne Binegar: simple thing to do would be to take the third row again and add the second row to it, then we'll get a cancellation of that lady leading entry.
Birne Binegar: Okay, so let's do it.
Birne Binegar: will have i'm going to.
Birne Binegar: Take the second row add it to the third and and use that some to replace the third row so i'll have zero Plus zero is equal zero effectively adding these two road zero Plus 001 minus one is 03 plus one is for one plus minus one.
Birne Binegar: I guess I got a mistake don't I.
Birne Binegar: yeah I did something here.
Birne Binegar: Why am I changing the second row end up getting changed somehow Okay, so I got i've got a typo on my slide let me.
Birne Binegar: Let me do this properly, then i'll work directly here.
Birne Binegar: This is kind of go to 11011 second row is not changed me 312 and then here what's happening is the we're going to change the third row by adding the second row to it so i'll have.
Birne Binegar: zero Plus 001 plus minus one is 03 plus one is for one minus one is zero and two minus one is one, and then the fourth row.
Birne Binegar: Is unperturbed by this operation that only affects.
Birne Binegar: row three.
Birne Binegar: All right, and so yeah and now we stop because we've got this matrix in at least row Echelon.
Birne Binegar: Alright, so real I salon for miss the first.
Birne Binegar: Special form that I talked about it's simply when you've got the pivot occurring in a downward staircase off to the right and you've got your URL at the bottom.
Birne Binegar: This is um.
Birne Binegar: When you reach Roche long form this is like the first half.
Birne Binegar: reduction.
Birne Binegar: Can we do.
Birne Binegar: Okay, so there's good, we still have some more work to do on such a matrix if I wanted to get it all the way to reduce your form.
Birne Binegar: But.
Birne Binegar: The subsequent steps at least the second The next step is really easy.
Birne Binegar: Let me just before I do that, let me just say verbally what I did in the last example I had a zero row in the matrix So the first thing I did is I moved that zero row to the bottom of the matrix.
Birne Binegar: easily done, and then I didn't have a pivot in the first row, the idea is you can start up in this upper corner and use it to entries to clear out the entries below and then you'll descend down.
Birne Binegar: But to get started, you need something up in that in that corner and so, in the second step, what I did is I flipped the first and second road so that I can get a pivot in this upper left corner.
Birne Binegar: That tape it was then used to clear out the entry in the third row, and then we repeated the process, looking at the second row and its relation to the third row, so what you do.
Birne Binegar: When you have pivot sitting in the same column you take the upper pivot and you use it to cancel out the three in a row below it and typically you do it something like this.
Birne Binegar: If I was working in the first row.
Birne Binegar: Let me describe what's going on here if some if you have a row, like a one one.
Birne Binegar: dot dot dot a one, and you have another row down here a I.
Birne Binegar: One a I am so this is the first row.
Birne Binegar: And here's the I throw.
Birne Binegar: How can I get rid of this guy well you what you're going to do is you're going to take a multiple this row and add it to this so that this term gets cancelled well what you'll have to do is, let me just focus on this third row what i'd want to do is multiply by minus.
Birne Binegar: One one over a I J.
Birne Binegar: Times that first row that i'd have a one one.
Birne Binegar: Now I have a I want em diva headed by a sorry a one a one times a one m the that those are the entries first entry last entry in the I throw what i'm trying to do is replace this row with a multiple of this row, so that this term cancels, and so what you want, on effectively doing here.
Birne Binegar: Sorry, I got the truth and wrong and I one and one one here so.
Birne Binegar: When I multiply every term in this row by divided by a one one i'll get that cancellation and then much survives the multiplication will exactly cancel out that term there so it's pretty easy to see.
Birne Binegar: What multiple on a per row, you need to eliminate a pivot entry below below the pivot and metro.
Birne Binegar: will be lots of examples, but dumb again it's not so clear when I write it out in prose exactly what's going on, but after we after you do.
Birne Binegar: Enough of these row reductions it's going to be P appear very natural what you do next.
Birne Binegar: Alright, so.
Birne Binegar: yeah and so you do that for the first road and then you just think of.
Birne Binegar: Having fixed the problem with row one, then you descend through the matrix and you do the same thing until you've gotten all the way down to this bottom corner and you can all the pivots as far as you can.
Birne Binegar: you've got this downward staircase formation of those credits.
Birne Binegar: Alright, so yeah and the steps here are just the steps I repeated before there's a zero you move to the bottom, if if there's a.
