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Math for Economists - Lecture 5

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We learned about determinant today determinants wasn't just last time 1st Foley did was we did the 2 by 2 matrix case and actually found the hole in matrix along the way we discovered this thing called determinant them spending a little bit of time looking at determinants of larger matrices may finally got a determinant Definition and we can ride out in In words it's a little cumbersome but simple way way to describe it is what's called factory Expansion of long any row or column remind you would cofactor as cofactor CD-I J. which was negative 1 died today and by day and then MIJ was this whole other determinants of a smaller matrix this was the thing where you crossed off a role McCollum and then you had some leftover numbers Egypt determinant of that so you can go back and look to that this is just an overview of the cofactor expansion what that means is that take the elements of the row or column that I choose and I multiplied Each 1 by its corresponding cofactor that if we had a few examples we did that 3 by 3 we didn't you different ways is using Roma column now Oregon and build on that and do a poor by
War Will start simple and then all of a sudden grow a little bit So 1st kind of intimidating the 4 by 4 matrix is a lot numbers in there since we can choose any row or column it's gonna be preferable that are row or column that we choose Has as many zeros as possible because what we do if I choose said pick this tomorrow right here all the motivated illuminates 8 1 times its cofactor plus 2 times its cofactor plus negative 1 candidate without means 3 computations if I choose instead of doing that they choose this column here Will Miller 1 Computation amenities 0 times whatever vertical factors that are needed computer go back because multiplied by 0 stuck in a matter that I have this negative 1 times its cool factor in an 0 times other cofactor so those going show up to this It is The smallest amount of computations that we can achieve having A single column with that 1 nonzero entry by the way I could have also chosen that 3rd World there would been just find that I have pick the call OK So and the other thing I needed to make sure I keep track of is the pluses and minuses The last time we talked about when you compute the factors that this negative once the again so it's better if you just keep track of that in your mind I was just that the whole matrix year here even have to do the whole thing when you're doing just 1 of Maybe refer to a little bit it always starts with a plus in the upper left-hand corner because that means 1 plus 1 even negative 1 where his father Then adjust alternates from there So by I just keep track of this while I picked the different elements here in fact when you're doing this in practice which you might do use want you pick your column just don't put down the positive data is next to its sister remind yourself so it could be plus-minus plus-minus that's the column that I'm doing so I could close plus minus plus minus there if you like and is showing a little tricks that you might develop as you do more these just Sure things are OK So the determinants of playing in this case it's given me 0 times it's called back to not show up some derided phenomena have negative 1 times its cofactor religious rented out for it Just a practice what what is the cofactor periods in the 2nd row 3rd column so I could write like that seek to 3 not necessary that you step For practice the Stuart case so that's equal to negative 1 at times now rate that with the cofactor is that negative ones to the tube plus 3 times MMBtu 3 Now this is what I say no you don't have to write this attitude leaders realize that the negative you just cancel the negatives essentially focused on the threat out what this says M 2 3 summers adjust them to figure negative want negative 1 knowledge right I them to 3 years and 2 3 a looks like it's a matrix because of the notation but we have to realize it's the determinant of a matrix is not just a major so I can use the street bracket notation compare that with the original matrix for 8 years The regular brackets OK in And 2 3 I didn't take the determinants of the Matrix where I've removed the 2nd row in the 3rd column in The so if you like you can rent it all out market for a 4 by floor Solana work on maybe do that the the notes and just 1 extra steps on and 3 is all matrix 1 0 0 1 1 2 negative 1 0 0 negative 1 0 0 Negative 1 0 0 1 Always dangers copies these things and lists of information OK so and duties not this it's the determinant of that checks were removed the 2nd row in the 3rd column so that the same as saying it's Equivalent to this 3 by 3 matrix 1 0 1 0 negative 1 0 The negative of 1 0 1 Now that he is chairman of the 3 by 3 major but see we have the luxury of picking Aurora columns on an epic once again the 1 that makes the fewest amount of computation here I have 2 choices it seems I could their picked the Column here that has only that negative 1 in it or I could pick that row on the column here is practised OK so it's right that out do when I do the cofactor expansion for this determinant multiply this it's negative 