Birne Binegar: take some time to set up.
Birne Binegar: till today.
Birne Binegar: actually begin solving systems of linear equations.
Birne Binegar: technique help describe is what the goal is to take a set of linear equation that means equations where you maybe have em variables and the way the berries was appearing equation is linearly meaning they appear in separate terms from other variables and.
Birne Binegar: At worst, are multiplied by a constant so doing explicit example.
Birne Binegar: To X.
Birne Binegar: To y Z three what makes it linear is there, your variables they occur in different terms and it works there multiplied by constant.
Birne Binegar: You can have turns that don't involve the variables at all, but those numbers are moved over to the right hand side of the equation.
Birne Binegar: One rewrite the equation standard form, so what i'm going to describe this how to solve it once you have equations like this it's not the same thing is solving quadratic equations or.
Birne Binegar: Other sort of equations that you more used to solving the barely that the simplest kind of equations but what problem is, we may have lots of them, and we want to develop a systematic technique for solving them.
Birne Binegar: So i'll start by reviewing what linear systems are and their solution set and then.
Birne Binegar: Then i'll show you how we solve them in linear algebra it's going to be different than the high school technique where.
Birne Binegar: we're in you start take the first equation use it to eliminate one variable or actually to express one variable in terms of the other variables.
Birne Binegar: That you substitute into subsequent equations and you pick another variable to eliminate you keep doing that, until you get down to.
Birne Binegar: Your out of equations and you can eliminate any more variables.
Birne Binegar: The variables that remained a call the three variables in your solution and everything can be expressed in terms of them all the proceeding variables can be expressed in terms of your free variable well.
Birne Binegar: That process works great if you have two equations into unknown to only take like one step to solve for.
Birne Binegar: The first variable, but if you have like 10 equations and 18 unknown, the first step, once you solve for state X, one would be this substitute that expression for X one into nine more equations.
Birne Binegar: And then you can eliminate a second a second variable so it's.
Birne Binegar: It starts off simple, but if, when you have anything other than a very small set of equations there's lots of substituting that goes on that's a good place to make errors.
Birne Binegar: What we're gonna do is develop a technique that's going to seem much more strenuous at first, but in fact our algorithm is going to make it really efficient.
Birne Binegar: Our strategy is going to be well, you have a given set of equations you want to get to a set of equations that describe the solution X one is equal to three x is equal to four etc.
Birne Binegar: So what we're gonna do is we're going to take the equations rather than concentrating on variables we're going to manipulate the equations until we get the equations of the solution so.
Birne Binegar: Instead of working in one equation, the time one variable time we're constantly operating on the set of equations.
Birne Binegar: Once I show you how that's done then we'll make it much more efficient will convert the whole algorithm into working with major cities and then, once that's established will have a real rapid way of solving lot large linear systems.
Birne Binegar: Okay, so, first of all.
Birne Binegar: A set of linear equations.
Birne Binegar: Just means, let me give you a typical example.
Birne Binegar: A one one X one dot.at 1am exam i'm using our notation we've developed coefficients are labeled by which equation there in and the variable that they're tied to.
Birne Binegar: Put numbers on the right hand side that would be one equation, I had any such equations it would look like this dot dot dot plus the coefficient and the empty equation of the empirical the M equals beacon alright, so that would be any equations.
Birne Binegar: unknown.
Birne Binegar: And what we refer to as an end by m linear system.
Birne Binegar: These four things for studying now and our goal, as we saw the preceding lecture is that.
Birne Binegar: solution sets of linear equations are always hyper planes solution sets of multiple linear equations are all intersections of hyper planes.
Birne Binegar: But intersections of hyper planes are just smaller hyper planes hyper planes of lower dimension so we're always be trying to express once we found the solution of such a system will try to express it in this geometric fashion as a point plus.
Birne Binegar: Bearing distances along a particular set of directions that will fill out.
