Math for Economists - Lecture 2

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Math for Economists - Lecture 2
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Were still in Section 7 . 2 it takes a little while to get through this section that sometimes I go Faster than other times But we have to really take our time allotted unfamiliar stuff 80 solving systems of equations is something you done before usually people haven't done 3 years 4 variables unless you GATT conquest and a little bit there till we need to do is now that we practiced diffuse better
And to get a little bit of an intuition for it where we wanted developer system the methodology for attacking these systems something that works every time in you can just rely on it as as a formula away so what or that it get here is a methodology for solving linear since
I would do this will codified at the end but Sometimes and math sometimes it's good to give the formula 1st and then do some examples sometimes as good to do an example and just cannot work your way through see what what you learn about it while you're doing example and then extract the important information from the example matter this time we take a look at him So there's a good linear system Edward linear applies Oliveri will only have powers or 1 so that were there but is it Describes that phenomenon the system means that multiple equations and I wanna find 1 solution that satisfies every equation simultaneously that the goal if it exist sometimes doesn't exist in this case it is resistant tell you what the solution is the solution of outright easily so that if you want worker mountain lecture tried while I'm solving them we get ahead of where we wanted so you solution here is that X as to why equals 3 R's negative thoughts that at that time I give you the solution now we work toward that I know what you do is just think about this system journey answer this question in your mind was the most Important part what where's the information contained Quest where's the information in this system contain there's a few choices here for the answer But about the variables are the important could I column a B and C instead X Y Z same system right so I column Mickey Mouse Donald Duck and Goofy doesn't matter is just do anything there for the variables ultimately the goalless defined The variables in terms of the number the solution but the variables They had Said do not contain any information so what I wanna do is when I saw all this was our method for what were some of the methods that we are looked at the methods that we liked from high school where we could eliminate a variable and then we could also substitute alluded to good method so really what's happening when I'm tired Substituting in 1 I'm eliminating is I'm just playing with coefficients here so the real information Contained in the coefficients ship so work with this what I'm gonna do is understood extract dollar coefficients psychic take this system and I and distillate And I don't get insulted eats away although Unimportant in what I'm left with Urges the coefficients that's really where all the information What's the label to this
And organize the coefficients unit
We need to see what's a matrix the matrix is a big rectangular swallowed numbers and is understood take all those important information a firmer that up there there's a 1 the upper left corner and then I'm gonna go across the 1st equation going create what we call the 1st row of the major 5 1 0 1 negative 1 and then on a draw a line in here indicate the equal sign
There a lot of different things and talk about what it's the 1st time through slowly here OK now among the 2nd equation extracted all the information from the 1st equation also what I'm doing this I'm speaking I'm writing and reading sometimes on coordinated of the wrong thing on the words of and since Ron known here help me out of a of directly from focus of the 2nd 1 there better to and X negative wanted from why Erica 1 in frowzy in negative 0 on the other side of the equal sign the 3rd equation negative 1 2 3 1 so the 1st thing is just realize these 2 are interchangeable I could write gave you a system like this it is unity like this there's a here's a matrix that represents a system of equations goes all to you could go back and forth by gave you this you should yield a right that equation system talking now let's that's also spend a little time and see what the solution looks like if I had this solution will wouldn't look like in the matrix