1 now in this configuration that's a positive location if you want The plus minus plus-minus rating You could see that this negative 1 occurs in a positive lot of this is equal to a negative 1 time negative ones Plus 2 and which is equal to a negative attitude to me they cross off the a 2nd round 2nd column for a 1 1 1 1 that's sequel to negative 1 plus 1 to negative to the completion of that determined So good news and bad news when you have a 4 by 4 matrix typically period Eliasson do automatically give you 1 that's got a bunch of columns Rose with zeros and you have to achieve that yourself but but that reducing it down to the smallest amount computations get it have to do much even I had a 4 by 4 so the next level would be give you more complicated for by foreign minister to use some of those Is that we won over last time 1 in particular so here's my new matrix a 30 day period Messier 4 you want 1
I wanna get the determinant of this
Don't just jump right in with these things pause and look at it for a 2nd and try to figure out your best course of action so when I since I have some experience dangerous shared with that share that with you so when I see this as I say I'm looking offers many zeros as possible the best I could do is this is part of the best column I get I know there's this column with a 0 in others that role with 0 those attitudes and rather have 1 so I see that as being the best column but I don't wanna just jump right Minster finding the determinant because that's going to require me to do 3
Different 3 by 3 matrices as I do the cofactor expansion so I'd like to do 1st is due tomorrow operation if you recall theorem arrived here only 5 years 9 . 5 says I can do role operations of the normal type that we've been doing right take a multiple added to another row There 9 . 5 says this doesn't change the determinant so I might as well take a step to do some role reduction to get myself in that situation where I have a single nonzero thing in the 3 0 so that it won't take as much to do to roll operations just to get rid of this goes Ones so what might be due for minute but I'm replace wrote To deal with Negative role 1 plus wrote to fill those things in the top rose market change that reduce it The 2nd row is change change but by multiplying the top row by negative ones get negative 5 efforts by multiplying by negative 1 that added 3 0 here by design and then negative 1 plus to me a 2nd row and I also won a modified this 1 right here Replace rose 3 with negative role 1 plus growth Every day when I do negative here negative for plus native to negative 6 5 made human 0 his negative to major 1 and 1 0 negative 1 2 is 1 that takes care of the 2 rose I was interested in modifying now understand reproduce that form so that the nice thing about June all at work is that the determinant of this matrix is the same as the determined that
So when I wanna computer determinant today I might as well use this 1 which would you prefer this is this is gonna be a column with
3 0 which just makes only 1 major really have to to do the determinant OK now just just 4 of your practice Since this is the only guy that I'm really Unitas let's just go and get the plus-minus his plus-minus plus that's a plus location for the cofactor of this determinant is equal to 1 time B C 1 3 cofactor on any of the others to be president and that's going to be Equal to the determinants of the matrix negative side negative 3 1 then made 6 negative 2 1 1 2 That original nature it's a same is just computing the determinant of for this 1 you could offer dual role reduction for this wanted if you like but there were 3 back Trees everything reduces them to buy to see muzzle just go for here good practice anyway OK but our pluses and minuses and this is a plus And minus and plus in Soweto plus or minus Plus seat he could follow this determined to make sure it has a thing where OK Some of them the 3rd column here but it's going to be 1 time the Matrix leftover once I eliminate the row and column that that 1 is better than natives 6 native to 1 2 And then the next 1 year this here this is minus 1 that's from the co factors He write this minors Israel That's why it's a mine is not a plus Kim operating on this 1 right here and across the its role and its column and what I have leftovers negative side negative 3 2 And the last 1 this is a positive location no need to modify 1 times the determinants of negative negative 3 native 690 If Slow days OK I've got negative 6 hours a native 12 Plus 2 that miners and make it if 10 Plus 3 Plus that Woods was negative to So positive 10 minor team group negative 10 Plus Prime determinant of that original 4 by 4 matrix yes 2 These here focus on computing this determinant here so it's negative 12 hits Natives extends to his native 12 minus negative 2 times once It's minus native to which is a plus to remember that that the determinant of a C D Is equal to A. D I see that's the determinant of to like determinants and that boiling down to to determinants of 2 by 2 you want the yes I do For the three-day peak You may reduce it more Withrow operations yet so it is that actually lowered about the way he said that that I would remove If I if I understand you correctly you want to use that makes zeros 1st by the way we did that the 1st step How to our tough for that No overnight is getting rid of Rose we did was 3 zeros That allowed us to have only 1 3 by 3 major it if I didn't have a 0 that I would've gotten 3 3 by 3 meters so that step appears to create a 0 so that I only get 1 was not is crossed out The and created at least that was the row reduction and we use that theory to justify that