Birne Binegar: In this case of law and ethics directions are independent of one another, this would ask a forum like this would actually create a hyper surface that's up a flat hyper surface over hyper plane of dimension N minus one.
Birne Binegar: OK, so now, and just remind you that these geometric objects that correspond to hyper planes are point.
Birne Binegar: When you just have that initial starting point in your construction you get line if you have a point and a direction from which you can depart from that point.
Birne Binegar: you'll get a plane, if you have two directions, from which you can depart from a given point but more general a general hyper planes just something that form.
Birne Binegar: Right and then the point of hyper planes is that.
Birne Binegar: When we solve systems of linear equations.
Birne Binegar: We only have two options we either don't have any solutions at all, or I solution will be a set of this form and our goal right now is to take us linear system and figure out if it has solutions, and if it does have solution what hyper plane is it.
Birne Binegar: Okay, so.
Birne Binegar: move this up a little bit.
Birne Binegar: So i've already marked how we solve.
Birne Binegar: linear systems in high school and back it's not restricted to linear systems.
Birne Binegar: You can tall non linear equations to buy the same method use the place you'd want equation, to eliminate the variable receptive to that into subsequent equations and.
Birne Binegar: figure out another variable that you can express we're not going to do it that way, what we're gonna do is we're not manipulate the equations rather than the variables and we'll keep manipulating them until we actually reach the equations of the solution.
Birne Binegar: Alright, so.
Birne Binegar: First question is well, what can we do to equations that will not change their solution.
Birne Binegar: i'll give you three basic operations, they should already be familiar to you because you know doubt solve lots of equations already but.
Birne Binegar: we're going to single them out now first so here's i'm just a pair of equations two equations to my known what can I do to this set of equations well, one thing I can do, I can change the order that's not going to affect the solution and it looks really trivial but eventually.
Birne Binegar: This operation is going to be more useful than it looks here, here it looks like i'm just reordering things it's not going to change the solution.
Birne Binegar: But when I get to a matrix rendition of this this is going to have the effect of changing two rows in a matrix so it's a little more substantial than, then you will get from this another thing we can do to us a set of equations look at the second equation it's.
Birne Binegar: Both sides are divisible by two so common that you do when working with equations you simplify them by dividing up common factors that appear on both sides.
Birne Binegar: That doesn't change the solution and so that's another sort of operation, we can do to a system of equations that will not change the solution here, I want you to understand this is we take.
Birne Binegar: Have a set of equations we look at one of them and we place, it is a multiple of itself.
Birne Binegar: And then finally.
Birne Binegar: here's another common operation.
Birne Binegar: This may be a little bit new to you the way it stated, but it's a familiar operation, let me, let me introduce it.
Birne Binegar: more slowly, the way that you typically seen this operation used.
Birne Binegar: If you have an equation like X plus y is equal to three x minus y is equal to two well one way to speed the solution of this is to simply add, not the first equation, to the second equation, then you would get to X, Y plus minus y is zero and equals five and so.
Birne Binegar: I can add equations together, and now I have three equations but the point of this operation is is that I can think of replacing that second equation with this equation that I derived from it, and the first equation.
Birne Binegar: Since since I derived this from these two equations it's also true, I did that by adding these two equations.
Birne Binegar: But the other thing I could have done, if I had these two equations I could recover the second equation, simply by subtracting this equation from the sun.
Birne Binegar: And so, these are also manipulations you can perform on set of equations they don't change the solution and hear what the rule, I want to give this today, you can.
Birne Binegar: Take any equation, and you can replace it, not only with a sum of another equation, but because you're also allowed to scale or multiple equations or multiply equations by numbers, we can actually replace one of the equation by it, some which are multiple have another equation.
Birne Binegar: Right.
Birne Binegar: Here, well, the first the first operation, we could do with changing the order, here we have multiply equations by numbers and we here at the more complicated operation, where we replace an equation by it, some with a multiple of another equation right So what can we do with these operations.
Birne Binegar: I mean just.
Birne Binegar: reestablish my notation if you see this, and the next slide it means i'm.
