Birne Binegar: Okay, so I guess we get underway us on Friday we'll have our our first exam today what i'm gonna do is i'm going to first of all give you a quick what happened here.
Birne Binegar: Okay, I guess we'll.
Birne Binegar: give you a quick overview of material to cover on the exam.
Birne Binegar: From a former point of view, you know what we'll do a practice exam but now shoot, this is just looking for some reason i'm not sure why.
Birne Binegar: Well, let me get started with with with a quick review, although maybe the first thing I should do is just pause and see if there's any questions coming into this review.
Birne Binegar: what's one thing I want to point out is you'll notice at the top of the practice exam it says no calculators no calculators will be involved.
Birne Binegar: will be allowed for the first exam either, because the only real calculations that you.
Birne Binegar: could possibly need a calculator for would be to add integers together all designed the problem so you're not adding fractions or other real numbers you shouldn't need a calculator.
Birne Binegar: except if you have one of the real fancy calculators where you plug in and matrix and they'll tell you what the inverse is that's that's not allowed in fact.
Birne Binegar: main purpose of the first exam is to drill you on row reduction, not so much because that's a particular calculation that is used over and over again that's true.
Birne Binegar: But the main reason I want you to do those row reduction calculation by hand is that you can interpret the calculation as you're working through it, you can sort of see how you're getting towards the solution.
Birne Binegar: You can see whether or not the fairly early on, if the equations have a contradiction that they have any contradiction etc so.
Birne Binegar: God.
Birne Binegar: i'll show you, if I work out the practice exam i'm going to do it right on the exam paper to show you there's plenty of room to do the role reduction calculations, etc, and all you have to do.
Birne Binegar: to approach these rub reduction problem with the right strategy you refer us to roll out one form and then you systematically.
Birne Binegar: proceed to produce or electronic form until you get your unique matrix and reduce your actual form.
Birne Binegar: Okay, so all that's coming let's let's be quickly step through the material we've covered i'll just done, I actually did this last Friday, but.
Birne Binegar: In the first week of course we concentrated on just the real fundamental vector operations addition scale or multiplication taking linear combinations of vectors to produce new vectors sorry I don't know why this thing is flashing this way.
Birne Binegar: Maybe it will help like disconnect.
Birne Binegar: reconnect.
Birne Binegar: Okay, well, I hope, it's stable, so the first week of class we just did this rudimentary sort of.
Birne Binegar: Basic arithmetic this underlying linear algebra we also introduce the fundamental geometric optics for the course the notion of what a hyper plane is.
Birne Binegar: A set of an infinite set of vectors in general, because these parameters, do you want to care allowed to range over the entire real line.
Birne Binegar: This is a common geometric construction and, in particular when we solve linear systems or systems of linear equations the solution sets are always going to be sets of this form.
Birne Binegar: Well, provided they have solutions until here's You see, for example, a single.
Birne Binegar: equation in em variable single linear equation and variables can be expressed in terms as expresses the hyper plane what I did here is I solved for the last component in terms of the.
Birne Binegar: earlier.
Birne Binegar: Earlier variables X one X N minus one and i'm able to express any solution of this equation there's a particular vector that lives on a hyper plane, because it has this form.
Birne Binegar: A constant vector plus a free variable times another content vector and similar terms, until we have last free variable time to laugh constant vector.
Birne Binegar: Right phil i'm I just address it my overview here is not to look that much like the exam because I won't be testing you directly on these things, but rather how they occur in problems.
Birne Binegar: So.
Birne Binegar: Once we.
Birne Binegar: Had vectors down I introduced matrices I introduced them, by the way, as a way of Encapsulating the data that goes into specifying a system.
Birne Binegar: of, say, any equations and Emma knowns we could capture that in a matrix and that, ideally, we captured in an augmented matrix.
Birne Binegar: or effectively we erase all the variables we race all the plus sign to life and we just put square brackets over everything to get this nice rectangle arrangement.
Birne Binegar: up on the left hand side matrix with n rosen incomes and this last call representing the in homogeneous terms the numbers that appear on the right hand side.
Birne Binegar: Then what we did once I was able to.
Birne Binegar: translate a linear system into a matrix reformulation i've told you that is not just writing the data down next to one another.