But
I want practice became just that take other roads or something you know just you know what the answer is seeking just keep doing it at a lot of mileage out of just 1 matrix by just picking different Rosa and manipulating it differently should always That means that the bid efficient way of studying
Who pay Should We went through all those years last time so what I wanna do is I wanted to a couple of proofs that required Sears while they're still relatively recent Indian music of remedies area their exercises there in the book had assigned suggested that I wanna do them for it because I don't want assume the United improved trading of the raid at a few cruises a couple of 3 cruiser unity responsible for I'm giving you 1 of Omar ready and then the tomb about to do are the other 2 not a complete list of the 3 proves that I might
Put on the mid-term review gone you learn did their good ones to practice what distiller proofreading tech talk at 1 approved this So if you were to get this problem yourself than it is in the books and be Vardi tried it what you would do it you would have to come through the book and find some here are some facts things Related to this end even if you don't really understand you just looking for that involved this symbolism symbolism like vertical lines in capital letters users look there's a very clear a capital letters and and uh Vertical line and there are so the 1 that would make that might stand out to you When you're looking through them around 9 . 8 which says that the determinant of baby is equal to the tournament day determinant of she found some kind of back that looks like the thing you're trying to to prove that doesn't mean you're you hit a home run by police That's what would start doing Easter collecting facts that looks similar to what we're trying to Alright that we have to discover followers so I need to start with the left side and then I need to do a series of logical steps and have my last that be this each 1 has to be just the OK Well this theorem allows me to say something about this allows me to break it up in the area years equal the area of the 1st times areas settlement We looked at for the determinant in area interchangeable terms Solis but that at my goal is to have nite might realize Wait a 2nd if I could or could I also say that is in this show that doesn't really matter that's that's what you really gain from this is to understand that the case may be viewed as dummy variables it just means I have a major extends a matrix equals Determinant the nature and terminal that just happened a call the song B 1st in this day it's still true doesn't matter that I call them in I can have I have Mickey Mouse and that man Right that the matter was Mickey Mouse and that as a matter of that sort Since I'm headed toward this I'd like the step right before that civilian deaths because that I can justify toast to some justification here here I would say this is just a straight Use of their 9 . 8 but then what about this is justified had which these 2 how how can I justify that what's that what's going on there What is a determined if it's the area what what else What is it is just a number Write is just like a 5 or 7 raises an area the number so with regular numbers we can switch things 5 times 7 packages rate that is 7 times That's fine so that is justified the way justify that you could say Real numbers All are commuted if that's when a fancy language for you could you could you just say that's why Equals x for number that tell you justify the switching of order and then or justify going from here to here will that's theory that if you don't like memorizing Theron 9 . 8 will you do in place of that is you put this back to that put it in there that they were using that fact here as well so that that's it's a three-step approved everybody can learn how do that he just memorize it is short enough read the tick marks on a page reproduces that way if that's what you want to focus of the 1st to proves that giving you are really really straightforward losers but the first one George Iike can't do those and you want try the 3rd on the little bit hard tho