Birne Binegar: Switching the order of equations you see this i'm scaling multiplying and multiplying each an equation by a number and here i'm doing that replacement of a matrix with it off an equation, with it, some with another equation.
Birne Binegar: Alright, so here's the first example two equations and two unknowns i'm going to manipulate those equations.
Birne Binegar: The first thing i'm gonna do.
Birne Binegar: Is i'm going to take this set of equations and i'm going to replace the second equation, with it, some with the first so X plus x is two X.
Birne Binegar: y minus y is zero so we're not going to appear one plus three is for So you see these these operations are easy to perform even this more complicated one.
Birne Binegar: And we get a new set of equations, which is actually a little bit simpler than what we started off with because, in the second equation, now the variable X is almost isolated.
Birne Binegar: Complete that isolation, by doing multiplication nothing the equation by a number I send the equation to to.
Birne Binegar: One half the equation to divide both sides by two you get X is equal to two.
Birne Binegar: All right, well yeah we know what know what X has to be.
Birne Binegar: To get this solution will need to figure out why as well, but i'm not going to substitute in here i'm gonna work strictly with this process of.
Birne Binegar: Doing operations that don't change the solution to manipulate the equations so next up i'm going to do i'm going to take this first equation and i'm going to subtract from it, the second equation, then i'll get X minus X.
Birne Binegar: But also have y minus zero y so i'll be left with why and then one minus two is going to be negative one sorry my cursor is on that.
Birne Binegar: So, how did I get the second equation X minus X zero y minus zero Why is why in one minus two is minus one so it's it's easy to write and now i'll do one more one more kind of Stupid operation, but just to get this.
Birne Binegar: My solution stated with the variables appearing in the natural order X, followed by why we get the solution Okay, so what you've seen i've done here.
Birne Binegar: Is i've sold this system of linear equations i've gotten to the solution.
Birne Binegar: Not by solving looking at the variables, but rather by manipulating the set of equations using operations that don't change the solution set and so yeah so these two equations this these two sets of equations have the same solution, but this one is readily identified it as the solution.
Birne Binegar: Okay, so.
Birne Binegar: Now what remains for me to explain, is how to make this process, efficient and how to how to.
Birne Binegar: Second, of all, to explain why I chose to do this operation at this point of time, I always have three operations, I can perform on any given equation what determines the sequence of operations that will actually lead me to the the equations of the solution.
Birne Binegar: Well that's going to be easiest to describe when I convert back into a matrix reformulation of the same algorithm.
Birne Binegar: So we already know.
Birne Binegar: And big part of what we've developed so far with respect to matrices and linear equations is that if we have a set of equations.
Birne Binegar: We can reformulate that those equations it's a matrix equation right you look at the coefficients that appear on the right hand side we encapsulate them.
Birne Binegar: In a you know n by n matrix if we started off with any equations in my notes, but then we also can capture variables in a in a matrix and we can capture.
Birne Binegar: The in homogeneous terms the numbers that appear on the right hand side, we can capture those in a matrix as well, and when we do so, and we know how to multiply matrices together, it turns out that this matrix equation component by component exactly replicates the original linear system.
Birne Binegar: Okay That said, I am not going to solve this matrix equation directly i'm still going to think mostly about solving the original linear system but i'm going to use the ideas that I can take the data that specifies a linear system, I can associate major matrices to that data.
Birne Binegar: So here's how we'll do that.
Birne Binegar: If you have a linear system i'm going to write it and matrix form but remember, if you want, you know the coefficient matrix and the numbers on the right hand side you can immediately go back to the original set of equations.
Birne Binegar: Rather, what we're going to do is we're going to form a new matrix it's going to be the coefficient matrix for the linear system and then it's going to have an additional column.
Birne Binegar: That additional column it's going to be formed from the numbers that appear on the right hand side of all the equations.
Birne Binegar: And it's actually this object that we're going to manipulate until we reach the solution.