Birne Binegar: The idea was, in fact, when I take a linear system, a one one X one.
Birne Binegar: plus one and X m equals say B one and we have similar equations a one.
Birne Binegar: And one.
Birne Binegar: X one plus dot dot dot plus a and and X equals B and similar such equations we could write this down as a matrix equation.
Birne Binegar: A X equals B we're a is exactly the coefficient.
Birne Binegar: A 11281 m a n one through a an m.
Birne Binegar: This column vector.
Birne Binegar: Is contains all the variables.
Birne Binegar: And on the right hand side we have this vector you want.
Birne Binegar: To the end and the point of introducing matrix multiplication aside from just taking the data that specifies, you need to specify given linear system and putting different data structures next to one another, in fact.
Birne Binegar: We can be fine this writing this matrix adjacent to this matrix is is the matrix product and we then define the matrix product.
Birne Binegar: As the new matrix you create by taking the rows of the first factor taking their products with the columns of The second factor and renting those results in another rectangular array so this matrix multiplication.
Birne Binegar: You have two matrices and they multiply together, you can create a new matrix by taking.
Birne Binegar: The thrill of a.
Birne Binegar: And that's going to be a list of as many numbers, as you have columns of a and taking its product with.
Birne Binegar: The jade column of the other factor be and again if.
Birne Binegar: you're looking at the column, to the matrix theory it's going to have as many entries in that column is you have rows so what you need in order for that dot product be fine it's a number of columns.
Birne Binegar: The first factor that will tell you how many components are in this factor has to equal, the number of rows at the second factor.
Birne Binegar: And then you can calculate all these all these inner products you'll get one for each row with a and one in each column of Jay and so effectively you get them multiplication rule that works like they have an n times n matrix because you have to have this match it can only multiply.
Birne Binegar: An n by n matrix can only multiply from the Left matrix with the same number of roses, we have columns there, so you might have and by P, and then what's going to get kicked out of that.
Birne Binegar: matrix multiplication is going to be a matrix with n rose and P calls, so you need to know these formulas.
Birne Binegar: One for explicitly computing the entries of product and the other one a quick and dirty formula tells you what kind of matrix is going to get kicked out when I multiply and N by n matrix i'm like pediatrician.
Birne Binegar: Okay, so.
Birne Binegar: you'll need to be able to carry out some matrix multiplication but they'll you'll see on the practice exam I only asked you to do this once but.
Birne Binegar: You also have some additional operations, you can add matrices together, just like you add vectors to you add similar entries of.
Birne Binegar: matrices have exactly the same size, so a two by three matrix can be combined with another two by three matrix simply by adding entries in the same slot together and, similarly, you can scale or multiply and you can take the transpose of matrix.
Birne Binegar: But then, once we introduce those objects, the idea was to.
Birne Binegar: Do something with them, and the first thing we did was we solve linear systems out, but the technique we use for solving linear systems is different from what you learn in high school instead of.
Birne Binegar: Taking one variable and using the first equation, to eliminate that substituting result in the subsequent equations and doing more.
Birne Binegar: selecting the variables to solve for and then substituting in the subsequent equation all that apparatus we avoided.
Birne Binegar: Rather, what we did is we thought of a problem solving a linear system as the problem of how to convert the given set of equations into a set of equations that could be interpreted as the solutions.
Birne Binegar: And the way we would do that it would we would perform operations on the equations that wouldn't change the solution set things like interchanging the order of equations or multiplying and equation by non serial number of replacing equation by a son with another equation.
Birne Binegar: So, but I showed you that that those three operate kinds of three operations could actually be effective and taking an arbitrary set of equations and giving.
Birne Binegar: and producing what could be interpreted as the equations of the solution, but then we quickly moved into a more efficient way of doing the same thing it just that, instead of operating directly on the equations.
Birne Binegar: We would instead to every system of equation corresponds to a particular augmented matrix.
Birne Binegar: The idea was to.
Birne Binegar: Rather than manipulate equations we would just manipulate the corresponding augmented matrix.
And and.
Birne Binegar: And then, what we needed then.
Birne Binegar: Is operations that would be sort of the image of what we do two equations but.