We want prove that this fact implies Please just that Some square matrix on just telling you that if multiplied by a chance of that happening at the identity and then it has this fact that If we have a situation that by means of determinants of the plus minus 1 They will really far removed from the result here it's really hard to just look at that discolored de obviously She had that you have this is really hard multiple statuettes was below the sea that so we have to do is similar to that we have today acted just sort of Pollard but if you look at what you're headed toward we wanna have determinant of iron ore this plus or minus is gonna come from but I gotta get a number involved over here I don't even have any numbers but somehow I have to to get a number involved So you look through the chapter in Easter to collect facts that seem relevant to this so you might find facts that have transpose because I see here Get some facts there was there's there's 9 . 1 in which said the chairman and today is equal to the Tournament of the transfer was nominees this but I just just collecting that look like and I might use them they might be useful but we also had 9 . 8 If you have the product of 2 the the same They just doing the product of the determinants The Dallas Stars this is why you need some help and you need to watch somebody do these new villages and yourself OK the 1st thing I'm gonna do It is start with the left The height And now I need to do a set of logical steps that arrives at this is my last conclusion with that little small start with this fact well the 1st thing I wanna do undertake this year phenomena take the determine what both sides and idea here is a B and that means that the determinant called tournament Why is that true well if a equals below me their innings as saying the same dimensions and every entries the same so clearly the permanent While this is good because what the determinant of Was Determinant of us over here if high I missed 1 0 0 1 and determinants of I just want just 1 times 1 minus 0 and determinant Embraer I told him that the determined that the identity matrix behaves like the 1 in this world so what's what it's going to be is that that the area is only 1 the volume is 1 or the determinant is 1 begin to see that with the with a couple of practice and I have the identities of 3 by 3 It's determinant is 1 thing you see Reddaway away near amid expansion along this column So over 1 times this determinant and that term wants which 1 times 1 of which is what pocket so that I get the number involved plus or minus but I can say right now that could determine is 1 have that so far The skull we're looking ahead here Time If I could expose season can again among themselves have this fact that this guy is equal that so let's break up into 2 I can break that up and a Times transposed was 1 What's the fact that I'm using here well it's 0 9 point is that allows me to break it up into 2 Bar I think just red and a B and pulls ahead the justification for that would Chris if Now I would say that this is equal a because we have the fact that the the term today is equal to a train this implies a league squared equals 1 grade because equals a But And now we can finally arrived at this conclusion because if I saw stop for the Izquierdo both sides forget the determine the variant plus or minus 1 that's the final solution if
The red here for my final application is that they see
A total of filled in box that indicates Finished your proof you'll see that in math books that we may be learned QED you're has not necessary but anyway in practice that Slightly more advanced than few unity of Phil Simms's satisfaction when you can write that out a clean piece of paper see how far you get look back at the solution keep going back and forth until you can write I now without looking and then that'll solidify for 3 questions about those steps Chu On the Section 9 . 3 official we finally in sellers of the more general matrix and IN This is what all this work was moving toward this just we had a detours at some time with determinant because it's a part of the universe Members of an environment he did to to remind you that minute the let's put right down the general situation heaven and and matrix there's is a and I can write down its entries it's a 1 1 1 and disposal it down and throw 1st column and road and call there's there's the most general and and matrix figure among them the new numbers Now we've been calculating these cofactor straight for every 1 of these guys as cofactor race if there's a 1 1 in their C 1 1 and theirs 81 to furnish C-1 Chistopher every 1 of these guys is a cofactor so 1 organize those in a matrix that we did we take information we put on a major says that's a natural place to store this monetary a what's called the cofactor matrix day or what That rights entries in the entries should be well the upper left-hand corner ashes C 1 1 and then that's the 1st row 1st column that I should have the 1st row and column in the hands of gopher Scotland and But basically just copying Maddox of season That The cofactor techno calculators each 1 of those is just a number now introduced a new Matrix which is called the ad joint airport And the adjoint today is going to be the cofactor matrix of a transfer It's it really this nature expected that extra little stepped in to take the trance In the notation for this we notation makers is a little tedious to be riding on this down cofactor matrix of a sound so we'll have this notation Yeah I 4 2 But good Take a look at the 2 by 2 kicks a busy Coulter a B C D Go calculate the agile Will the adjoint its the cofactor matrix The practice that C 1 1 3 1 2 2 1 2 2 Answered the transfer of the services to do up there got for things to computer users not too bad to compute on us just remind ourselves that the negative with 1 guy plus J. accuser plus 9 minus plus locations and so what what is the and when I do a cofactor plus times across that D Sosa's as refugees at the scene 1 1 that's equal