Birne Binegar: Okay quick example three equations and three unknown.
Birne Binegar: here's the augmented matrix that corresponds to this in your system.
Birne Binegar: reader all the information contained in the equations is awfully contained in this form, here, you can read off the equations.
Birne Binegar: One so think of these three columns is corresponding to the three variables one times X one two times X two minus one times X three is equal to one times X one times X one minus one times X two one times X three x is equal to one so.
Birne Binegar: Not only can you take the data about specific system of equations and encapsulated in augmented matrix you can go, you can go back to equations as well.
Birne Binegar: So here's what we're going to do we'll take a linear system will convert it it's data into an augmented matrix, then what i'm going to do.
Birne Binegar: This second step is that it's a difficult one takes a full lecture to describe i'll tell you how you take this augmented matrix.
Birne Binegar: systematically manipulate it until you reach the augmented matrix on the solution now what's nice about that is we're variables have disappeared and all i'm doing is.
Birne Binegar: manipulating numbers in a rectangular Ray so the arithmetic it's gonna it's just going to be arithmetic that carries out this calculation.
Birne Binegar: And then, once we get the augmented matrix of the solution will convert back to equations and will that the equations of the solution that will actually have the solution expressed in the simplest form possible.
Birne Binegar: And then finally we'll take those equations and we'll try to realize the points and variable space that satisfy the equation will try to identify them as a hyper plane.
Birne Binegar: This this will be pretty straightforward it's well it's actually step two is the.
Birne Binegar: Is the part that it requires a lot of explanation, but it's it'll be easy to understand it just it, you can see how much work is going to go you'll have to do.
Birne Binegar: This solving linear system, via this matrix technique is much faster than the substitution method, but it also requires a bit more explanation.
Birne Binegar: So.
Birne Binegar: The things that we in this step here where i'm changing an augmented matrix to an augmented matrix the solution.
Birne Binegar: The underlying idea here is that we've seen that we can make change equations and not changed their solution.
Birne Binegar: But equations corresponding augmented matrix ease, so we can equally well change augmented matrices so that the when we convert back to equations we still have the same solution set, what I want to describe yes, so if you go from.
Birne Binegar: equations.
Birne Binegar: You go to an augmented matrix.
Birne Binegar: And then, but you can also.
Birne Binegar: Do what we call an elementary operation.
Birne Binegar: Your elementary operation on these equations and you get a new new equations with the same solution.
Birne Binegar: But you can also convert that to an augmented matrix they get a new augmented matrix.
Birne Binegar: So what you could do you want to know how the augmented matrix change, you can revert back to the equation, to the operation and then you get a new augmented matrix.
Birne Binegar: What i'm going to tell you.
Birne Binegar: is how to do that directly, so we don't have to go back to the equations where you're just manipulating these augmented matrices.
Birne Binegar: But that's that's going to be really easy the analogs of these operations interchanging equations multiply equation by numbers, etc, those are going to be really implemented directly on these augmented matrix.
Birne Binegar: Right so.
Birne Binegar: And that augmented matrix the way we get the various rows of it is we scan across a given equation, and we just pull up the coefficient put it on.
Birne Binegar: On the left and the number that appears on the right hand side we put in right column if we change the order of equations, then what happens is we just change the order of the rows of the matrix.
Birne Binegar: We have X plus y is equal to X minus y is equal to one well, that would correspond to the augmented matrix one one to.
Birne Binegar: One minus one if I interchange the order of these equations X minus y is equal one plus y is equal to two well.
Birne Binegar: That just can be one minus one one and 112 alright so here I.
Birne Binegar: took two two sets of equations and their corresponding augmented matrix ease, what I want to point out here is, I could have gone directly from this to this, I simply interchanging the road.
Birne Binegar: Alright, so that should that's natural enough.
Birne Binegar: and
Birne Binegar: Also, when we.
Birne Binegar: replace an equation by it, a multiple by a number, what does that do to the augmented matrix well every coefficient if I multiply this equation y to.
