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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
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
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
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
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
Offene Menge
Mathematik
Güte der Anpassung
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Zahlenbereich
Korrelationskoeffizient
Gleichungssystem
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Term
Ausdruck <Logik>
Linearisierung
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sinc-Funktion
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Phasenumwandlung
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Wurzel <Mathematik>
Physikalisches System
Entscheidungstheorie
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Abstimmung <Frequenz>
Rechter Winkel
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Komplex <Algebra>
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Inzidenzalgebra
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Ordnung <Mathematik>
Nichtlinearer Operator
Bruchrechnung
Punkt
Sterbeziffer
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Vorlesung/Konferenz
Kombinator
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Schnitt <Graphentheorie>
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Familie <Mathematik>
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Zählen
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Eins
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Einheit <Mathematik>
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Klasse <Mathematik>
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Richtung
Verbandstheorie
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