to its negative want the Lund plus 1 times that made the determinant of the Matrix when I across the 1st row the 1st column outages Group D A good deal this spot but no I get Yet It OK So what do you do this guy here see 1 2 That's going to be made 1 of the 1 plus 2 times determinant of the Matrix leftover when I crossed off beat this role in this column so just get C in here that's Eagleton negative The transfer and then the other 2 you think of this won't be mine SV and this 1 will be a day Here like seem to want sequel that might be consigned to chew is equal to And would take the transpose solely to the transpose amid the rows and columns D minus C minus a Now that look familiar Where we seen where this was A tonight and I switch ANC India switching Indiana make this 1 minus in one-liners that is now worry that the universe is so were they use that
Get our general formula the injures So a inverse with this was equal to 1 over a D minus BCE time D minus B minus a
And then we can write this in another notation know this we now learn is called the determinant write that down my fancy notation and then this guy right here happens to be the yet
Rates A calculated the adjoint separately and I got that so I can say that China and lucky for us that's the formula for an Bayern reaches have user Chubais to seats that's the formula nursery so couple of things at 1st I need I need to say that a deal might be seen as 0 and then also say that the determined and is not 0 hero as the inverse doesn't exist
Also is the formula for Way I'm just sort But are formula To use hung on a day earlier 3 by matrix
But To me There's a lot wanna find numbers so we need to things in my formula you I need the determinant and yet joined the adjoint is a matrix of
Determinant the number to thinks salutes 1st get this determinant that's determinant of 0 2 Negative will erode the ability to reconfigure favor row or column might favor would probably be this column right here OK so I've got 0 times its cofactor minus 1 times the Matrix left over when I crossed the 2nd row setting columns 1 2 Negative 3 Chu That's the first one here be careful you aircraft things out you might lose your minuses differences UCI across the top in my head The negative their careful OK this is actually a negative location was plus minus Q us down here this it's my niece around minus Plus minus This is a native location so I'll put in a minus for that 1 and then it's cross opted to road across its column 1 negative to has had a negative in there With that negative is not to be careful appear where have won 2 in the negative 3 native 1 3 a native reach back up Original 1 I'm crossing off if this And that's why I had 1 2 3 's Believe that there is down about 2 things here make you believe me poker I get negative times Chiu plus said It's to minus negative 6 where 2 plus 6 and then minus 0 Minus So that gives me a negative lead plus 6 which is negative to that system to turn yes What wait for the lady Docket Yelets highlight that so No I'm just trying to save show you how you start to use short so what what I'm doing is I'm doing expansion along here and when I do the cofactor This 1 is a positive location because it's it's the 2 2 locations so this was positive and that means this will be a negative review the cofactor so when I calculated on a rented out bosses it's some negative ones that location is the 3 choose but I don't you At 1st Add in all the negatives where poorly Cheadle Look it's only when I'm back when I'm doing the cofactor doesn't affect these numbers these come from the original sister number in front of this negative rating here is that negative 1 but I don't modify it that positive location and this is a positive 1 but I do modify The negative location on and only when I'm calculated cofactor of the finish writing of this that negative is I did finishes its negative wanted from close to it was negative for the 5 which was negative 1 so that's why I have that That's when calculating the cofactor it gets a little tedious derail these things as he wanted develops workers kind The 1st on China do the A little bit of a jump there so feel free to ask OK Do we all believe that that's the determinant of that nature
I believe it is that's what counts But now we need to get this adjoint so what I'm doing is I'm a calculator cofactor in every location is
The 9 calculations every single 1 of the guys I have to go get the cofactor for reach 1 owner make a distinction between calculating the determinant in getting the cofactor when you doing the determinant your multiplying these numbers times the when you're just getting the cofactor you're not multiplying by these numbers that's a different Here are multiplied by the negative one-time mystical factor and 1 mystical when I go calculate that here is not including the multiplication of how we see that the Serbs will eventually led to write in well-written due here none in a calculator factory what that's going be his enemy the negative ones and I plus Jake and then also a little determinant so let's go put all the negatives in 1st negative here There's a negative here was a negative negative those are all the locations where there's going to to be a negative so now that Now what I need to do is ready in all little determinants of that original nature we produced a here so that I can look at it will do it
