Birne Binegar: morning.
Birne Binegar: Okay, so i'm sorry about last Friday I didn't get the project to work you have to class I figure out how to turn it on.
Birne Binegar: film.
Birne Binegar: Even the recorded version was was was not so so great, so what I did is I on canvas I posted my 930 lecture for Friday, rather than the one who's getting class.
Birne Binegar: But I think we got things working again today.
Birne Binegar: let's see a couple of administrative things first of all homework assignment to do today.
Birne Binegar: And then i've also assigned homework set web assign homework problems at three that will be do a week from today.
Birne Binegar: And then coming up.
Birne Binegar: we're just about through with the material that's going to be covered, on the first exam.
Birne Binegar: Introduce sectors matrices solving linear equations i'll talk about.
Birne Binegar: Planning finding the inverse of the matrix you know that will probably be on Wednesday and Fridays lecture and then generally because.
Birne Binegar: there's take some time for me to lecture and then also assign homework and have you do homework problems.
Birne Binegar: you're not likely even though we'll have covered the material that will be on the birth exam by the end of this week, it probably will not be until two weeks from.
Birne Binegar: early as possible, I think, would be two weeks from today for the midterm.
Birne Binegar: So that will give us not i'm not only to get through all the material, but also to review in class and typically the way I what I do is the class meeting before the exam I dedicate to it to a review session and depending on how we schedule it exam.
Birne Binegar: I may move forward on the material, but it won't be covered on the exam so once I get past in verses of matrices.
Birne Binegar: If there's more than one class meeting before the exam then i'll just go on to new material but but i'll still say that review, for the first exam to just before the first exam.
Birne Binegar: Okay, so now, what I want to do today.
Birne Binegar: Is I want to finish up our discussion about how we saw linear systems using real reduction augmented matrices so what i've concentrated on so far is this process or reduction, how you take a matrix and you convert it into it reduce real excellent form that reduce wrote Echelon form.
Birne Binegar: When the augmented matrix is in that form it's going to correspond to the solution of the linear system and what remains has to convert back to equations and then.
Birne Binegar: realize the solution set as a hyper plane is a geometric object, rather than values for the particular variables so that's going to be topic of today's lecture.
Birne Binegar: So i'll remind you of how we solve systems and then we're going to focus on this last step in the process, but we take a matrix in reducing long form the augmented matrix and we.
Birne Binegar: realize the solution set as a hyper plane and we'll do lots of examples as well purpose of the examples this part you'll see it's pretty easy that just require some explanation, but in the examples concentrate mostly on this process of row reduction.
Birne Binegar: Okay, so the method we're following is, we take a linear system, a system of linear equations and we attached to that a certain matrix matrix.
Birne Binegar: is actually split into two this first factor here on the left hand side is the coefficient collection of all the coefficients that occur in the linear system and that last column of this matrix correspond to the numbers on the right hand side.
Birne Binegar: The way we solve linear systems is we systematically convert this augmented matrix to a new form the so called reduced row Echelon form.
Birne Binegar: Now the way we converted if we perform operations on the augmented matrix that are equivalent to doing operations on the original set of equations that don't change the solution.
Birne Binegar: And the idea if we were working only with equations would be to take the original equations systematically manipulate them until we got the equation, to have the solution but.
Birne Binegar: The augmented matrix method is much, much more efficient, and so we use augmented matrix season, we minutes like that, rather than the equations themselves.
Birne Binegar: Once we get to the augmented matrix and reduce relational on form we convert back to the equations that will give us the equations of the solution.
Birne Binegar: And then the final step that has yet been has not yet been explained, is how to express that solution set as a geometric object, maybe a single point, maybe a line, maybe a two dimensional plane or high dimensional analogs of that so called hyper plane.
Birne Binegar: Alright, so we've covered up up to about step three we've we've gone through and so let's.
Birne Binegar: quickly review what we know so far tickets system of linear equations you can attach to that and augmented matrix and you have an augmented matrix You can write down the corresponding linear system fairly easily.
Birne Binegar: thing to note is that each column on this coefficient side of the argument and matrix corresponds to a particular variable this first call it consists of all the coefficient of X, one in each of the end equations of the original linear system.