Birne Binegar: Operating directly on the role of a matrix so each row of the augmented matrix corresponds to an equation dope I do something to an equation it's going to have a.
Birne Binegar: Corresponding action on this augmented matrix and that's exactly what these elementary.
Birne Binegar: row operations are.
Birne Binegar: There were three of them.
Birne Binegar: And the way they work was first of all, we could change the order of equations and that corresponding to changing the order of rose has a major get interchange two rows or we could take a row.
Birne Binegar: and multiply it by a non zero number or replace it by its multiple by a number zero number or the third thing we are allowed ourselves to do is take a row and replace it with it, some with a multiple of another row.
Birne Binegar: So just using these three operations.
Birne Binegar: We weren't able to take an arbitrary matrix and convert it first in the row Echelon warm then in to reduce cholesterol, so this is the first step.
Birne Binegar: In rubbers option.
Birne Binegar: But I split these apart and the two steps, simply because your strategy changes as you try to go from row excellent form to reduce your initial, and this is typically the second step.
Birne Binegar: And so here what you're shooting for.
Birne Binegar: Is a matrix where the pivot.
Birne Binegar: All current this downward staircase so everywhere to the right and below a pivot or sorry to the left and blow a pivot you should have nothing but zeros, but you can have non zero entries sitting above the pivots we're off to the after the right of pivots.
Birne Binegar: That this is your first goal and the way you achieve this, for the most part, is by using this third row operation your principal task is to.
Birne Binegar: give it a pivot up in this corner here is to clear out the entries below so you take multiples of the first row precisely so that when you add it to a lower row, you can cancel that leading entry.
Birne Binegar: And then.
Birne Binegar: Once you've reached robust one form your strategy changes, you have two more requirements in order to be in reducible Echelon form, you need to pivot one, so you.
Birne Binegar: convert pivots to one.
Birne Binegar: And you also have to carry out the entries above pivot.
Birne Binegar: But once you once you reach real Echelon form you use Rotary scaling to convert the value of the pivots equal to the numerical that you want, and then you.
Birne Binegar: Work from lower rose to clear out entries that sit above pivots and little rose and then again you're using the third row operation to achieve that.
Birne Binegar: Once you get to a matrix and reduce your national on form, then you have a unique matrix and we interpret that as the augmented matrix of the solutions.
Birne Binegar: What renamed then just follow the linear system is to go from the matrix augmented matrix and reduce row Echelon form and write down the solution sets of hyper planes.
Birne Binegar: But in between here, going from the make augmented matrix and reduce cholesterol on form to the hyper plane.
Birne Binegar: A step in between this bigger thing out which variables are going to be free variables and which variables are going to be fixed variables that you.
Birne Binegar: That information you get from the renewal Echelon form fixed variables we're going to correspond to columns of The.
Birne Binegar: augmented matrix and reduced lunch on porn that contain pivots and a free variables are going to correspond the columns that Duncan painted it.
Birne Binegar: Once you've made the decision of what variables are free and what variables are fixed.
Birne Binegar: you write down the equations correspond to this matrix you move the three variables over to the right hand side and then you can now begin to formulate us a generic solution vector the form of any vector that would actually satisfy the original system.
Birne Binegar: So i'm going to step through this and we're going to do, several examples of this well actually one example of each step, I think the way that practical exam is worked out.
Birne Binegar: And then the last thing.
Birne Binegar: We talked about is.
Birne Binegar: How to find it inverse of the matrix.
Birne Binegar: inverse of the matrix you would be the unique matrix that well, first of all, the only just for square matrices but if the unique square matrix of the same size as the original matrix such that whether you multiply from the left or the right by this matrix you've.
Birne Binegar: got the identity matrix so I guess what I want to say here is that what I mean by inverse is that.
Birne Binegar: Say be is a the minus first if either.
Birne Binegar: The times a is equal to the identity matrix or a times B is equal to the identity matrix because these two conditions.
Birne Binegar: I told you all the difficult to prove but, in fact, if one of these equations True, the other is also true so really to identify an inverse of a of a given matrix you just got to find some matrix that when you multiply from either side.
Birne Binegar: By that matrix you get the identity matrix right, and then we then we developed up a means for calculating these inverse matrix sees.