That is look at this is good practice to receive sing-along predicted said I look at the first one is I'm crossing off 1st row and 1st columns of this little nature There have determinants negative 1 0 I want to And then move over a minute the understated take care all the 1st row of registered 1st row 1st column 91st 2nd call 3 0 negative 3 to please Follow me do this and tell me if there is something wrong then and cast off the 3rd row this little block here we were negative
That's the 1st row now we move on to the 2nd row 2nd Roman across it off phenomena 1st the 1st
The 2nd 2 0 2 1 2 So far so good OK 2nd row 2nd column that these Israel 1 to negative 3 to the 2nd row 3rd column to be careful here so 2nd row 3rd column at 1 0 negatives 3 1 And now will move on the 3rd Reproduces shoots Starting here filled cross of the 3rd row another 1st column that these me was 0 2 negative ones then 2nd column of the 1 2 3 in the last 1 for free here 1 0 3 negative For just take a 2nd year in which you copy and see if you believe it is area Let me add to it's done with deceptive do those little determinants now successfully 9 locations OK the top left school and I forgot somebody will you say Want focus this worry OK So that comes from a 2nd row 3rd column a 1 0 3 1 negative 3 1 and using That but I forgot something really important here if you look back a period that had your formula it's the transposed I forget He really does not worry about it but it's gonna we have to take the transpose began to get our final OK upper-left its native to minus 0 . 6 months but the 6 of them negative but greater than this 0 3 Minus 3 0 This 1 is 0 minus 2 defenders another negative or positive this 1 is too Minus 6 so that a to minus negatives to minus negatives And this 1 is 1 minus 0 but then there's a negative in front of negative 1 this 1 is 0 Minus negative to positive 2 0 minus 6 but then with a negative 0 minus 6 major with another negative so positive this was negative OK so finally we can take their transpose now to negative negative 6 0 To a negative 1 and then to 6 negative that there is the edge of the day that a lotta work But a challenge you see see if you could take a Anders right this could you do that in your head as you can really concentrate Take a That challenge for it even if you just write this 1 down and see if you can see that the determinants that you're supposed to take as you as you cross amount with your mind really good practice just plain strength
If
If Wealthy micropenis I checked their member of goal was defined the injures who got that Determinedly 1 got the adjoint now I can break down this A embarrassed that Sequent don't 1 visitors time agile This so I have negative one-half that's the that's 1 over the determinant times the adjoint that's the 1 to 2 2 6 1 1 And then we can incorporate the negative ads 1 make more negative 1 3 4 native 3 0 cents a curse them by you Multiply this matrix times original matrix and see that he gives you the identity and maybe even China that you're head good practice To get the answer that have on the page Any questions about that do to me examples as soon as you can appreciate why thing you can also appreciate that down so I just do not have to go beyond a 3 by 3 it's too much is then we have to do 16 3 It's a 4 by 4 major adapted to 16 of those call factors in each 1 of those the 3 that you make too much this is the most that you really have to be responsible even another was true From The nice thing about this too isn't it That If I ask you on German I will Here's a fine inverse when you get your answer wants least mentally make sure that it is and always check your answer Don't Israeli Danny could always multiply and see if you get back to the original if you get the identity when you multiply which are right So what will you do now is I don't wanna map out a lotta machinery available What a minute do His Oregon and look at it and economics example uses a lot of the stuff that we've been talking about all basically have No 2 or 3 economics examples that you have to be responsible for 1 from the 1st part of the course which is this part and then 1 later on not a focus on it but there are a couple examples they were there is a shift Those lawsuits will break from the dry mathematics Really know this guy Leontief says economists but he won the Nobel Prize and what he did was he studied The economy the industry sector and we realizes that if you're in 1 particular industry you use all the things from the other industries so as an example of 1 of the basic raw products Got things like a electricity and water and timber and maybe I N users and and then there's different people out there producing each of those the let's say I wanna produce I want and need some lectures city I'm Pollyanna needs some build myself of building on Pollyanna need some water rates so when I'm involved in 1 aspect of the industry of an industry I'm gonna need something from all the other industries So that their related and then that creates a cock complexity and citing studies Of this guy decided to stay in effect I think would really did was 50 different things from There Industrial 50 