Over here the solution Well Let's just follow the model here whatever Coefficients added finally X Y and Z and put him in the matrix so what 2nd look like here well on the For the X I have a 1 but then there's no why nosy X is equal to that That winery their represents x equals 2 because ready It said order put X plus 0 why 0 z that's the same sex equals 2 0 that's an adolescent or China learned early on is that zeros represent missing variables if my goal is to eliminate variables and my goal is to create zeros in the nature of the next 1 y It's like adding 0 x plus Y Plus 0 Z and then that equals 3 and the last 1 0 0 1 negative about the solution looks like away 1 of figured this is this is a jumbled up Rubik's Cube and this is solved Rubik's Cube need a learn how to go from here to here if I can manipulate this in the proper way to tell you what the rules are manipulated the proper when I arrived at this and I get my solution To just like the Rubik's Cube of ING the Rubik's Cube URD had operated by but there's certain rules greatly concerned the hedges and turn the top but you can't Trinidad features that little thing breaks open so there's certain rules I have to follow when I'm manipulating those rules come from what we already know how to do with equation where we do with equations assist us to see 1st 2nd year how how would I eliminate that to exercise my whole focus was to give X in the 2nd equation seated policy I could multiply the Tapie quit in my native to write that doesn't change in this remote but both sides but native to Annan added that would get rid of that 2 acts so what is that he was being here that means I would multiply this role by 2 and add it to the 2nd row and that would cause a 0 here OK For any use of the Tools that we know adding that We already know how do up the System of equations will reply that to the matrix situation Not every year where this 1 is to be convenient for us to get started notice over here what my goal is I'd like battle won the upper left-hand corner like Al zeros so let's try this chatting in this 1st and the wind energy that is and eliminate these This is this is the X in the 2nd aquarium eliminated eliminated that Germany get you might think that is the 1st side of the river skewed and just like what the Rubik's Cube once you get 1 side you don't mess that unity on how do you appreciate that if you get 1 side they chanted in the rest of you at that sell you went backward you have to keep the side and finish the rest of you sitting with somebody of 1 side and then to arrive at that OK this little guy here this has named call at the pit is a little bit like you're pivot foot in basketball
You know what that is in basketball you have your European and is that the foot the you're allowed the move around on it stays fixed and you use that will allow yourself to move around that you who were they used this number to move around in the matrix in the sense that I am many users want to get rid of these 2 numbers that's my goal
It's let ominous start off a commoner rewrite the whole matrix again just I have been here will start little string of Operations It's always dangers recapping the Matrix and leader minus sir something he attractive it can 1 minute due to change its nature In my mind just thinking it's the same as eliminating variables of their If I want to say however that 2 x multiply this negative to had it so I give me that notation to depict what's going on here what I'm gonna do it is I'm gonna replace wrote to pay wrote Tuesday they had replaced and that I'll give you the language afterward direct the symbolism 1st row to win replace it with a winner going negative to 1 plus wrote That's the symbolism wrote will be replaced by negative to row 1 plus wrote it I don't care what else happens all I'm focused on his that that to hear make it is 0 only 1 side of the Rubik's Cube OK this 1 can be gotten rid of this by adding the 2 Solomon replaced Roche 3 with just grow 1 plus and if you like us record set in English Adger math English dictionary this year says replace row 0 3 with 0 1 cluster of 3 you could make that out The Red Group so the arse scans were roads you at the new matrix that I get really when you doing he said something stays the same go get that done right away I'm changing wrote to a changing rose 3 so row 1 isn't changes relate that immediately there's role 1 thing and then wrote to his bid become negative to row 1 plus to song that multiply by negative to an added that makes a 0 here multiply by by negative too Negative 2 plus 1 is negative 3 multiply by negative to makes positive to add 1 that makes positive 3 and you do the same over here a multiplying by negative to negative 12 that I know what I fear that we 2nd row and then there's the whole goal was to start treating zeros and I accomplished something that Caro 3 disk and add these 2 0 0 3 gets replaced by negative 1 and 1 added these to get added a three-year these to get added I have to And then we need to get added Agassi at see if you agree with that book a chilled what I've done a sensors and on 1 side of the You have made some progress here have essentially reduced these 2 this is now 2 variables of our direct this backed out in terms of X and Y widely negative 3 y Plus 3 Z and then 3 choosy equals 7 have had reduced to damage to equations 2 variables which you could go solved but we wanna practices matrix met here so let's do a few more things much less notice what's next what what else would I like him a Rubik's Cube I'd like to have 5 0 1 0 4 that the column so what I'd like to do that like to make this a 1 and let's do that by multiplying by negative one-third this is essentially an equation that I can multiply equation by constant Solomon next that than of replaced wrote