Birne Binegar: And to go from the augmented matrix to the original equations you just take a one one time fee associated variable X one a one two times X 2813 times X three etc, and you could easily break down the original system of equations.
Birne Binegar: All right, and then what we're going to do after once we convert to the augmented matrix is rather than work with the equations further will work directly with the augmented matrix.
Birne Binegar: Using so called elementary row operations there'll be things like interchanging the order of the equations.
Birne Binegar: But in this case, it will be interchanging the rows of this augmented matrix or multiplying equation by a number would just be scale are multiplying a particular road after by a number.
Birne Binegar: or adding a multiple of one equation to another that's going to be implemented by adding a multiple of a row to another replacing a row by itself with a multiple of another round.
Birne Binegar: And then.
Birne Binegar: now become to the news part, and the first thing I need to explain eventually we're going to convert the hyper planes, but one thing to explain is the distinction will make between our variables fixed variables versus three variables.
Birne Binegar: So when you have fewer equations and you have known you're not going to have enough restrictions on your variables to get a unique solution.
Birne Binegar: you'll generally get infinitely many solutions and but what what in fact happens is you're able to express some of the variables in terms of other variables, the variables.
Birne Binegar: That you use to express the other variables are called three variables and the variables that are expressed in terms of the free variables are called fixed variables, let me give you an explicit example.
Birne Binegar: Here we have a single equation, and two unknowns if I solve for why i've got why express it's a function of X and so once I know this equation tells me that once I know what X is.
Birne Binegar: Why is determine the whale interpret this is all interpret X is a free variable in the solution, and why, as a fixed variable it's fixed by the value of X, because the equation is true.
Birne Binegar: So yeah so, on the other hand.
Birne Binegar: I can also drive this equation from the original equation, but all interpret the variables differently now and solving for X in terms of why so regard X, as the fixed variable it's being fixed by the value of the free variable why.
Birne Binegar: Right now, this is common whenever we have systems of equations how we solve a linear system we have to decide what are we in a regardless of free variable and what's going to be regarded as a fixed variable.
Birne Binegar: In this instance it look like, we have a choice, but what's nice about the row reduction method is it leads to a consistent and natural choice of what you regard is three variables and what you regard as fixed variables are indeed sorry yeah fixed variables.
Birne Binegar: Okay, so here's here's the way we do it.
Birne Binegar: In our linear algebra technique, we take an augmented matrix and reduce it to its reduce relational on form that.
Birne Binegar: What happens when you have a matrix in Russia on form is they, in renewable Echelon form is that given column if it has a pivot, it has to be one and it can be the only entry in that call.
Birne Binegar: And so now they're going to be certain entries of the matrix and reduce your national on form that have pivots those we're going to associate with free variables.
Birne Binegar: And the columns that do not have pivots are going to be associated are going to start the fixed variables are going to be the columns that contain pivots and the free variables are going to be the columns of The augmented matrix that do not contain.
Birne Binegar: This will become clear on the slide where you have visual example.
Birne Binegar: here's an augmented matrix.
Birne Binegar: Here are the pivots one in the first row, and the first lot second row, and the third slot and the third row in the fifth slot so.
Birne Binegar: According to what I said on the last slide since columns one, three and five contain pivot we're going to interpret the corresponding variables X one X three x five as be fixed variables in our solution.
Birne Binegar: The columns that do not have pivots notice this second column, it has non zero numbers here, but they don't appear as the first non zero number in the row, and so this column doesn't have the pivot nor does the fourth column have a pivot.
Birne Binegar: So in our interpretation of the solution.
Birne Binegar: we're going to use X one X three x five as the free, there is a fixed variables.
Birne Binegar: And since we have no pivots in the second and fourth column we're going to regard X two X four is three variables.
Birne Binegar: Right, let me, let me do that and i'll show you why why this works well for us.
Birne Binegar: here's the same augmented matrix i'm just going to read off from the form of the augmented matrix the corresponding set of equations.
Birne Binegar: One time to X one two times X two zero times X minus one times X four zero times X five is equal to one that's how I got the first equation.