Birne Binegar: X is going to be multiple like two Why is going to be multiplied to and and the number on the right hand side is going to be multiplied by two so.
Birne Binegar: Each number in the corresponding row is going to get multiplied by two, and so, but now, if you think of this row as a vector what we've actually done if we scale or multiplied that vector by a number.
Birne Binegar: We multiply each component of that vector by the same number and that's that scale or multiplication so.
Birne Binegar: that's how we'll implement this this operation.
Birne Binegar: And then finally.
Birne Binegar: Another thing we could do two equations is, we can replace an equation by some with a multiple of another equation.
Birne Binegar: And what that's going to mean since each equation corresponds to a row vector we can replace any equation, or we can move place a row of the augmented matrix I vector some with some scale or multiple of another row.
Birne Binegar: Alright, so let me give you a notation for that.
Birne Binegar: This is row interchange it corresponds to changing the order equations this is going to be scale or multiplication of the row vector corresponding to the I throw with a matrix and that corresponds to multiplying both sides of the equation by a number.
Birne Binegar: And then, finally, this operation here, where we replace a row with it, some with a scale or multiple of another row that corresponds to any replacing an equation by at some with a multiple of another equation.
Birne Binegar: Right so what's nice about this, though.
Birne Binegar: Maybe it's a little bit strenuous to see the connection with the original operations, we performed on equations but.
Birne Binegar: Given that these are really simple to perform directly on an augmented matrix we don't have to manipulate variables at all, let me just show you real quickly here's a two by two linear system.
Birne Binegar: let's read off the augmented matrix one times X one time two y is equal to.
Birne Binegar: One times y minus one times X minus y is equal to one right.
Birne Binegar: Now i'm going to manipulate this guy rather than manipulate the equations.
Birne Binegar: So the first thing i'll do is all subtract.
Birne Binegar: The first row from the second row now that's row two plus minus one times row one, and what will happen, we can do this in place.
Birne Binegar: Working on row two and i'm going to subtract one from it, so the first component one minus one is zero.
Birne Binegar: minus one minus one negative to one minus three is negative two so you can work directly on the paper and you're not moving variables around you're just working on this augmented matrix.
Birne Binegar: Alright, the next step.
Birne Binegar: Here we have a common factor here, so this basically is one one multiply the entire vector multiplied by the number negative to build the simplify that let's divide out that factor of negative two from both terms I multiply the whole row by minus one half and I get that.
Birne Binegar: All right now.
Birne Binegar: let's see now i've got a new augmented matrix what i'm going to do to this now is i'm going to clear out this entry above.
Birne Binegar: That bottom one and i'll do so by subtracting the second row from the first all right let's do it one minus zero is is one.
Birne Binegar: notice i'm operating on the first equation, so this is the guy i'm going to replace one minus zero and one one minus one is 03 minus one is two okay good now I got another augmented matrix which if I converted to equations would have the same solutions as the original system.
Birne Binegar: So let's do that take this equation or this augmented matrix and convert it back into equations.
Birne Binegar: First column corresponds to the first variable second column corresponds to the secondary, so we have one times X plus zero times y is equal to two or simply X is equal to.
Birne Binegar: zero times X plus one times y is equal to one or simply y is equal to one and you see what i've done i've taken the original system and i've worked it out to get the equations of the solution, rather than working on the equations directly and always writing the variables down.
Birne Binegar: i'll get the data and i'm going to collect that data until I get it get it in the form that all regardless the augmented matrix of the solution and I just convert that to equations to get the actual answer.
Birne Binegar: Right now.
Birne Binegar: The last step in my program which actually to express that solution as a hyper plane in this case it's kind of trivial because we only have.
Birne Binegar: One value of X in one value why that's going to be allowable if the tariff equations to be satisfied and that solution set and where X is equal to two and y is equal to one.
Birne Binegar: That corresponds to a single point to one inside my space of variables, which is, I started off with two variables, so I have.