To by negative one-third wrote a novel lots happen in this step row 1 doesn't change row to does change by design my whole goal was to get a 1 year multiplied by negative one-third cars that now see what else happens to be a negative 1 this slight then native one-third of the scale But But And the last didn't change is that they just did what I said Now I'm in a situation that I was just in over here you see this 1 here and this I used it to eliminate these 2 2 guys now let's use This 1 here to eliminate those 2 I notice what happened is that this is not going to change any there are these Because as Addison's canoe 0 times 0 plus that's like change and then I'm a deal that make these zeros but none of the others things in a change of keeper that 1 side of my Rubik's Cube intact over here are reproduced here 1 play negative 1 6 0 1 4 0 3 So what lead you now my whole goal is to make a 0 here 0 here the way I do that is a For this 1 I'm going to replace role 1 where might replace it with Negative wrote Feud blast reward And the whole goal is to make that a 0 see would also yes I don't want to reduce the 5 PM ET at below they carefully would you want to your incorporating this when in this war yet really is row 1 minus 4 2 The same thing but that the way The way I always draws be consistent with this is I'm always put the 1 that's replaced in the 2nd term Yet to be careful doing stuff like that in multiple steps because sometimes you change it in the new and use it again later so I just stick to the pockets of Afghan lets Mrs. do that right away wrote one's gonna change And on doing as I multiplying this by negative and having it So that's 1 0 multiplied by native who makes positive 1 so that it is 0 but I got a free 0 there I was nice multiply this by negative 1 added a few to isn't going to change Is reproduced at and end and that the last thing I wanted his only era this three-year song literal replace Roche 3 ways Negative really wrote to plus rose 3 7 who might be stable why don't you use this 1 here that's a legitimate question why why wouldn't use this 1 answer that in your mind Why am I using this 1 to eliminate that and that and not this 1 So what look ahead and see what happens if I do this and this is something that you might have to work out when you're practicing all you might do this and then you realize what happens if I were to use this 1 what's gonna happen on the ruins 0 by multiply this by something and added a negative want to get rid of this And in a case that 0 so once used this role to eliminate step on move to a different road that's just my experience talking chided Part my experience that that's all I can do some of you have to go practice yourself let's get our row 3 new row 3 when a multiplier wrote that you buy negative 3 and added 0 here another 0 and multiplying negative three-year so I get positive 3 and to 5 And then I'm going to multiply this despite negative 3 that's negative 12 and that it get native to almost home free Were almost at this situation that this column this column take care of us to this now below More nimble than a few times a day on Hinduism multiplied by one-fifth here on a replace rose Really but 1 that rose 3 rolling 1 doesn't change Grow to doesn't change had a 1 year in native what goes so far I've gotta get this straight this part The only thing that's missing is this a straight there by getting rid of this negative 1 and I'll be done with my roots here so let's do that what's gonna happen I'm gonna be replaced wrote To what should I replaced 0 2 with but I add rose 3 to route Roh 3 doesn't change this as a positive wound up food If they wrote to you Is going to change that become 0 1 0 that these 2 3 The answered woes and analyze the swans odd So I removed due solve The 1st phase The first one there is always the hardest 1 of many questions about that Show Now The Foster said this was the complete solution see we always have to go all the way to the end that at 1st I wanna take a few Italy and you'll learn as a more complicated you wanna learn how to stop part way they go back to the equations and get the job done without having to rewrite of these matrices but we still need to practice this a little bit OK so let's talk about what we just is dig here said Carter had some of that method that I just so what I've done is I've just reduced in matrix to what's called row echelon former matrix sometimes called reduced row excellent form an Wannamaker distinctions on scholar were left alone
Are excellent As I've always donors investigate vocabulary We've got a matrix of talkin about the rectangle of numbers on we've got Arroyo here so let's talk about these rose we've been playing around with rows of panel rowers and then what let this Echelon award is that play a role here somebody's knee I sometimes think they're named after people but not in this case the Echelon yet that is aware English-language