Birne Binegar: Similarly, I can read across the second and third rows of the augmented matrix they get two more equations now, if this is where we come the word the interpretation of variables as be the fixture free is come in.
Birne Binegar: You look at this first equation there's only one fixed variable right because there's only one pivot in any given row.
Birne Binegar: And so here's our fixed variable what i'm gonna do is i'm going to move the free variable to the right hand side of the equation similar in the.
Birne Binegar: second row X four is the free variable i'm going to move it to the right hand side and X five doesn't involve any variables X five has to equal three.
Birne Binegar: And so, when we move the free variables, to the right hand side, what happens is, you see that the fixed variables X one X four X five or now expressed in terms of the free variables and that's generally the way if you.
Birne Binegar: Generally, the way you would write.
Birne Binegar: A solution, set up a linear system expressing the variables that you can express in terms of free variables you do so, and the variables that are left undetermined you put on the right hand side of the equation.
Birne Binegar: Okay, so.
Birne Binegar: Making the distinction between three and six very both like distinguishing between columns that have pivots and columns that don't have pivots that's what allows us to write down the solution in this manner, it helps us to decide what we regard it's free.
Birne Binegar: If I didn't look at that augmented matrix well there's not a clear choice as to which of these three variables, I think, regardless it's fixed and which I regard is three is a reference to the augmented matrix and reduce your last line form that we make that decision.
Birne Binegar: All right, and then, once we've written down.
Birne Binegar: Once we identified the values of the fixed variables, we can now use that to actually formulate the hyper claim expression for the entire solution set.
Birne Binegar: Alright, so what i'm gonna do is first of all i'm going to write down a generic i've dimensional vector but now i'm going to use the equations.
Birne Binegar: of our solution, they now are expressing the fixed variables in terms of the three variables or substitute where I can in the spectrum so X one on the previous slide was equal to one minus two X two minus X four.
Birne Binegar: And so all substitute for X one its expression in terms of the free variables X two is a free variable by leave it alone X three can be expressed in terms of explore, so I make that substitution.
Birne Binegar: explores the free variable I leave it alone and X five we've seen it has to equal three if the equation is satisfied.
Birne Binegar: All right now, what we have is a solution director that's completely determined by the values of X two X four.
Birne Binegar: we're getting getting close to the hyper plane solution at least now we're.
Birne Binegar: realizing your solution as a vector rather than concentrating on the individual values of the variables.
Birne Binegar: Okay, so i'm going to.
Birne Binegar: express the solution factors in terms of the free variable the last step is to.
Birne Binegar: Take this vector expression for the solution and rip it apart we're going to look at the constant terms that appear in each component and how the free variables appearance each component.
Birne Binegar: So he's concept terms i'm going to pull off.
Birne Binegar: In a single initial vector and then i'm going to have terms.
Birne Binegar: In each component that are proportional to X two i'll grab those and put them in a single bunker and then there'll be terms that involve X for i'll grab those and put them in a single vector.
Birne Binegar: it's like taking my solution vector and i've ripped it apart, depending on how it depended on the free variables and then because the second factor.
Birne Binegar: is apparently X two times this constant vector this last vector is X for scale or multiplying this constant vector.
Birne Binegar: And now we've reached my goal and you may not recognize it, but this expresses a solution vector as a point.
Birne Binegar: plus an arbitrary number X two is a free parameter, so it could be anything at all in which still produce a legitimate solution, so we have a point plus some number times one vector and another number times a different vector.
Birne Binegar: That means we got a high proclaim.
Birne Binegar: first week of class, we talked about hyper plaintiff setup vectors that we can form like picking a starting point.
Birne Binegar: And then straying from that starting point in a given list of directions in this case B one and B to be very these numbers to you wanted to so that we actually fill out and an entire plane like very just two parameters.
Birne Binegar: This expression houses form that correlation with the this constant vector P zero if that constant vector there this constant vector V one.
Birne Binegar: corresponds to this constant vector and this parameter that very well, it corresponds to our ability to vary the free parameter X two.
Birne Binegar: And the second parameter that we can vary in constructing this hyperplasia it corresponds to our ability to vary the three parameter export so indeed.