Birne Binegar: What it boils down to is writing down your original matrix and the identity matrix a join to it reduce this double matrix to we do show Echelon form and then, if.
Birne Binegar: The left hand side real reduces to the identity matrix then Whatever happened to the iPhone the ice side the right hand side that would be your new.
Birne Binegar: At minus first, on the other hand, in the reduce cholesterol on form of this double matrix the right hand side it's not the identity matrix then your conclusion is that the a does not have an inverse.
Birne Binegar: And then last thing we covered.
Birne Binegar: I guess on us.
Birne Binegar: Oh Friday reflection, a little bit on Monday was this fundamental theorem matrix in verses which showed you relationships between.
Birne Binegar: A matrix having an inverse solve ability of linear systems and, in fact, the ability to factor is a matrix in terms of elementary nature sees.
Birne Binegar: Still intact, I don't think i'm on the practice exam this fundamental theorem does not appear explicitly and.
Birne Binegar: I will adhere to my policy of writing exams that are very, very close to the practice exams so maybe you don't have to worry so much about.
Birne Binegar: This fundamental theorem although as soon as we get through through with the exam our next.
Birne Binegar: Our next collecting of material it's going to be more theoretical more like the statement of the serum than the calculations that we've.
Birne Binegar: been working on so far.
Birne Binegar: Okay, so let me now.
Birne Binegar: I mean i'll just pause and ask you any questions about what might be covered on the exam then then we'll step through a practice exam.
Birne Binegar: Okay, so let's proceed with practice exams again I want to emphasize no calculators will be allowed on the cam you should not need them, as long as you can add two to five and see it that seven you won't need a calculator yes.
Birne Binegar: Oh sure it's like the inverse of this matrix but as.
Birne Binegar: Well yeah yeah you can don't be don't be upset by variables you just have to remember that they're very both and not try to evaluate them explicitly so.
Birne Binegar: let's see what is that matrix.
Birne Binegar: I think it was.
Birne Binegar: You can still reduce this thing, so what you might do is take the matrix it from the matrix a joined with.
Birne Binegar: What that will be.
Birne Binegar: Sorry, I don't like cancer.
Birne Binegar: it's going to be one a minus a one a joint with 1001.
Birne Binegar: Alright, so we want to get this first row Echelon form so we'll need to get rid of this.
Birne Binegar: Entry there, so what i'll do is i'll same row to row two plus eight times real once.
Birne Binegar: I that's that's exactly what I need to add to the second row in order to get rid of that pivot in the second row, and what will get here first row is not changing one zero and then.
Birne Binegar: minus a plus eight times, one is going to equal 01 plus.
Birne Binegar: I guess it's going to be a.
Birne Binegar: yeah be it's gonna be a squared there.
Birne Binegar: Because what i'm gonna do is i'm going to add to the second row eight times the first row and eight times a is going to equal a square and then i'll have.
Birne Binegar: A plus zero is a zero plus one is one.
Birne Binegar: All right, and let's see next thing we might do is we have it in real Echelon form.
Birne Binegar: next thing we'll do is get the pivots equal to one for second row two to one over a squared times.
Birne Binegar: let's see Oh yes, one.
Birne Binegar: When I add yeah multiplied the one plus age squared that's so that would be one over one plus a squared time to that, and this is going to produce matrix.
Birne Binegar: What am I trying to do i'm trying to get rid of that guy.
Birne Binegar: And so I guess I need.
Birne Binegar: Sorry for some just re normalizing this guy one a 1001.
Birne Binegar: A divided by one plus a square and then one divided by one plus x squared.
Birne Binegar: And then the last thing you do is to clear out this entry here and so sorry.
Birne Binegar: work here i'm going to send real one.
Birne Binegar: Real one minus.
Birne Binegar: minus a time to the second row and now it's going to produce a matrix one zero.
Birne Binegar: And I let's see we have one.
Birne Binegar: minus eight times row two so that's going to be.
Birne Binegar: One minus a square divided by one plus a squared.
Birne Binegar: And then, here we have.
Birne Binegar: Zero minus.
Birne Binegar: A one plus a squared.