Maybe like the upper echelons of society would that conjure up in your mind that city of fresh on different echelons of society which should conjure up as they each time you move up the ladder in society there's fewer people so creates kind of pyramid Each time you go up in the Echelon echelons there's fewer people and so it's more elite is another Place the biscuits used in the military echelon formation Echelon formation is boats or planes ships they they travel like this A war setting the reason why Here's because if you're China fire on them if you hone in on this guy if also that there are like this thing decision at this there and used trainer guns and you wipe out all week break and then If you do in single file then When you a minute guns you'll hit the Horn in on this scale than I do users Like the guy out all so To make a map after recalibrate every single time they do in the staggered was called Echelon formation planes do it plans at manager they can actually do it vertically as well it even harder OK It anyway so that's where the word comes where we see echelons here here is usually national There's actually so that the word Dallas talk about what it is exactly what tavern example here for 1 0 0 who have my example from above Burma had to a little bit had on the role of zeros on 2 negative 3 of the 1st things 1st attention to It is Just drew their that staggered effect and that's what enabled us to solve the system adding that the 2nd and the other thing that New Year's as rose 0 so 1 write down what all the key features an essentially what we're doing is were worth establishing a method for Heidi had solved the roots he is
Notice what I did was I picked that 1 made everything in its column 0 removed moved over the next column I 1 and their thing 0 moved over the next column and I want and their made everything 0 betrayed that down a little bit this didn't happen last but I have to put it in the features could you have to be able to do it when you're doing different problems OK so rose with
Or only we'll zeros are at the bottom Show The 1st nonzero entry This is a that's right in a row G The 1st nonzero entry in a row is 0 1 this
The 1st nonzero is 1 this
The 1st non-zero entries or voters were the 1st 0 entries 1 here is on doesn't have a 1st non-zero treated as violate OK let's set Looks identified as with vocabulary is called the leading water
If OK so that's a sector 1 of those called leading 1 vocabulary we used to describe those 1 of the 3rd feature have described the fact that each 1 moves in that this I have to describe that that feature there were Gabel 1 million moves in So that the leading 1 A 3rd 1 placed to the right of the top of the role but you have a leading 1 the new sees you move down on the leading 1 has to move over to the right that describes Echelon and then here's the way we were solving Rubik's Cube is
Each Column with a leading 1 has zeros elsewhere so notice that that's what happened here add a leading 1 and that column and everything else was 0 years a leading 1 Hello Ontario was if there's a leading 1 3rd column everything else is you that force for me ask you this camera person can you see this damn near at the bottom of the board That OK to go Firm So the 4 features those practice it again that watch somebody do is a little hard to watch somebody didn't because it's a arithmetic and directing shiny new if he outfits differently if his 2nd race here anybody need this board here were reading the So each time I do system that adds some new features that In effect along the same 1 being injured too much to add different thing this last 1 I I made it very convenient batting this 1 in the upper left-hand corner allowed me to just get started right away Here amid a high did a little bit to move things around it it is the solution X equals 1 1 8 0 0 or negative But Who's Gordo clean board rare matrix But
But At germs was
That there is a copy critically I'm ready to go through before use dive right in our goal is to get in this form here so we have a solution pretty far from that there are no 0 so 1 wages 7 tuition for this It is time to create 0 especially beginning gets zeros going in the 1st example I solve this side over here and I got a 1 0 0
But I don't have to do that in fact is not so convenient to do that here what was convenient last time was as 1 so what would I do that researcher anyone anyone's gonna be good even a negative ones so I can use this 1 to eliminate the negative but how I'm old find that I can use this 1 Eliminate that negative to how multiple to So that's really my 1st as I'm gonna replaced road to watch a replaced level bulletin makes zeros here so replace it specifically what I enrolled 1 plus that's going to create that 0 see what else happens I wanna get rid of this negative use on their replaced Rose 3 with Tune row 1 last The I get 1st of all of the top row doesn't change as it is used it The 2nd rose It is scheduled to do that 1 multiplying area thing by 5 Homer multiply this by filing is 15 added I get 17 multiply this by 5 negative 5 and that that's negative view And this 0 by design Multiply this by 5 Agate said 10 at 780 said was due rose 3 . 4 0 3 on multiplying by 2 and adding it gives me a 0 yeah but by 2 Erie 6 and added later 3 Multiply this by 2 I negative to additive for And but despite negative to that's negative forward and that a negative ritzy believe me all the steps there believe myself I don't think they believe myself This is a and multiplying by positive 2 years as positive for amending the 3 But with Do double check work when you do that because once you mess of 1 thing here lost soccer but I made some progress yes what did you start sort of
It would Yet because of questions I tried addresses a little bit But the results as my experience now why why not wanna do this will how would I make this