Birne Binegar: This is a single point in the entire space of variables that will satisfy the equation just put this in parentheses.
Birne Binegar: And then, it is a hyper plane, but it's a trivial kind of hyper plane it's where we have a single starting point.
Birne Binegar: Zero but we don't have any additional direction from which we can depart from this point is a.
Birne Binegar: simplest example of a hyper plane, but in general, we could have additional terms here what those additional terms, which would mean is that we have more than one solution to our linear system.
Birne Binegar: Right, mostly plenty examples of that.
Birne Binegar: let's.
Birne Binegar: now concentrate on the meat of the problem, which is for me to explain why did I do this operation first Why did I do this operation next and Why did I do that.
Birne Binegar: i'll explain that the other thing I need to explain this okay I got to this augmented matrix how did I recognize it as a stopping point for the all these.
Birne Binegar: manipulations of the augmented matrix I stopped at this point and converted directly to equations how did I recognize this as the augmented matrix of the solution.
Birne Binegar: Okay well.
Birne Binegar: we'll have to explain that explain the order and choice of operations and i'll have to explain how we recognize when we've reached this augmented matrix of the solution.
Birne Binegar: Well i'm going to answer the second question first i'll tell you what the what the goal is and how to recognize it and then we'll worry about how to how to get there.
Birne Binegar: Okay, so.
Birne Binegar: What this is going to entail, of course, is recognizing a matrix of particular form is be the augmented matrix of your solution.
Birne Binegar: So i'm going to now highlight some digital elements to look for in a matrix that will tell you once we organize things exactly what the augmented matrix of the solution should look like.
Birne Binegar: So first concept is the notion of pivot it's not a hard concept but it's a it's a term that will be throwing about all the time, a pivot.
Birne Binegar: Is the first.
Birne Binegar: Non zero.
Birne Binegar: entry.
Birne Binegar: In a row.
Birne Binegar: So.
Birne Binegar: 0121100211 minus 100 Okay, this is a.
Birne Binegar: Three by four matrix let me underlying the pipit the first row there's the first non zero entry in the second row there's the first non zero entry in the third row that's the third.
Birne Binegar: The first non zero entry alright so that's that's all that word beach.
Birne Binegar: Your three pivots in this in this matrix there's one in each row.
Birne Binegar: And we just pick out the first one that.
Birne Binegar: If we read from left to right we pick out the first non zero number.
Birne Binegar: To describe the the augmented matrix of the solution.
Birne Binegar: The key point is to recognize the pattern that these pivots for fall into issue in the matrix.
Birne Binegar: And that pattern, the central one is called rope exelon for.
Birne Binegar: All right, so yeah so here I just gave you an example of identifying pivots and here's the crucial.
Birne Binegar: Well, the first half of the crucial identification a matrix is in real Echelon form if the pivots in the upper rose always occurred, the left of pivots in the Rose below.
Birne Binegar: This matrix here it's not enroll Echelon form because there's a pivot in a row.
Birne Binegar: I go down to the second row and the pivot is moved off to the to the left, rather than the right or let me phrase it exactly stated here here's a second row if I go to the upper row this pivot is occurring off to the right rather than off to the left, that would match this definition.
Birne Binegar: So what we're stipulating here.
Birne Binegar: Effectively, is the pivot if you have a matrix in relational and form the pivot.
Birne Binegar: form sort of a downward staircase.
Birne Binegar: makes have large, you may have large skips, but if you have a pivot here if it's relational on form, you can have a pivot over here it's got to be off to the right, that would be the next candidate.
Birne Binegar: So you have nothing but zeros up to that point, and then the pivot and the real beyond that it's got to occur off to the right.
Birne Binegar: And so, you would have nothing but zeros up to this point when you get a matrix that it's mostly zeros in this corner and mostly not zero numbers up in this corner, then you get a matrix in real Echelon form.
Birne Binegar: Okay, so.
Birne Binegar: This matrix general Echelon form.
Birne Binegar: Yes, yes it is so notice notice the staircase.