Birne Binegar: This expression here expresses all the points on the of the solution as lying on a particular hydroplane hyper playing this specified by these three vectors okay so that's the goal that's the goal and.
Birne Binegar: What all.
Birne Binegar: I guess all quickly review this and then what i'm going to do for the rest of today so like she was just worked out examples.
Birne Binegar: Okay So how do we go from the augmented matrix to the hyper play.
Birne Binegar: First of all, we look at the augmented matrix and reduce your salon form and we decide what variables are going to be fixed and what variables are going to be free.
Birne Binegar: The variables that correspond to columns of The augmented matrix that have pit it those are going to be the variables, you can solve for in terms of the free parameters, the so called fixed variables.
Birne Binegar: And then the variables that correspond to columns on the augmented matrix to do not have tickets, those are going to be the free parameters in the solution for the three variables.
Birne Binegar: Right, then we write down the equations.
Birne Binegar: Once we got the equations written down we systematically move all the three variables from the left hand side, where the first appear over to the right hand side, leaving the fixed variables stuck on the left hand side.
Birne Binegar: Then we write down a solution vector using the equations we got in this step to replace the fixed variables by their expressions in terms of the free variables and last step is to.
Birne Binegar: write down a solution vector or you substitute.
Birne Binegar: You substitute for the fixed variables in terms of the free parameters and you expand that solution vector in terms of the free parameters.
Birne Binegar: All right, though.
Birne Binegar: verbally when I write it down it doesn't look like it not.
Birne Binegar: That obvious what actually going on they'll become more obvious when we just do one example.
Birne Binegar: Okay, and in fact in this example i'm going to show you not just how to carry out the method, but what happens when.
Birne Binegar: When, in fact, our system of equations are not independent of one another that maybe there's some connection between these equations.
Birne Binegar: So, if you remember in our initial discussion about solving linear equations then naive expectation that you get simply on.
Birne Binegar: From your understanding applying school algebra if you have five equations and foreign owned well Let me pick a different example may have five.
Birne Binegar: nodes in three equations each equation could be used to eliminate one variable and so your solution set would be five minus three or two dimensional solution set.
Birne Binegar: Well that's true only one the equations are actually independent of one another, if these two equations absolutely applying imply the third equation, then.
Birne Binegar: Imposing the third equation doesn't put any new restriction on the variables, and so my the actual set of independent equations it's going to be smaller than your original list of equations.
Birne Binegar: The right way to count the size of a solution set is to take the number of variables and subtract the number of independent equations but.
Birne Binegar: When you're given a linear system it's not always easy to see that that there's two equations might imply the third what i'm going to show you is how the row reduction technique.
Birne Binegar: Things that information out makes it obvious.
Birne Binegar: Alright, so.
Birne Binegar: Here i'm just repeating what I said all already.
Birne Binegar: If I had three equations and three announced, you would expect a zero dimensional solution, a zero dimensional solution does not mean that there's no solution.
Birne Binegar: but rather what it means, is it the solution is zero dimensional, and so it consists of a single point.
Birne Binegar: All right now.
Birne Binegar: let's just.
Birne Binegar: apply our technique for solving linear systems using again and matrices I take the a system of equations and I convert each equation, to the a row of the augmented matrix.
Birne Binegar: Then i'm going to apply my role reduction, the goal now is to get to the augmented matrix of the solution.
Birne Binegar: Right this three by four matrix is not enroll exelon form.
Birne Binegar: robot show on form means the pivot entries or whatever they are, they have to be occurring.
Birne Binegar: In a sort of downward staircase as you move.
Birne Binegar: from left to right in this matrix the pivot should be dropping steadily down they never changed direction and appear off to the left of a pivot in the row above it.
Birne Binegar: So if we're going to get a matrix.
Birne Binegar: In row Echelon, for the first thing I have to do is get rid of these pivots that are fitting directly below the pivot in the first row.
Birne Binegar: Well, we have a row operation that will do that for us, pretty easily what we try to do is take us multiple this row add it to one of these lower row, so that the leading entries exactly Council.