1 at half the multiplied by one-third the NASCAR because a fraction here Correction you like that so I could if I wanted to allow 1 involve fractions of what you said is widening make a 1 year and then use that as it if you could
The union after multiply this row by one-third That's a treatment of fractions moral void fractions of year fraction G most of us trial attraction I did make some progress I got there I got something I get some 0 now the think about this could I could I just take this column moved over here figures which 2 columns with that change the system legally They were my really do and just removing the disease over the front Exs over to the right there wouldn't change a thing the order of the variables so I wanted to I could actually switch these are not going to add some complexity you have to remember that that the 1st column represent Zeddie in the last column represents an all I wanted to that but you could do not want to get comfortable with the kind of keep going here so now what could I do some I don't really see anyway Create 1 without creating some fraction so were honored do is understood notice this is a multiple of that 2 So let's just make this 4 0 by multiplying by 2 9 use this 1 because of their happen is as I had this one's cheeses years I worked really hard to get the 0 sought only uses rowing moving now move on to the 2nd air made deal to replace rose 3 with 2 road to plus Road 3 wide used to do that and just trying to make some 0 and China void fractions also To get only do that with the 1st row doesn't change reproduced there in the 2nd row doesn't change a lot of work done on the 3rd row is illegitimate multiply this by 2 and Adelaide 0 here and is 0 they're multiply this by that to that's 34 and then add the 3 37 and then on but this the same thing but this by 2 34 and 3 that's 30 set and that's convenient now by could multiply this row by 137 would do that The replace rose 3 by 1 37 rose 3 and opera doesn't change 2nd rd hasn't changed The 3rd row becomes But That could look back at what were what we want we'd like to have a 1 0 0 in the top rate this 1 0 0 let's move this to the here
That's just like switching the equations wouldn't have mattered if I had written the bottom equation that the switch the equations so here's my new I'm gonna So which rose 1 and row 3 that's notation double are there to indicate that and I just switching to 9 1 0 0 1 on top on the bottom I have 3 negative 1 1 2
In the 2nd of the said now could nite is this 1 nugget released to nite do that Ray news this 1 to 17 in the 3 bullets do that now Montana replaced chew with Negative 17 wrote 1 blessed The was do that the top row doesn't change by there when I do that again 0 here I get negative too Here and I get a 0 here Look at that so I say this to become 0 renewing negatives 17 plus this is 0 And then 0 plus negative to negative tube and then I get 0 here 0 What do the last 1 here Bumgarner replaced roast 3 by A negative 3 1 plus 4 It is 0 the 1st but notice also been can't do some short because when I have 0 the market a change anything because no matter what I multiply this by that it doesn't change The for free and then multiplying multiply but by negative 3 remote but here by natives 3 and added a negative 1 pretty close stood Final solution here
Pretty close There are about a lotta zeros This will be the last 1 what we do completely although I just wanna show you all the different role operations Can deal 1 example So we would we a Next time let's make a 1 year ago won there along with a new With a structure with making it back wrote 1 0 0 1 0 1 2 0 0 and then 0 negative for 1 negative 1 we have 1 last step to get the this negative 1 that'll be the salt routes due now so all do with replace rose 3
With throw to plus rose 3 . 4 areas of change said road hasn't changed the last 1 0 0 1 Now we can read off in the office solution that says X Gold won this 1 here said why 0 0 1 says Z most negative toward their solution Dallas take a 2nd look or we could have saved ourselves sometimes I claim all of this stuff was extra does to Gilligan here you need any of this
Very all that could have been avoided if we have to do it wants to appreciate that you can avoid hand and then want to chat Right now I know you probably 1 our voices so you see when I was here I was The had experienced enough when I'm here because this is 37 X
Equals 3 so that I can find next door that X I put here that 17 Thanks but I know X and the I consult for life could Back substituted their let's just We have those not do another example like this us just do that from here from the point where we had this matrix here solve this is really what you're doing is major negative said They'll glitter So here we were done because We we doubt done what we wanted to do wanted eliminated variables right that's eliminating 2 variables so that I could solve that so we do his work your way up the nature of rose 3 that tells me that 37 had equals there he said and which tells you that dagger wrote to 1 wrote to say it's says 17 x minus 2 205 equally 17 but then x equals 1 so that implies negative 2 I was just ready for you that implies 17 miners to equal 17 which means y 0 and then the last 1 rose 3 that says 3 x 9 x y z to and X is 1 of the 3 minus 0 's people to Fellsmere But negative negative from British The Better I don't really want to others that I just want to show you different moves like Switching to 2 rows multiplying rose by Constance but mainly unity just get away with this move radio just doing a few these moves reduces it down to the point where you consult for 1 day you go back and back said that's the real way to do this they questioned the value of that sort of stuff their