Birne Binegar: there's a entry in a pivot below the pivot in the second row what you do issue.
Birne Binegar: You just do a.
Birne Binegar: addition for multiple have another road to clear out that pivot entry in the lower row, and you just keep repeating this process moving down the matrix row by row.
Birne Binegar: And then the stage getting to reduce your life long form it's just going to terminate naturally when you reach the bottom of the matrix.
Birne Binegar: All right now to get a matrix is we do show Echelon form it's got to be enroll Echelon form with two additional conditions, you need the pivots the all equal one.
Birne Binegar: So here's a matrix it's in row excellent form but it's not in reducing Ashkelon form.
Birne Binegar: Primarily, at least the first instance, because these pivots are numbers that are not equal to one but that's easily resolved right.
Birne Binegar: To make that pivot one i'll divide the entire row by one half to make this pivot one I can multiply the entire row by one third, and similarly third row we multiply by minus one alright so.
Birne Binegar: Getting the pits to equal one that's easy you just look what what you pivots are and you pick a suitable number the multiplicative inverse of the entry multiply the entire row by that I don't just change this leading entry I multiply every term in the row by one half.
Birne Binegar: And then I get a new matrix or now the first two conditions of the three conditions required for a matrix in reduce your actual on former now satisfied.
Birne Binegar: Right, then the last step is to zero, the entries that are lying about pivot remember that when a matrix is in reducing electronic form.
Birne Binegar: rotational on form pivots or equal to one and, in addition, the pivots have to be in a in a given column, they have to be the only non zero entry in that call here, we got a two sitting above a one sitting above the pivot one, so I have to get rid of this guy.
Birne Binegar: This next stage is a lot, like the first stage, except the direction which we work is the opposite, when I was going.
Birne Binegar: For real Echelon form, I started in this upper corner and work my way down to the lower right hand corner.
Birne Binegar: What we do in this last stage is we start in this with the bottom rose and we try to clear out the entries above and then we proceed to the next, higher row and clear out the entries above there.
Birne Binegar: But the mechanism by which we do that it's essentially the same as we did in row reductions, all I do let's see I must will get to an example.
Birne Binegar: Okay, so here's a matrix that in row Echelon form the pivots are all equal to one.
Birne Binegar: To get it in reduce your national unformed I need to empty out the non zero entries that light above tibbetts.
Birne Binegar: we've already got zeros below the pivot, we need to clear out the entries above pivot so we start with the last row and we'll use it's only non zero entry to clear out the entries and last entries in row to enroll one.
Birne Binegar: So this is.
Birne Binegar: right on the crack the blackboard let me see if I can move it a little bit.
Birne Binegar: Alright, so.
Birne Binegar: To get rid of this to what i'm gonna do is i'm going to replace row to.
Birne Binegar: Another typo.
Birne Binegar: i'm going to replace the second row like it's.
Birne Binegar: Different with two times row three so let's do that i'm.
Birne Binegar: Starting with the second row now and then i'm going to subtract two times the third row from it so zero times three times 00 minus three times two times zero is 01 minus two times zero is one and two minus two times.
Birne Binegar: One is going to equal negative two or that term will be zero until we get we get this matrix and good Oh, and we also did something else as well i'm also.
Birne Binegar: Wonderful take care of the first row, at the same time, so what i'll do to clear out this entry sitting above.
Birne Binegar: The pivot in the bottom row what i'll do is i'll subtract one half times the bottom row from the top row.
Birne Binegar: that's that operation and then what will happen is one minus one half times zero is one two minus one half time zero to one half minus one half times, one is zero.
Birne Binegar: All right, good.
Birne Binegar: This pivot.
Birne Binegar: And this pivot those columns look good, the last thing last issue we have is that above the pivot in the second row, we have a nonzero number and so all we do is take we got to get rid of this guy so we'll use the bottom row.
Birne Binegar: it's a some with the first row in order to get rid of this entry, and so what we end up doing is subtract from the first row two times the second row and there there we have it.
Birne Binegar: In.
Birne Binegar: Maybe three steps we've gone from this matrix to reduce row Echelon form.
Birne Binegar: Okay, so yeah.
Birne Binegar: we'll do lots of examples of this, but.
Birne Binegar: The most important thing to understand the process it to know the definitions of the forms of the matrix you've got to be able to visualize them to see what to do next.