Birne Binegar: So, to get rid of this guy what i'll do is i'll subtract the first row from the second row and use that as a replacement for the second world and to get rid of this.
Birne Binegar: Number three in the first column here what i'll do is i'll take three times the first row that will give a component three up there and i'll subtract it from the third round and i'll use that to replace the third row.
Birne Binegar: So.
Birne Binegar: i'll do maybe.
Birne Binegar: This operation.
Birne Binegar: One minus one is zero minus one minus one is negative two one minus one and zero and zero minus two negative two, so these can be before really rapidly you just need to space just need to figure out the pivot you want to use to eliminate.
Birne Binegar: pivots that are below it and then all you have to do is multiply your pivot row by an appropriate scale or factor, so you can get the desired cancellation.
Birne Binegar: All right now I end up with a matrix it's not yet in relational on for because, up to the second row, it is in row Echelon form, but now this third row, it has a pivot directly below the pivot in the second row I need to move the pivot and the third row off to the right.
Birne Binegar: So i'll try to cancel this thing out by adding multiple the second row to the third row.
Birne Binegar: And what happens in this instance, is that the second and third row exactly cancel.
Birne Binegar: All right now.
Birne Binegar: getting a little bit ahead of ourselves, let me look at the equation that corresponds to this last row it says.
Birne Binegar: zero times X one plus zero times X two zero times X three is equal to zero well left hand side simplifies to zero, and so.
Birne Binegar: One of the equations of of the system is equivalent to set just saying zero is equal to zero, which is not putting any restriction at all on our veritas.
Birne Binegar: These row operations are not changing the solution at all, so, in fact, what these what these row operations have done so far has revealed that there's a triviality in the original set system of equations.
Birne Binegar: What were the real restriction on the variables come from is from these first two rows this last row in the augmented matrix is.
Birne Binegar: not saying anything mathematically except an identity that zero happens to equal zero doesn't put any restrictions on on the variables.
Birne Binegar: What happened is we used operations that makes simplify the equations, but we did not change the solution set and what we found is that one of our equations somewhere amongst our equation was a triviality that could be removed.
Birne Binegar: Alright, so, in other words, we started off with three equations and three are known, and we see that that's effectively have the same solution set as two equations and three.
Birne Binegar: Right and and that way.
Birne Binegar: In that way, the.
Birne Binegar: Reason redundancy amongst the original equations was exposed.
Birne Binegar: Alright, so let's.
Birne Binegar: let's continue.
Birne Binegar: we're gonna keep going until we reach reduce your last one for the distinction we're now want to impose on this matrix is that the pivot have to equal one.
Birne Binegar: that's the first condition, and the second edition is that above and below pivots I should have nothing but zeros well getting it into row epsilon form we've already got zeros city below the pivots here's our problem and non zero entries that sit above pivots.
Birne Binegar: So we'll fix the pivot first by multiplying this row by minus one half that gets the pivot equal to one.
Birne Binegar: And then last thing we need to do is to get rid of this entry above puppet.
Birne Binegar: So let me remind you.
Birne Binegar: Reduce row Echelon for me three things.
Birne Binegar: needed to row Echelon.
Birne Binegar: It means pivots.
Birne Binegar: have all been normalized one and it means above.
Birne Binegar: And below pivot.
Birne Binegar: Only zeros.
Birne Binegar: Okay.
Birne Binegar: And so.
Birne Binegar: What prevents this matrix from being in reduce your last one for is the appearance of this non zero number above the pivot in the second row but it's easy to get rid of this guy we just take a multiple of the second row we added the first room will get the desired cancellation.
Birne Binegar: Now we simply send the first row, to which difference with the second row and now using just elementary row operations, we recast the original augmented matrix into reducing national on form.
Birne Binegar: This matrix is unique up for give you start off at the given matrix there's only one matrix and reduce all Echelon porn.
Birne Binegar: You may have many different ways of reaching that final form that destination is unique Okay, so this, in fact, is what will interpret as the augmented matrix of the solution.
Birne Binegar: Right now.
Birne Binegar: Having reached the augmented matrix of the solution we convert back to equations here it is.
Birne Binegar: First, this augmented matrix says well here's written out in full glory both playing each.