Birne Binegar: And then seven rows not changing 001 and then a divided by one plus three squared one divided by one plus a square Okay, then this matrix here I mean you could simplify maybe these entries, but this matrix here is going to be your inverse.
Birne Binegar: Okay yeah that's yeah It may be that it really wanted you to simplify these terms I don't know if you tried to simplify these terms.
Birne Binegar: didn't give us a regular.
Birne Binegar: class.
Birne Binegar: Oh shows it's maybe it's just a problem with web assign well yeah i'm so i'm sorry about that yeah but remember when you when you do enter matrices as as a result, the idea is that you.
Birne Binegar: know only going to give you entries for one row at a time, and you put in one in one row each entry in each box.
Birne Binegar: And then you there's a whole arrow below that you click on to bring up a second row that you can then put the individual entries in so it could also have been for some students just entering entering your answer correctly.
Birne Binegar: So so.
Birne Binegar: What happens on weather time I.
Birne Binegar: said, if you want to input show you want to give.
Birne Binegar: Two by two matrix as an answer.
Birne Binegar: Then what you do is rather assignable.
Birne Binegar: First of all, you'll show you two blanks well I need for entries were two by two matrix what you're supposed to do is.
Birne Binegar: Assistance your matrix a the CD you put the a and the B to get the first row, and then there is a little symbol here.
Birne Binegar: Like a little down arrow or something that you click on that after you've entered this know bring up a second set of empty boxes and then you put the remaining entries there and then you you fed web assign the the your answer, but unfortunately, when you first look at.
Birne Binegar: The entry boxes that were assigned displays you only see two boxes, for even when your answers the two by two matrix this second row doesn't appear until you filled in the first row and click this little.
Birne Binegar: arrow button at the bottom.
Birne Binegar: There were four banks, all it could have been it could have been that they were expecting the full augmented matrix until you, you would enter the four entries of this first row with the augmented matrix and then you'd have to enter the second row separately.
Birne Binegar: Almost 220 okay.
Birne Binegar: yeah well in any rate yeah.
Birne Binegar: Okay well yeah.
Birne Binegar: Okay, well, let me touch it it did give you the four entries you needed, yes, so getting it wrong they kept getting it wrong yeah.
Birne Binegar: Exactly actually a little bit of a problem to for a computer to do symbolic algebra now if these were these had been numbers i'm sure there would be no problem at all.
Birne Binegar: And, but when when you have symbols now that's that's where computer systems, well, it should nowadays nowadays it's really common to do.
Birne Binegar: To do symbolic algebra on computers, but apparently with a sign it's not good at that.
Birne Binegar: Okay well anyway don't don't worry.
Birne Binegar: don't worry too much about this particular problem i've shown you how you do it, and in fact on a on an exam I promise you, you won't have to work with variables, except when you're trying to write down solutions and distinguishing between free variables and fixed variable.
Birne Binegar: Okay, so i'm gonna have I have 20 minutes let's see if I can get through this exam I think I can actually I worked it out.
Birne Binegar: Prior to class, and it was fairly simple, so in this first problem you're multiplying matrices together just write them down next to one another few minus one one and 1010 minus one zero now what I suggest you do before you even get started calculating.
Birne Binegar: figure out what kind of matrix you can be produced, so this is a matrix three rows two columns and it's multiplying a matrix with two rows.
Birne Binegar: And three columns it was the inner dimensions match so that's good the matrix you're going to be.
Birne Binegar: The product is going to be defined, and what it's going to produce is a three by three matrix right it's the outer dimensions that give you the dimensions of the products and the inner dimensions have to match, but now we can compute.
Birne Binegar: First entry in the first row and first column of the product is going to be the product of the first row vector.
Birne Binegar: first factor with the first column vector of the second factor, so that will be one plus zero is one, and then to get this entry, the second column I use the second column at the second factor, so I have zero Plus zero.
Birne Binegar: And then one plus zero which one.
Birne Binegar: Now I can move to the second row nice continue to take that product two plus zero is two.
Birne Binegar: Zero minus one times minus one is one will get one there and then two plus zero is two and then let's see three plus zero is three zero.
Birne Binegar: minus one and then three plus zero is three all right there's a quick matrix multiplication I hope I just following my fingertips you're doing, you can see how I could do most of the products in my head.