different and each 1 used all the others maybe some of them didn't what that created was a 50 by 50 matrix down our understand what he did get to simplify and just to buy has so will the sad part is a model 2 industry model we see this word a lot in economics think about what what it is what is a model its list a version of something you see in the real world trade if I wanna build a bowl model I have the real blow but then isoscalar reeling down tonight I make a miniature version So when economist looks in the real world and find something is interested in Hershey this guy saw that as this particular aspect of the economy interesting to him so he started studying it and we wanted usual create something so you can predict what's going on the world map for a model does force it's a simplistic version of the real economy but it reveals some aspect of it allows you to predict that the That's the goal focused our to industry model or in a simple could have electricity and water and just relax created comment he in Dublin so we want a four-minute produce electricity and the water that's necessary for society we see it's hard to store electricity and store water to be used later so it be really efficient I would wanna produce the exact amount that I need that the ideal way to figure out how to do that we want we want our output the demand may I know there's certain population out there that need some water there is a need some electricity and through them Historical records I figure out what they're gonna need so only much to produce but not quite Because the demand you see there is the amount of the population needs but then there's the amount that I'm in a need order produced that's the whole point of this land to a model is that if I'm gonna produce electricity and indeed both electricity and water to produce so I might Set in the Society might need one-year electricity but if I only produce 1 year electricity it cost me something to do it on can end up with wants to create a little bit more era have won for the society so that that that the level of complexity but we can handle it to this demand a get broken into 2 parts there's this internal demand an extra money and this is from the population those 2 things equal the demand that I want that We had to spend a lot of time explain all these things and set of the notation and that's why I wanted to make his economics example if flat with a lot lot of time but when you're calculating recently and it doesn't take long to set up
This is missus
Production and this is from population That's a sign some variables 1 of electricity Please just numbers I'm giving you see that there's something to be given the problem so 1 electricity needs 30 cents electricity and 10 cents of water I could And if I'm there produced 1 of water then However it might in turn this is all my internal demand and need Set up 20 cents Of electricity 40 cents of water Israel said OK and then let's let's take a look at the external demand the external demand so let's say the population needs 12 million dollars worth of electricity and 8 million dollars those numbers that yet have water would have 12 million electricity and a millions of water An analyst at organizes information is isn't years with Mister using all the matrix mathematics of the study so my collect this my demand D. OK it's the demand for electricity is the demand for water collected just right that I know those numbers are 12 of demand that at external demand and the public output output is of the call that had want is equal to the output of electricity and Studio output of water was organized that in the same way I have my ex is my output and then it's a vector consisting of the 2 things that I'm interested in doing the output of that that's 1 of my goal here is defined this assault for this X wanna find out what the hell output that I should have from the plants in order to to satisfy the internal and external the call OK last thing we need to organize this information in a matrix and we did that Other to put that into a matrix who aim for this forces called the technology tricks this is the thing that was studied by Late into the nite as get a 50 by 50 thinkers My memories right now down this moment OK is this Gaggin and be well On this idea of the input of electricity and water and the output of college Input Output Input Output analysis electricity and water so what should I put in here Well I've got 4 pieces of information here is that the amount that I need from electricity to make electricity use the amount of water that I need to make 1 of electricity so if I wanna make Was ready That in English So this is the electricity needed for 1 of electricity Course In this dial be the water me for Mandala electricity so this is all the 1 of electricity and this is the water needed for that Dog skews may I got This is like This is the input here He this is the electricity needed for 1 electricity and electricity needed for 1 of water so that tall Pot rose electricity This whole bottom rose the waters of the river water needed for life that 1 of output of electricity and then this is deep water needed for 1 dollar of water She has the matrix helps to that alleges electricity but that number and then in electricity and water water electricity water water the list of those numbers in now we have those I need 30 cents electricity for 1 of electricity and I need 5 10 cents of water that goes down here for 1 lectures to Call that em so you act That's 3 panel there just to set up the whole problem If you're given a problem like that used to immediately start the sectors it takes a few seconds after that said at 1st but always alert time going now it's a on how But Try So we start with this principle that that would be caused the demand had a we get the output of the output labeled as x x and then the internal demand depending on how much output and multiplied by that nature except matrix tells me how much I need to spend on each side so the internal demand as B M X and then I have external demand