Birne Binegar: entry in the row times the corresponding variable and then setting it equal to the number on the right hand side but.
Birne Binegar: The system simplifies two X one plus x three is equal to one X two is equal to one and the last row simply says zero is equal to zero, so I really only have two equations on my three unknowns.
Birne Binegar: What we'll do is.
Birne Binegar: we'll make the distinction between fixed variables in three variables, so that we can proceed to get that hyper plane solution.
Birne Binegar: Okay let's look at this matrix.
Birne Binegar: We have a pivot in the first column so X, one should be regarded as a fixed variable pivot in the second column X two is going to be regarded as sticks.
Birne Binegar: No pivot in the third column, we have a non zero entry but it's not the first non zero entry so it's not a pivot so X three corresponding to a calm, without a pivot is going to be interpreted as a free variable in the solution.
Birne Binegar: Right and now.
Birne Binegar: From this matrix we interpret X one X two X six X three is free and we simply take the free variables coming from the equations attached to this augmented matrix we simply move the free variables, to the right hand side.
Birne Binegar: And now, each of our fixed variables is expressed in terms of either a constant or content and the free variable.
Birne Binegar: and
Birne Binegar: We can write down a typical solution vector substituting where we can for the fixed variables so X one with equal to one minus the three variable X three x is equal to the constant one and X three is a pre variables, will you leave it alone.
Birne Binegar: And last step is to stare at this sector and see how we can rip it apart sort of expanding it with in terms of powers will not even powers it's just linear terms in the variables, so this the specter here can be thought of.
Birne Binegar: As a piece, that does not depend on any variable that all one one and No dependence on the on no constant in the third one, which 110.
Birne Binegar: Then we have terms that involve X three it's going to be minus one in the first component and no X Ray and the second component, but one times X three the third component and we've got our solution, and in fact it's expressed as a line it's a single point.
Birne Binegar: And then we stray from that point and arbitrary mount X Ray and the first line.
Birne Binegar: Here I just call that parameter T and the second line I create a geometric line by starting in a point.
Birne Binegar: departing from that point along the particular distance and along a particular direction but an arbitrary distance on that direction and it generates align.
Birne Binegar: What we're effectively doing here is constructing align get a coordinate system here like plotting a point that would be my P zero choosing a direction D and then straying from that point.
Birne Binegar: One way or the opposite way alone that particular direction and that will fill out the entire infinite line.
Birne Binegar: Okay, and so there we have it, we had a system of three equations and three announce our role reaction process expose that we really only had two independent equations.
Birne Binegar: And so, three on three variables minus two independent equations not three minus two is one, so we should expect a one dimensional solution, and in fact there, it is a line.
Birne Binegar: Okay, so that's that's how it works out in this case.
Birne Binegar: We do another example where another.
Birne Binegar: problem, so to speak up within a linear system becomes apparent.
Birne Binegar: When I was talking geometrically about.
Birne Binegar: Solutions of linear systems.
Birne Binegar: I phrased it at every equation corresponds to a certain hyper plane if we have.
Birne Binegar: got the dimension that hyper plane is one less than the number of variables, you have because you can think of it as.
Birne Binegar: Using the equation to re express one of the variables in terms of the others, so if I have 10 equations or, if I have one equation antenna knowns my solution space should be a ninth dimensional hyper plane sitting inside of 10 dimensional space of variables well.
Birne Binegar: In that interpretation one thing that's possible geometrically is to have two equations that are parallel or to hyper planes that are parallel to one another and never intersect that would that's going to happen, whenever your equations in take container internal contradiction.
Birne Binegar: here's how that.
Birne Binegar: That situation, what it looks like when we're doing the rub redemption process.
Birne Binegar: Alright, so um I set up this example feel that it has a contradiction, let me show you how I constructed this example, I took the first two equations and I am.
Birne Binegar: I subtracted one from the other and I get a new equation, so the first two equations actually employ this third equation, but we didn't have that.
Birne Binegar: This would not be any new information because it uses the information the first equation to produce this equation.
Birne Binegar: When I added this equation, you see it conflicts with the the equation I derived from the first two.