Birne Binegar: All right, then in.
Birne Binegar: The second problem on the exam, this is no calculations that all my point of giving you these matrices it's not to have you do.
Birne Binegar: any real reduction or look for the solutions, but rather what I want you to do in this problem is interpret this matrices tell me what kind of.
Birne Binegar: equations they came from, and then tell me a little bit about the solution just from the fact that these major cities are already in row Echelon forms.
Birne Binegar: Right, so what I asked is how many equations were there, how many variables does this system have a solution, and if it does have the solution, how many what's this what's the size of like solution set how many free variables are there.
Birne Binegar: Okay, so let's look at this first ones.
Birne Binegar: Well, actually let's look at this second one first, so I can address this second questions here we have three rows in this matrix so we came from a system with three equations and there's four columns on the left hand side, so this particular example.
Birne Binegar: came from.
Birne Binegar: Three equations.
Birne Binegar: In 12344.
Birne Binegar: that's, the first question does it have a solution, no because i'm there's no solution.
Birne Binegar: Because the second row.
Birne Binegar: You convert back to equation.
Birne Binegar: It says that zero has to equal.
Birne Binegar: minus one.
Birne Binegar: Up zero times X one zero times X two zero times X times X four X equal to minus one or that simplifies to simply zero is equal to minus one which is a contradiction.
Birne Binegar: So, because one of the equations associated to this augmented matrix is contradictory there's going to be no solution and the problem, and here it doesn't make sense, you don't have any solution it doesn't make sense to talk about the number of free variables in the solution.
Birne Binegar: All right, and then for the.
Birne Binegar: Through I don't want to write up there.
Maybe i'll.
Birne Binegar: bring this down a little bit.
Birne Binegar: For this first augmented matrix.
Birne Binegar: always look what's going on with the pivots first before we do any interpretation Oh well, aside from the fact that we have four rows so we have four equations and we have.
Birne Binegar: Four columns on the left hand side to that means we have four associated variables, we do have a solution.
Birne Binegar: Because we don't have this contradictory situation we don't have any pivots in this final code in this case, we did and that's why we didn't have a solution.
Birne Binegar: But we don't have that problem with this matrix, so we are going to have a solution and the count the number of free variables, we look for the number of columns on the Left that don't have PID it it's only X three.
Birne Binegar: X Ray is.
Birne Binegar: The only free variable.
Birne Binegar: it's gonna be a one dimensional solution set, but whatever linear system this augmented matrix came from, and then, here we have four equations.
Birne Binegar: In three variables.
Birne Binegar: Have it does have a solution.
Birne Binegar: right we don't have this problem and then.
Birne Binegar: Each column on the left hand side contained to pivot, so there are no free parameters in the solution.
Birne Binegar: That means you have a unique solution.
Birne Binegar: In fact, the equations that come from this augmented matrix say that as one has to equal three x to have well.
Birne Binegar: Actually there's a term here, but you can basically see that you can reduce this to reduce relational on form you're going to get a unique number for X one X two X Ray anyway, this solves the problem to get after that last question what you're doing, you should counting the number of.
Birne Binegar: columns that do not have pivots that will give you the number of free variable so give you the dimension of your solution.
Birne Binegar: And that's All I want you to do on this problem, no calculations, I just want you to interpret these matrices.
Birne Binegar: All right, and so, in particular, well, I always have some students who see an augmented matrix and they try to write down the hyper planes solution not necessary for this problem you'll get opportunities to write solutions in terms of hyper planes later on in the exam.
Birne Binegar: Now.
Birne Binegar: The next thing I like to do on my exams, at least the first exam and linear algebra.
Birne Binegar: If I want you to be able to solve linear systems, but.
Birne Binegar: that's kind of a long process right, you have to convert the augmented matrix you have to reduce rush on for me after.
Birne Binegar: Further over do should we do should wash long form and then you have to write the solution.
Birne Binegar: corresponding to the augmented matrix and reduce wrestling form, you have to formulate that solution is hyper plane I don't do that in all in one problem or another, I do I break that whole procedure up into steps and I test you on the individual steps.
Birne Binegar: So.