Birne Binegar: consistently as I move down the matrix the pivots are moving off to the right.
Birne Binegar: This matrix is not because, as I moved down in the matrix here, we got a problem this guy if I see send in the matrix the pivots has moved off to the left, not like what happened, he does so that's not relational on form.
Birne Binegar: And that's it that's all we're looking for the patterning which the pivot occurs, we need it to be this descending staircase going from the upper left to the lower right.
Birne Binegar: All right now.
Birne Binegar: Changing a matrix into real Echelon form is the first half of the algorithm well i'm i'm about to describe, but let me give you the second half to, because this will be the final form of the augmented matrix of a solution.
Birne Binegar: A matrix is in reduce row Echelon form if, first of all got to be in real excellent form, you need this downward staircase but.
Birne Binegar: We have we have some additional requirements, the pivots can't be arbitrary numbers like here here the pivot is two years pivot is one is a pivot is three what we require for reduce will Echelon form is the pivots have to have value one.
Birne Binegar: and, moreover, if a column contains a pivot we want that pivot to be the only non zero entry in that column, so if I look above a pivot or below a pivot in that matrix I don't want to see nothing but zeros except for the pivot itself.
Birne Binegar: let's do some examples Okay, so this matrix is in row Echelon form, you see how the credits are descending.
Birne Binegar: it's not.
Birne Binegar: It all the pivots are equal, the one that's good, but if you look above hip it in the third row I have some non zero entries, so this matrix is almost in reducing want form, the problem is these entries there.
Birne Binegar: And if I simply made the zeros, then I would get a matrix in reduce all Ashkelon form but be aware, though, is that if you're.
Birne Binegar: Thinking about changing this matrix into another equivalent matrix that would have the same solution set you can simply say Oh, I have one here, let me just make it zero and think you you.
Birne Binegar: you're working towards solving the equation know you've changed the augmented matrix and you've also changed the equations you're working with as well.
Birne Binegar: So if we want to take this matrix that not in reducing last one form and get to a matrix and reduce realized one born way to do that is to apply the operations that don't change the solution set not just Zeroing out things that you don't like.
Birne Binegar: Okay, but at least now, you can see.
Birne Binegar: how one might do this.
Birne Binegar: If a if a pivot was not equal to one what could you do to make it equal to one.
Birne Binegar: Let me give you an example let's look at this matrix what could I do to this this row, so I can get that pivot equal to one.
Birne Binegar: multiply by one half right that one half punch to would equal one now allow that new pivot is one, and this just becomes one half but i've done an operation that does not change.
Birne Binegar: doesn't change the solution set so that would be a legitimate way to actually if I had a pivot that wasn't equal to one I could easily make it equal one.
Birne Binegar: Alright, so the strategy, then it's going to be done well, I guess, I got more examples of this, this matrix is in reduced form and it's.
Birne Binegar: again.
Birne Binegar: For this particular matrix is easy to see why this should be the solution the augmented matrix of the solution.
Birne Binegar: Because if we write out the equations corresponding that all get augmented matrix it's going to look like a solution X one they'll think of.
Birne Binegar: First column corresponds first variable second variable third column corresponds third variable reading off the equation first row is X one plus zero times X two plus zero times X three.
Birne Binegar: equals to a second row is telling us that zero times X one plus one X two plus zero is equal to one or simply X two X equals three and the third row is telling us that X three x equals four so.
Birne Binegar: Here we have.
Birne Binegar: A matrix it's been reduced real Echelon, for if it's are you sending in the right way they're all equal to one and above and below pivots I only have zeros and when I convert back to equations yeah Indeed it does look like a solution vector we specify numbers for each variable.
Birne Binegar: Alright, so there's an example.
Birne Binegar: here's an equation equation that's not in reducing Echelon form, it is in row Echelon form there the pivot but the pivots aren't equal to one, and indeed above this pivot I have a non zero number.
Birne Binegar: And this, this is.