Birne Binegar: First, two are telling me that y minus two Z is equal to one last equation says y minus two z's equal afford all these cannot both be simultaneously true because four equals y minus two D does not equal one.
Birne Binegar: Okay, so.
Birne Binegar: i've constructed instead of equations, for which I know there's no solution let's see how about its exposure when I just night easily apply our linear algebraic technique for solving linear equations.
Birne Binegar: right down the augmented matrix corresponding to this linear system.
Birne Binegar: And we start reducing.
Birne Binegar: The first step is to get rid of this pivot below the pivot in the first column that's done that row operation, and then we have.
Birne Binegar: A pivot below a pivot and we'll do another elementary real operation to clear out that entry level should track the third row will subtract the second row from third, and now we end up with this augmented matrix.
Birne Binegar: Let me convert back to equations to show you what's going on.
Birne Binegar: That third row.
Birne Binegar: When I write it down the corresponding places I get zero times X one zero times X two zero times three is equal to three well the side evaluates the zero.
Birne Binegar: On the other side, we have three so.
Birne Binegar: The equation that corresponds to this last row it's contradictory unless you figure out some system of mathematics or zero is equal to three it's gonna it's going to be contradictory and so.
Birne Binegar: What you do.
Birne Binegar: When you uncover phenomena like this and you're robbing options you've stopped because this in fact tells you that there is no solution to your original in your system somehow.
Birne Binegar: Your equations have an internal contradiction that cannot be resolved until there's no solution.
Birne Binegar: what's striking, though.
Birne Binegar: Is that when you look at the original linear system if you don't see that contradiction rather it's been exposed by our process or production.
Birne Binegar: What the road reduction is doing sort of intuitively it's it's we're acting on the augmented matrix but the augmented matrix always correspond to systems of equations were systematically.
Birne Binegar: manipulating the augmented matrix so the corresponding equations are getting simpler and simpler and here they got so simple that we actually saw explicitly the contradiction that was contained in the original equations.
Birne Binegar: Right so.
Birne Binegar: in general.
Birne Binegar: During the process of Robert auction, you can have zero rows that appear when a zero appears that means somewhere in your original set of equations you had some redundancies and that.
Birne Binegar: You had more equations and you actually needed to find us the complete solution.
Birne Binegar: But that's Okay, if that happens, you just means you have fewer rows where the action is actually going on whether zero doesn't mean there's no solution it just means that one of your original equations.
Birne Binegar: That was actually redundant information rather than a further restriction on your variables.
Birne Binegar: But if a pivot appears in that last column.
Birne Binegar: If you have a row.
Birne Binegar: were on the other side of this dividing line, you have a non zero number, this means it somewhere in your original set of equation, there was a contradiction.
Birne Binegar: And if you find a pivot.
Birne Binegar: See, this is the first non zero entry in this last row, if you have a pivot in the last column, it means your original system with contradictory nurse, no solution at all.
Birne Binegar: So now that's the way that row reduction is going to expose the redundancies amongst your equations and you're hitting contradictions.
Birne Binegar: But it also.
Birne Binegar: exposes that.
Birne Binegar: Those two phenomenon, it also allows you to detect right away the dimension of your solution set.
Birne Binegar: Because effectively as long as you don't have zeros like this and you've got it in real Echelon form.
Birne Binegar: The equations that the roasted do survive is non zero rows of the matrix and reduce electron form those non zero rows are going to correspond that independent equations.
Birne Binegar: So you can get your dimension of the solution, set by simply taking a number of variables and subtracting the number of non zero rose and the reduce your actual on.
Birne Binegar: Okay, so.
Birne Binegar: me step through this program how we solve linear system theory.
Birne Binegar: You take a linear system right down and augmented matrix and then we use the three elementary row operations to systematically converted first row Echelon form.
Birne Binegar: that's clearing out the entries below pivots and then we convert it to reduce cholesterol on form like forcing the pivots to all equal one and then clearing out the entries that sit above pivots.
Birne Binegar: And just using these three operations.
Birne Binegar: Know there's a matrix in relational on form these Asterix is here means it can be any any number you want, as long as the.