Birne Binegar: In this first problem, for I give you an augmented matrix you don't have to find the corresponding solution, all I want you to do is.
Birne Binegar: I must be on problem three All I want you to do initially is is take the system of equations right down the augmented matrix and get halfway through the road reduction process right down the augmented matrix for this linear system and then reduce it to just roll Echelon form.
Birne Binegar: Let me see if I can carry that out here.
Birne Binegar: Alright, so let's look on screen here.
Birne Binegar: This work alright, so we can have.
Birne Binegar: zero times X one.
Birne Binegar: minus one times X to two times X three is going to equal one that's the first row.
Birne Binegar: The augmented matrix corresponding this in your system, then we'll have to 110 to that line down and then minus 1113.
Birne Binegar: All right, that's the augmented matrix now will reduce So what do we got to do we need to hit it up in this corner.
Birne Binegar: or reduction process to real life long form starts by getting a pivot up in this upper left hand corner, then clearing out the entries below it so what I might do is switch the first and the third row.
Birne Binegar: Then i'll get minus 111321100 minus one to one.
Birne Binegar: Now i've got a pivot up in this corner and I can use it to clear out the pivot below, and so the next step would then be to focus on that guy and get rid of it by sending row to row to minus two times for a one or I guess plus because I already have a minus sign there.
Birne Binegar: And what will get his first row is not changing.
Birne Binegar: Third row is not changing.
Birne Binegar: On to one and then what happens second row i'm going to.
Birne Binegar: multiply the first row times two and then add it to the second row, so I have two minus 201 plus two is three one plus two is three and zero plus three is three.
Birne Binegar: Alright, so we've.
Birne Binegar: gotten that far now let's see.
Birne Binegar: Be six you're right yeah three plus three is six.
Birne Binegar: Right.
Birne Binegar: Right and then.
Birne Binegar: Then I guess we'll quickly.
Birne Binegar: Well, we can leave this.
Birne Binegar: As.
Birne Binegar: minus 1113 let's change this one, just to make it simpler i'll divide the second row by to Ali Baba third row alone, because all i'm trying to do is get through row Echelon four minutes, the last step i'll do i'll do is i'll add i'll replace the third row by at some with the second row.
Birne Binegar: row row three is going to go to row three plus row two and then i'll get a matrix minus 11130112 and now 01 plus minus 101 plus three is three two plus three is three all right, and then the problem and because.
Birne Binegar: i've now reached.
Birne Binegar: A matrix where the pivots are occurring in the proper downward staircase so that this matrix is actually in real Echelon for.
Birne Binegar: All right, so that that that's typically going to be one kind of problem will you'll see on the exam.
Birne Binegar: But it's not the end of Robert afternoon, if we were trying to find the linear now if the solution set up the corresponding.
Birne Binegar: set of linear equations The next step, you would do she would try to take the augmented matrix in Russia long form and could.
Birne Binegar: massage it further until you reach we do rush line for but rather than work from this matrix where you may or may not have gotten the correct answer I give you a fresh matrix and start off and I asked you to.
Birne Binegar: proceed and convert it into reduce the washroom for well that's means you've got to get the pivotal equal to one, so the first row is going to have to be divided by two second row is going to have to be divided by three.
Birne Binegar: And the third row.
Birne Binegar: divide that by minus two.
Birne Binegar: Then you'll get a new matrix 11231100122100001 minus one last row remained all zeros.
Birne Binegar: OK and OK, so now we've got it in row Echelon for in the pivots for all equal to one last thing to do is clear out the entries.
Birne Binegar: That are lying about pivots that one there in those two entries, let me put a squiggle here, these are actually a problem in trees and will you now.
Birne Binegar: Lower bros to clear out entry sit sit above pivots and we started this lower right hand corner, so the route, the entries above its pivot then we'll move over to the next pivot.
Birne Binegar: In a second step will clear out this entry here.
Birne Binegar: So what we would do first maybe ascent.
Birne Binegar: row one two row one minus wrote three that's so that.
Birne Binegar: This entry will can be used to get rid of the entry in this ticket above it or this entry above the pivot and third row and will send row to.
Birne Binegar: row to minus two Times wrote three that will clear out this entry and what yet is 1123 okay now here's where the action started.