Birne Binegar: But once once you do, that the first stage is going to be the start in the upper left hand corner and use the approach to clear out entries clear out pivots that occur in rows below it you keep doing that slowly descending to the matrix until you get the matrix in relational and.
Birne Binegar: Then you can quickly reach scale the pivot entries so that they're all equal to one.
Birne Binegar: The last step will then be to start in the lower right hand corner and then figure out what to do with lower road in order to eliminate the entries sitting above the pivots in the lower rows.
Birne Binegar: And it's pretty easy to spot, for example, what do I have to do to get rid of this guy well i'll take minus two times this term will cancel that so I simply.
Birne Binegar: Take this row and I summit with minus two times the third row, and that will get rid of that as well or give me a new row well we don't have that problem.
Birne Binegar: Okay.
Birne Binegar: So that's that's the process.
Birne Binegar: that's now.
Birne Binegar: implemented.
Birne Binegar: will do the full shebang we'll start with a linear system.
Birne Binegar: will convert it to an augmented matrix we're real reduce that all come Nick augmented matrix then i'll convert back to equations and you'll see that he did i've got the solution in my hand.
Birne Binegar: Oh, oh here's three equations and three unknowns.
Birne Binegar: first step.
Birne Binegar: Take the data, the coefficients the number of equations and the number number of variables use that information to encapsulate or.
Birne Binegar: to rewrite the data that goes into specifying this system captured all in a single augmented matrix.
Birne Binegar: Remembering.
Birne Binegar: That, although I didn't write them down at least implicitly the variables are still there it's just that they're imprisoned really attached to certain columns of this augmented matrix.
Birne Binegar: right, then, once we do that i've got an augmented matrix will start to roll use it.
Birne Binegar: Right so here's what we start off with this is the augmented matrix corresponding to the original linear system.
Birne Binegar: Alright, was one of the problems we've gotta pick it up in this upper left hand corner great I don't have to do any manipulations to get started.
Birne Binegar: I can immediately start eliminating pivots below the pivot in the first row to get rid of this guy offend row to to row to minus row one right and then these leading entries are going to end up canceling and i'll send row three.
Birne Binegar: Two row three minus well.
Birne Binegar: i'm gonna have to track two times the first row in order to get a cancellation here minus two times, Rule one.
Birne Binegar: And I think that's what I did.
Birne Binegar: That.
Birne Binegar: Yes, so that's what you do and then.
Birne Binegar: i'll do one of these let's see one minus 101 minus negative one sorry one minus one is zero negative one minus one.
Birne Binegar: here's the here's the air I just committed it for you, but be aware, remember your frequently or subtracting moving up rather than downward so one minus one is zero.
Birne Binegar: Negative one minus one is going to be negative to one minus one one minus a negative one it's going to be plus two and two minus zero is going to equal to okay so and then similarly we Catholic what happened to the third row and now we've got a new matrix where.
Birne Binegar: We need to eliminate this guy.
Birne Binegar: Right, this is in real life one form up to the second row, but as soon as we get down to the third row, we have a pivot below the pivot in the second row.
Birne Binegar: But.
Birne Binegar: i'm not going to do that for directly, I mean I could I could take.
Birne Binegar: To the third row I could add let's see minus.
Birne Binegar: minus three halves times the second row to get the cancellation, but I i'm I recommend you don't do that, because then.
Birne Binegar: Once you start multiplying rose by fractions of fractions are going to play you for the rest of the calculation.
Birne Binegar: So what's easier to do is go ahead and make that pivot equal to one we're going to do that eventually anyway let's do it now, because then it's easier to see what I have to multiply the second row by in order to get rid of that term.
Birne Binegar: So i'll just reschedule that second row.
Birne Binegar: And now it's much easier to describe what we get what we're going to do to get rid of this term will simply multiply the second row by three and then add it to the third row.
Birne Binegar: And so what will pop out of that if good we've reached row.
Birne Binegar: Echelon for.
Birne Binegar: Because the pivots are now in that Nice descending staircase and that's that first half.
Birne Binegar: rho reduction.
Birne Binegar: To reduce.
Birne Binegar: As platforms that's we've completed.
Birne Binegar: All right, there's more to do because we got to get all the way to reduce your actual on form.
Birne Binegar: Alright, so.
Birne Binegar: The next step is to take that matrix in row excellent one, that we just computed.
Birne Binegar: And this is the result of the last line on the last slide what we'll do is we'll convert that pivot equal to one.
Birne Binegar: All right, and now we got a matrix in row actual on form pivots are all equal to one.