Birne Binegar: The pivots are occurring in this downward staircase from the upper left to lower right you'll be in a row Echelon.
Birne Binegar: get it and reduce your actual inform us this operation.
Birne Binegar: multiplying rose by a constant factor to re normalize these entries these pivot entries so that they're actually have value one.
Birne Binegar: notice how our that this operation does not change the position of pivots in the matrix they're staying the same location if you're sort of their value is be readjusted.
Birne Binegar: And then.
Birne Binegar: Then, once you've got the pivots re normalize to have value one your next objective is to get rid of the entries that sit above pivot and use this operation you Hello rose.
Birne Binegar: use these templates to clear out the entries above it.
Birne Binegar: I should point out that there's another distinction to be made here the process by which you reach row Echelon form.
Birne Binegar: The ideas you start up in this upper left hand corner remove all the entries below that leading pivot, then you move down to a lower row you clear out the entries above so you start in this upper left hand corner you systematically work your way down and off to the right.
Birne Binegar: The process by which the matrix in row Echelon form is converted to reduce cholesterol on form.
Birne Binegar: After you read normalize the pivots you're working upwards until you start in the lower right hand corner clear out these entries, then you move to the next pivot in an upper row we're actually the next row.
Birne Binegar: The next pivot because a laugh you work with that gun you use that to clear out the entry there and then you move here use that to clear out the entries here.
Birne Binegar: So the first half of your road we got to you, moving from Upper left to lower right in the second half, you moving from lower right up to towards the upper left, but you follow that pattern you'll systematically convert in arbitrary matrix into its unique.
Birne Binegar: row, you know, reduce row Echelon form.
Birne Binegar: All right, and then.
Birne Binegar: Once we reach reduce cholesterol on form we decide what variables we're going to interpret this six variables, the variables that we can express in terms of the free variable and the free variables themselves.
Birne Binegar: Fixed variables look for the columns that have pits three variables are the columns that do not have it is.
Birne Binegar: Alright, so here's an example matrix and we do show actual inform you have your pivot variables i've underlined them X two X three x.
Birne Binegar: X two X three x five I guess i'm like three variables are so fixed variables corresponding column with pivot see this column here most common mistake on the first exam.
Birne Binegar: there's no variables here in fact X one does not appear in any of the equation, so do we forget about X one no, it remains a free variable and the solution.
Birne Binegar: will still will still take.
Birne Binegar: Each row with this of this augmented matrix and will convert it to equations and we'll still move the free variables over to the right hand side.
Birne Binegar: But we don't forget about X one.
Birne Binegar: It hasn't appeared yet, but on the next slide it will appear.
Birne Binegar: you're going to write down the solution vector well there's no equation in determining X one so i'm going to leave it as a free variable.
Birne Binegar: There is an equation determining X two X three x five so i'll use the equations to replace the fixed variable X two X three x five.
Birne Binegar: In terms of the free variable X four X six and well don't need it, but X one is also a free variable now i've got my solution vector expressed in terms of the three variables.
Birne Binegar: What remains is to expand the solution vector in terms of the free variables here's the constant term 321 there are terms proportional X one, the first component.
Birne Binegar: that's the only place it occurs, so you get this simple factor there explore appears in three different components as coefficient minus two and the second component.
Birne Binegar: coefficient minus one, and the third component and coefficient of sorry young coefficient one in the fourth component and, similarly, you can see the dependence on X except.
Birne Binegar: And now we've got a typical solution vector expressed, on a point on a three dimensional hyper play here, we have three different ways we can stray from this point and.
Birne Binegar: Each of these vectors is a six dimensional vector so, in fact, our solution set is a three dimensional hyper plane embedded in a six dimensional space, the six dimensional space of variables.
Birne Binegar: OK now let's see oh i'm about a time, I have two more examples on the slide but I I don't think i'll go through them, you can look at those yourself and you should be looking at the homework problems as well, that will give you practice, the thing you really need to.
Birne Binegar: become efficient, it is this process of reduction, because what you'll see is, we not only use it the status to.
Birne Binegar: find solutions of equations will use it over and over again in other aspects of this course.