But Crew Those The 1st couple or heart is apt explain everything now seen a couple thank Internet to heal the can also go a little more quickly yes Although the constants in the equation to you have Expos wipers E equal something The line represents the equal sign yet is was called a hard minted coefficient matrix alright that that and wants a rate generalization of anyway so it would we don't hear much right down stop Practices in a go on and you to generalization more arithmetic offers little generalization hair
OK so what are we going to doing solving systems of equations and in this case this was a former variable system with 4 quake Asians so Obama generalize this situation where I Manhattan's normally equations and our nominee variables would say we have Mn equations and came in the area There is a of a problem in terms of notation salutes sir to write down a little bit years this is the equation 1 I'm gonna have Have Any variable so I have that X 1 X X 2 and then I have to go all the way down an access and Hannah nasty equals say that these are my variables but let's just before But in coefficients was just get all the equator tho equation too It's X 1 X to I have all the same variables equal was something slightly different too And then OK what I wanna see here agrees to equation for equation 5 amyloid on the last equation is sincere and equations so I good on the last 1 that equation em I still have all the same variables X 1 X 2 X and that's not legal to be set them and now I have to try put in these coefficients in front of each 1 of these guys there's a letter grade incident Florida that's a number of underrepresented by a variable or a letter represents the constant so This is equally In 1 of the scholars Hey now This is the coefficient in equation 1 form of the 1st variable I have to cause the double subscript a 1 1 6 Coefficient in the 1st equation the 1st variable than the next guy A bullet still the 1st equation renounce the the 2nd variable and then This guy is the 1st equation and now and there's a double subscript you can avoid it I've gotta tell build identify where I am in equation which coefficient and the only way to do that is a city which equation And which variables so you have that to subscripts this 1 here is equation Variable 1 this is equation Chiu we're able to an equation to variable and Then we have But gave no you've got that that holiday on the last equation that equation em 1st Variable equation and 2nd variable equation M and OK so that was a little bit a New is that the coefficients of all the and everything At Last put this in the matrix of what would you get out of this mistrust coefficient nature and I usually do here it is Baker said last time Part of this classes to get some mathematical maturity you have to build a deal notation deal with subscripts and things like that so whatever coefficient matrix of light over here and then I have all the constant Cecil B. B 1 2 down to beat him and then the numbers and friend here in a 1 1 and all along this road row 1 And then the variable that I could a June 1 June Although down A Tamil gone A man there's coefficient matrix They called the Automated coefficient matrix because were augmented by putting the constants on the all of these are the coefficients of the had that that's called the argument that what we do it would organize manipulate this in order to solve the system and what we saw on the previous examples as it were doing well operations list identify exactly what we can do is where are legal moves to manipulate this matrix the row operations that we can do is I can replace some he sees this is why doing a general takes practice because I have to specify to say just some role I can't wonder wrote to its some role on the replace it cannot take Some other and had a may baby true Vermont all of it if this is what we've been doing that role operates that's the only do over over again we take a multiple of 1 row and added to the other restrictive finished that example he had had a little bit more of a complicated 1 and think I will continue to
OK row operation number to what else did we do and some of those other examples I have 1 of their yet you multiplying a role by constant just to make it better rate Mr. Havel want her to get rid of the fraction so we can have a place
They by a constant This would be a good place a row over the use of fancy language here With a a combination of blows this is a linear combination of a linear cut will talk more about what linear combinations are laid toward get too bogged down and that but It's just tumult a constant mob but I can't say that much of a linear companies and this 1 here His multiply A bureau A constant that's a legal mode and the last 1 of us had switched to switch to show the 3 things that did in doing so that got us down to the row excellent form in practice do need you really is use this would be a because what happens is If you use this 1 and reducer downed a certain point you back substitute after always due on the steps but these are the legal moves that year allowed him to try and stop