Birne Binegar: We got to get rid of these guys.
Birne Binegar: Because they're sitting above these are non zero entries that are sitting above pivots and lower rows but it's easy to see what to do, you get rid of this guy I would add a third row to the second row to get rid of this guy I would add the third row to the first row.
Birne Binegar: note here on starting in the.
Birne Binegar: The first half of the production, I work it from this corner down to this corner the second stage what you do is you work from the bottom right and you try to work your way often up to the upper left.
Birne Binegar: still working first throughout the entries above this pivot in the last row.
Birne Binegar: see if we can do that alright so easy to see what we need to do, for the first row to get rid of this guy well, I had the Third Road to it and, similarly, will do exactly the same thing to the second row.
Birne Binegar: All right, good this columns okay that columns Okay, but here we have a problem.
Birne Binegar: So we'll add will end up subtracting the second row from the first to clear out that entry.
Birne Binegar: There we go and now we've got good a matrix that now in reduced row Echelon form because it's in relational on form.
Birne Binegar: pivots are all equal to one and above and below pivots we have nothing but zeros good alright.
Birne Binegar: The real reduction.
Birne Binegar: stage or on the road action part of the calculation of solution to the original into your system is now finished.
Birne Binegar: What we do next, if we take this augmented matrix which by the way, it's going to have exactly the same solutions as the original linear system because we performed operations that don't change solutions what we'll do is we'll convert back to equations.
Birne Binegar: To do that.
Birne Binegar: See there's the augmented matrix.
Birne Binegar: call that X one X two X three remember the columns on the coefficient side of the augmented matrix are attached to particular variables until once we.
Birne Binegar: realize that attachment we get the corresponding equations be the first row we get one times X one zero times X two zero times X three is equal to one or.
Birne Binegar: You simplify that simply X one is equal to one, and the second row.
Birne Binegar: You can construct the equation from the augmented matrix and then you can simplify it to simply saying X two X equals two and the third row basically tells us X three has to equal three and we're done.
Birne Binegar: The.
Birne Binegar: we've got three unique numbers.
Birne Binegar: That.
Birne Binegar: This equation this set of equation has the same solution as the original system, but in fact this is something that you would just anyway call the solution of the system, because each variable is set to a particular number.
Birne Binegar: Okay, so that's our technique.
Birne Binegar: and
Birne Binegar: I guess what i'll try to do next is i'm.
Birne Binegar: Oh there's one last step here the last step that will employ uniformly is once we get the equations of us have the solution is to realize that those.
Birne Binegar: vectors that satisfy those equations we want to think of them as vectors not as as in terms of variables, so what I found on the last slide is that X one had to equal one X two had equal to X three had equal three so my solution director is three components are one, two and three.
Birne Binegar: And we can think of that is a hyper plenty.
Birne Binegar: If you think about hyper plane as being up this particular forms, generally speaking, you start with on point, and then you start straying from it and a bunch of different directions, however.
Birne Binegar: That parameter T that I raced that's usually a free parameter, and the solution means you have more than one solution, but when you have a unique solution is in this case that hyper planes solution is just going to be this vector.
Birne Binegar: encapsulated in a set because i'm talking about the solution set rather than the solution.
Birne Binegar: But.
Birne Binegar: The case when you have a unique solution there's not that much difference between a set of solutions and the solution itself.
Birne Binegar: Okay, so.
Birne Binegar: we've done one complete example.
Birne Binegar: We proceeded if we.
Birne Binegar: took our linear system wrote down the augmented matrix.
Birne Binegar: US road option, the first reach.
Birne Binegar: row Echelon form so at this stage.
Birne Binegar: you're starting here and you're trying to move downward through the matrix until clearing out pivots as you move along until you reach this bottom row.
Birne Binegar: Then, the next step.
Birne Binegar: The next step will be.
Birne Binegar: In a row reduction to reduce your last one form and that's consistent to sub steps first you convert all the pivots equal to one by multiplying the row by inappropriate factor, and second, of all you use.
Birne Binegar: A process of adding multiple to have one row to another to remove the entries that are sitting above the pivots in the lower rose up.
Birne Binegar: When you reach real Echelon form below the pivots you can only have zero, but in this stage what you're largely trying to do with getting rid of the pivots that are sitting up entries that are sitting above tickets.
Birne Binegar: Then you convert back to equations and the last step is to reformulate the solution as a set of vectors.
Birne Binegar: Okay now we're going to do some more examples, the purpose of the.