Birne Binegar: I did I can write down those first four entries right away because anyway, going to be additive to zero here's where the action occurs and we'll get zero there but we're going to have.
Birne Binegar: First row minus the third row is going to produce I guess zero.
Birne Binegar: For.
Birne Binegar: or she is going to produce a zero there and then this real miners that is going to produce it to.
Birne Binegar: And then, when the second row 0012 nothing happens up to that point, but then, when I subtract two times the third row from the second i'm going to get zero and one minus two times that's going to equal 300 sorry it's it.
Birne Binegar: That help so 00001 minus 100000 okay.
Birne Binegar: All right, let's see last last thing we got to fix is this to that sitting above the pivot in the second row until will send row one two row one minus two times second row and now we'll get 1103 this will be minus one, there will be three minus four zero minus 002 minus six is.
Birne Binegar: yeah sorry.
yeah.
Birne Binegar: Two minus six it's going to be negative four and then you have 00120300001 minus 1000000 all right and that's your reduce row excellent.
Birne Binegar: Okay, so.
Birne Binegar: that's the process.
Birne Binegar: always work systematically when you're doing the row actual on form you work from the upper top.
Birne Binegar: and work your way towards the bottom when you're going from row Echelon form to reduce your national inform you start in this corner and you clear out the entries, and you move over and clear out entries.
Birne Binegar: If you don't follow sort of that pattern working from Upper left the lower right and then from low right to upper left you end up going around in circles and you'll never going to get to your usual.
Birne Binegar: it's just by chance you end up at reduce your salon form, but if you follow this two step procedure you'll be naturally led to reduce your actual art form.
Birne Binegar: Alright, so.
Birne Binegar: This problem oops i'm bad at times.
Birne Binegar: But this is an important problem now I wanna I wanted to explicitly, so let me put it up on on screen and i'll just work directly on the on the screen here.
Birne Binegar: Once you get to a matrix an augmented matrix that can reduce your lash line for that's going to correspond to the augmented matrix of the solution.
Birne Binegar: So what you'll want to do, then, is first of all, always look at where the pivots that I have to pivots in this augmented matrix until that's going to mean that X two X three are going to be fixed variables.
Birne Binegar: And the other, the variables, corresponding to the other columns on the coefficient side are going to be free, so what i'm going to do first of all, so i'm going to write down the equations corresponding this augmented matrix X two minus two X four plus x five is equal to one.
Birne Binegar: second row says that X three plus x four minus X five is equal to two and the last two rows.
Birne Binegar: simply say zero is equal to zero well, these are the really the effective equations that come from this augmented matrix.
Birne Binegar: What i'm gonna do is i'm going to use these equations to express the fixed variables in terms of the free variables.
Birne Binegar: So this is going to tell me to X two is equal to one minus one plus two X four minus X, Y and X three is equal to two minus X or plus x five as you move these variables over to the opposite side of course they switch times and then write down a solution vector.
Birne Binegar: substituting where you can for these fixed variables so X one is free there's no pivot net first call, so we leave it alone X two, we have an equation for one plus two X four.
Birne Binegar: minus X five X three x two minus X four plus x five and then X four X five are free variables last step is to pull out the constants in each coefficient or each entry of this matrix and then.
Birne Binegar: terms that are proportional X one well only hear the.
Birne Binegar: term proportional to export.
Birne Binegar: that's going to be nothing in the first slot.
Birne Binegar: To in the second slot minus one, the third slot one and the fourth slot nothing in the fourth slot and then X five times.
Birne Binegar: Zero minus one plus 101 and now I express a solution vector as a point on a hyper playing this use your P zero.
Birne Binegar: This is your parameter times of constant vector all that the one call that vector V2 v3 and then basically.
Birne Binegar: Every solution vector is a point plus some parameter times one vector another parameter times a second vector third parameter times the third vector so it's going to be a three dimensional hyper planes.
Birne Binegar: Okay, well, so I ran two minutes over but.
Birne Binegar: The last problem it's computing inverse all you have to do is write the three by three identity matrix adjacent to that the given matrix and reduce to reduce your electronic form.
Birne Binegar: You can.