Look at under determined to Watson under German is I have more equations there our fury equations and variables studies of like Playtex plus minus seed 3 2 x while supporters and negative 3 so I at more variables
Then equations to a matrix this 1 1 negative 1 3 2 negative 1 more negative 3 here we have fewer steps to get going here I just wanna put it this introduced in this row Echelon for the 1st step years never used This 1 year to eliminate that 2 of the road to replaced and made it 2 0 1 and blessed with Multiplying by negative to that makes mixes 0 here more of a negative curious negative 3 multiply by negative to positive to Adelaide 6 multiply negative to negative 6 in Adelaide and negative phenomena Make a 1 here here will go replace Road to with negative 1 3rd forming a one-year further easily add here for the undetermined system here Reduced to promote national on this is the 1 where you have to reduce don't have to that it's helpful to reduce it all the way because you end up with infinitely many answers to look back this is the 1 acetate you yourself before you came a classic style so were practicing it again But It's helpful to reduce this 1 all way don't count on that doing back substitution appear focus on and The place wrote to With negative 1 3rd Road to this scare P And 1 last step on the use this 1 to eliminate that that'll be row actually do not replace role ones with negative wrote to reward home free 1 0 PM multiplying by negative here so it's positive to added I don't want more but a negative to native 6 and had it A negative 3 to believe me I'm just multiplying But the negative 1 0 and then that's it book it so this is your role Echelon formal notice It solves all the conditions leading ones and then also Everything units call 0 so this doesn't matter to try to eliminate the surgery in a room zeros so that's a reduced form and lots of folks just see how we get our final solution auto worker So from to get y Time 3 for why pools 3 of Tunisia and then row 1 I had X plus VAT 0 4 x called negatives so you see what happens is I've got everything in terms Aziz so now let's do that parameters thing we did last time said Steve legal team duties any real number And then I can say would X and Y are in terms of a negative equals 3 plus duty and many you same Asia solution that takes practice writing things down in terms of parameters takes practice with He is a scan of abstracted 1st new a By last time to practice that yourself caters watch somebody do that and expect build a master of the craft Take questions are About 20 s It won last little concept 1st solving racist is another special kind of system Oregon a need to look at who got a new word here we know its systems of equations are thought about linear systems here 5 Homogeneous is the the word It's not a modernist like your mill it's homogeneous Know what that really means is a site same family same family of collegiate this is This is a homogeneous system example Set an an example The key to the homogeneous system is that all the constant 0 on the right side at a special case Is going to come up later in studies We have That is a little bit now lower in this context the 2nd equation is the first one on the 3rd 1 is all can't what are clear That's the key to that that word here this is Can you do this in your head I claim you can find a solution Find a solution your head What do you do get 3 simple rate 0 0 and 0 follows all work the matter 5 billion X Y and Z 0 all through those equations work and that's because you have 0 on the right side So Mexico 0 0 Z 0 Is a solution So we done well known there might be other solutions we might this might actually reduce down to the To EDS Under determined system so we might get into only so we have to worry about that years the question are there Others loose The interiors gives 0 there are still yet if there is another solution is actually infinitely many of you still good salty is likely to be other ones will do that if this is a product of void that board effective for brief
Village cwt many analysts also another chance to practice their organization of bogus were OK So it put in a matrix 0 1 1 2 0 0 0 And then I have 3 negative want to make to By 3 practices a bunch now so look