Birne Binegar: Examples that i'll conclude today's lecture with is just to make contact with our geometric understanding of.
Birne Binegar: equations remember last week, when I was talking about equations I said, you can think of any equations that the solution hyper planes.
Birne Binegar: And that, when you have periods of equations there would be intersections of hyper planes and that, for example, if you had equations that were redundant, with one another, that would mean that the hyper planes athlete coincide with one another, well, but I also said that.
Birne Binegar: In three dimensions that could be kind of obvious when you plot the hyper planes, but when you have more variables and.
Birne Binegar: That Those sort of redundancies amongst the equations are can't can be hidden.
Birne Binegar: And I mean look at this example.
Birne Binegar: yeah I don't I don't see any obvious redundancies in this in this set of equations but i've worked this out example out beforehand, and there is going to be a redundancy i'm going to show you how it's exposed as we do reduction.
Birne Binegar: Alright, so, first of all, since we have three equations and three unknowns.
Birne Binegar: I evenly we're going to expect the solution space to equal, the number of variables or leftover so you take the number of variables you subtract the number of equations.
Birne Binegar: And so we should have a unique solution, no be no for should be no free variables in the solution if we have a solution at all, but this is naive and what i'll show you, is it doesn't actually happen in this case.
Birne Binegar: Alright, so let's convert to an augmented matrix to easily done.
Birne Binegar: And then we'll roll reduce this augmented matrix.
Birne Binegar: To just follow our procedure So here we have a pivot in the first row it's only off to the left, which is a good spot for it so i'll use it.
Birne Binegar: Effectively, this guy to eliminate these entries below go to the row operation simple achieve that, then once we've.
Birne Binegar: done that we see we've got in row Echelon porn up to the second row, but here we have a problem in the third row so we'll add a multiple the second row to the third to eliminate that entry.
Birne Binegar: And now we've got to make sure in reduce rational on porn good first half of reduction is complete.
Birne Binegar: Now we'll move on to the second stage first thing we'll do is we'll fix those tickets i'll make that equal to one.
Birne Binegar: And then the last thing I need to do your my two pivots don't worry about this guy here it's not sitting above any pivots so it's okay to be non zero, but in this entry here it's sitting above the pivot in the second row, so I have to get rid of it.
Birne Binegar: And so, simply subtract the second row from the first good now, I have a matrix that can reduce show Echelon form.
Birne Binegar: And i'm noticing that I have the zero at the bottom.
Birne Binegar: I started off with three equations and three unknowns.
Birne Binegar: If you walk down the equation that corresponds to this last row it says that zero times X one plus zero times X two zero times X three is equal to zero, which is simply saying.
Birne Binegar: Zero is equal to zero it's not putting any restriction at all on the variables, so, in fact, if you have a zero row it corresponds to a vacuous equation.
Birne Binegar: And so attack my linear system, it has the same solution as as the original set of equations the.
Birne Binegar: equations corresponding this matrix you're going to have the same solution set if the original in your system, but one of the equations corresponding to this augmented matrix is trivial, and so I could ignore it, and my system is now a system of equations and three unknown tactically.
Birne Binegar: And so redundancies somehow hidden in the original set of equations have now been exposed and trivialized.
Birne Binegar: Okay that's good I mean it just means that our procedure is effective in getting rid of the redundancies that sit inside the equations.
Birne Binegar: Right and then we've reached this equation you reduce your actual form now will write down this.
Birne Binegar: The corresponding equations.
Birne Binegar: First column is X one X two X three.
Birne Binegar: I write down the equations for this augmented matrix it works out to be this or simply.
Birne Binegar: Just dropping these zero terms X one plus x rays equal to one X two is equal to one and zero is equal to do so, these are the equations of the solution, but.
Birne Binegar: they're not in a nice form, where every variable is set to a specific number here, the first equation involves two variables.
Birne Binegar: and
Birne Binegar: Unless sorry, let me, let me step through this quickly the idea here is that we regard X three is the free variable we take the pivot variable.
Birne Binegar: X one and we use we if any other terms appear in the corresponding equation i'm going to move them over to the right hand side and think of that as.
Birne Binegar: A free variable and what will happen in this case, you know just move that X three over do it really Okay, so I guess i'll just stop here because i'm already a minute over the idea is that.
Birne Binegar: When you have non zero entries your row the pivot entry is going to be the variable you can solve for and the other the other term it's going to be moved over to the right hand side is the free variable.