Birne Binegar: This is not in a row in even in row Echelon form this matrix is in row Echelon form but it's not in reduce rational and form because my pivot here is not equal to one nor is equal to one below.
Birne Binegar: And then finally here's an example of a matrix that is in reduced role Echelon form, you can have numbers appearing in columns but now, since.
Birne Binegar: These numbers here there's no pivot in this call here's a pivot here's a pivot here's a pivot the sport column doesn't have a pivot so it's allowed to have non zero numbers.
Birne Binegar: Let me yeah.
well.
Birne Binegar: I guess i'm going through more examples here.
Birne Binegar: Here, this isn't raw short form check the criteria in roseville on form tickets are all week of the one above and below pivot I have.
Birne Binegar: I only have zeros that's Russia one for this isn't relish one for me as well okay so.
Birne Binegar: we're big we're gonna be looking for augmented matrix of this type.
Birne Binegar: Nobody said this matrix, unlike the preceding examples there's no identity matrix here, let me think of it as an augmented matrix.
Birne Binegar: let's say it's um.
Birne Binegar: 100001 i'm going to think of an augmented matrix song in a shirt this vertical line there and then I had 0112 in the last row is 0000.
Birne Binegar: All right, so.
Birne Binegar: This is an reduced row Echelon form, it should be the augmented matrix of the solution, so what is the solution look like.
Birne Binegar: Next, one is equal to one that's coming from the first row at two zero times X one one times X plus one times X three is equal to, and this last row was just saying zero times X one zero times X two zero times X is equal to zero, or simply zero is equal to zero.
Birne Binegar: All right, so that's my solution but it's probably not it's not the way I like to see the solution presented, because all three variables are are occurring.
Birne Binegar: So what we tend to do is, we take the variables, with the higher index and we put them on the opposite side.
Birne Binegar: And now i'll write down a solution vector X equals well X, one has to equal one.
Birne Binegar: X two has to equal to minus the third component and the third component I don't have any equation for specifically so i'll leave it alone.
Birne Binegar: And then we can write this vector as 120 plus x three times.
Birne Binegar: One zero no X Ray here we get X three with a minus sign in the second component and the third component we get one.
Birne Binegar: notice, we have a line right every vector here can be every doctor in the solution set can be expressed.
Birne Binegar: As a vector where we started a given point and we stray from that vector X three is undetermined it's a pre parameter in this solution so every solution vector is a point plus some number times another vector, and so it is a hyper plane is in fact it's along.
Birne Binegar: So, in a nutshell, this is this is basically the technique we're going to follow we're going to take no said if equation convert to augmented matrix manipulate that augmented matrix using operations that do not change the solution said, until we reach a matrix in reducing electron porn.
Birne Binegar: Then what we'll do is we'll convert back to the equations and try to express our.
Birne Binegar: Our solution as as a point on a hyper plane, rather than isolating on variables, we want to think of our solution to be particular vectors rather than particular values of the coordinates.
Birne Binegar: Okay, so let's see.
Birne Binegar: Now we come.
Birne Binegar: To.
Birne Binegar: The actual algorithm the algorithm by which i'm going to take an arbitrary matrix and i'm going to put it.
Birne Binegar: First, in row Echelon form and then i'll do for the manipulations to get it in reduce your actual on form when I reached reduce your last one for that's when I stop and go back to the equations.
Birne Binegar: Right so here's the idea will convert equations to augmented matrix will use real operations that are have the same effect as the elementary operations, we used on equations to convert systematically.
Birne Binegar: augmented matrix we start with to the augmented matrix of the solution, and then we convert back to equations and then the last step will be to reinterpret our solution vectors as as vectors.
Birne Binegar: Alright, so what am I got I got three minutes left.
Birne Binegar: i'm going to stop here because.
Birne Binegar: it's.
Birne Binegar: Once once we start I just start describing these operations you'll see that you can do this, all visually just manipulating the numbers in rows of the matrix but i'll delay that presentation until Friday.