Hone in on the that there nearly put it there that 2 steps to start with a minute to replace wrote to his career this 3 summer multiply this by negative 3 Matters Some replace wrote to negative 3 role 1 Lusk Road to top row doesn't change What do this right now it is 0 here multiplied by negative 3 native What led 3 here is Negative 6 and added So far so good they now want to this negative Susana replaces growth with few world 1 plus road theory that creates 0 here Do the rest now doing 2 times this 1 plus 5 7 2 2 times 2 4 to 7 notices Eros never change I can multiply those an item all day there always is So we can't stop radio again actually done well above the 1 little extra step which a 1 here so let's do not replace row 3 with 1 set and rose 3 . 2 0 0 negative We can also make this Multiply by this by native forced to do that is to make it so it replaces ready to with a negative 1 4th Road will do to operations there and wants that and I you can see what's gonna happen is this this Rocha me eliminate now all do replaces wrote freely with negative Roach plus wrote relief Tobruk doesn't change 2nd doesn't change and the become rogues Sir And now we can do 1 more step to get into wrote a short form will do that replaced wrote 1 with negative road to flush 1 On multiplied by negative 1 and having it what clever negative waters if so there's a reduced role Echelon form and now we can extract be necessary information from them
This same just did in the previous or where they get a parameter for the road shield is that give us that gives us that why plus Z equals 0 implies Hawaii and row 1 says
X plus VAT Equals 0 that lies
Now that last step to organize all this was set of parameter what's let Z equal and then we have X that's negative
Why As people from negative to Is a real solution to you give me a value to you like 5 the negative 5 negative by providers a solution to the original equation
System would agree That's a lot of stuff a lot of watching somebody do arithmetic it's hard watch somebody do that you really to practice that
But I can I went through all the different possibilities has switched gears A little easier because there's a lot of stuff
The switching Years ago it clean water here if if we need to get into Chapter 8 So
But The chapter is all about matrices novel you Lombardi seawater matrix billion formally introduce it we just use it as a tool now organist talking about the mathematics just matrices alone As I said before what a matrix is is it's a rectangle of numbers of formal language for that is it A rectangular A reaI Of numbers and we'll use capital buttered traditional hip couple of things to do I've got the the plural A singular version matrix is singular matrices world If you look of what's here that the matrix notice I'm is lot of numbers were not attaching it to an equation that is behind the scenes you could think of this as having come from the coefficients of an equation they often that's how you get matrices to start with but in this case Just In a steady matrices by themselves another 1 than using capital letters that registered use capital letters to denote matrices denied his collar capital on a little a capital B I said rectangular and was square so we show you a rectangular ones we also had a rectangular ones last Clearly there for the 2 matrices Boone Apple learn what to do with them and was also write down the general major face so if I have I have to specify here this 1 here has 2 rows into columns this 1 has to rose And 3 columns so in general I might have and columns and might have em ballots right in what all these little entries of the very first one is in a row 1 column so I used this is this the same notation using before so I introduced before so this is a row 1 column 1 that I now in Row 1 Column end as is and how down near the last 1 this is row Emma column 1 and then I have row and column
Her is too big Sir We organize this
Saying that a M by that tells you that the size of the rectangle 2 by 3 this will be a 2 by 3 this is a chill in English we say so a Then we say that in English is close to match in situation where you have em rows and columns of the order that we we we decided we do rose 1st as you could say we could switch a Soviet convention we decided that Rose come 1st but you could also say this a couple different ways he could say A has dimensions and by and that's an equivalent statement has dimensions and my and could also said A as order words Those are the words we use to describe the size of a matrix cases lives on in on some special if I have a 1 by end tricks What 1 and right away we have to know which is which is this is a 1 row and column so what's some examples of 84 . 2 3 got 1 row 3 columns was won by 3 or we could have I want to 3 4 or 5 that won by 5 so 1 by end is just tend to lose some examples but we have a particular Ocatillo that we use this this is called a rope stood still a matrix you could call it a major design wrong is just that a convention is We call things that just a single room and also have won for a single call Yukon call sectors will be used to this term by the end of the quarter The course because we're see it all volatile it was the opposite of that is going to be an and by 1 matrix So that should be pockets that M. Rosen only 1 column so these guys a written like this that's 8 3 by 1 make ever 5 by 1 1 2 3 4 5 and guess what those recalled vectors for this he mentioned should just a couple more things than all call Review His ability to stop will stuff here at the start of a meeting after Section 8 . 1 will finish at 8 . 1 I'll be quick next time and then a point to a lot of stuff to do there yet The I let that to the teacher told you tell you section after class to going beyond section 7 . 1 7 . 2 specifically with piranha ask him if he makes a new request Direct Agassi Arizona alleviate discussions after this I get to leave