This is an internal and texture and now we get to use our mathematics so this guy looked at in the world he saw this phenomenon happening he tried to describe it to organize the different things OK demand I get a vector for that I've got help vector It is setting up the model to describe what he sees in the real world but now the whole goal is to solve a X we need mathematics not let me ask you if I have those who were here to see if I have on the X equals M X plus D little backed our high schools situation how what I saw Well I go X minus M X equals deed and what I do here if this was a number a combined number 1 like that all of the best I can do this offer Exs too Factored out so that 1 minus and X D for the warmest offer X X is equal to 12 . 1 minus M fingers We usually is right 1 over that overrated specifically like that because now we translate that into matrices when I saw this matrix equation I wanna do it the same way lips The that implies that X minus Them Tax equals D I wanted you this same step here were a factor out and Acts terms Member when multiply the start I need to get back where it started with so the question is will it with race here over here we go 1 so what plays the role of 1 in the matrix were the identity that plays the role of war if you like you can write a little higher a year and then you see the factorization is remember the identity times anything gives you think back again the doesn't change our than I could say that X is equal to or hire minus M members D And we can go Elite minus that sequel to 1 0 0 1 2 minus M features . 3 What to do . 1 4 will get that point said in and negative point to negative . 1 . 6 and then I minus and inverse As is the interests of a matrix we gotta do that I need 1 over the determinant that'll be won over . 4 to minus . 0 2 The noon said frantic 42 with 2 decimal places that ambulance 2 times 1 with to get that key and then what'll I do with these guys I switch and . 6 . 7 and then make those positive he Reasoner Matthew rate would forget the economics Mahler using her map I just tell you what it says This year it is 1 over . 4 Which is 1 or for tents which is can over for us is right that at 10 over 4 . 6 . 2 . 1 centimeters use a calculator to Felicity like not on my exams The reason I wrote and see the tender gets rid of all the death of the pretender decibels are that the attendance animal about everything by get 125 3 halves the one-half and then . 2 5 is 1 force and then 7 over Forrest said I And so were home free our whole goal was defined acts show Tho X is equal to that matrix Times d you want to put agreed here so you don't lose track of those numbers Worst 12 over radio X water and electricity they need to be produced in order to satisfy The demand of the public That to do more than the 12 and 8 million dollars worth it because I use someone I produced And the amount the minute I get is 22 20 to This multiplication 1 . 5 is 18 plus fours 22 A quarter of that history a 1 . 7 5 letters 14 6 22 million electricity is needed and 17 million dollars worth of water In order to produce the amount of the people need There you go still wanna study economics has Karolyi can't are now having seen this step for man at the edge of teachers some of you
Why did you By Will 8 12 yet OK so see exit there when I saw that I X is I'm understand 1st D I when worked and found out what understand Rice's I got this matrix and this
He followed his Dietrich
It's if Before With that Would Key me questions chess Yeah probably but the you
I like said before I start time explaining on so if you got a problem I would just say Here's the demand 12 million and then find out output with this technology nature of his
Give me the major marketing speak a word problem and make use of their read anything others give you the whole set at and then which'll basically have to do this equation has won a major you can the math You that this is a great comparison a great example of what we're really doing in this class you learned that it is this when you were younger than you did not do this this is Big boy equation that matrices rather than numbers It It's a much higher level so that's really what I need test you so I can even do without the context that it is give you a matrix is solve this equation like that had done that was something from this will be on the should start getting nervous about the mid-term a week from Friday
I'm nervous about it could have have always nervous that had that type of them but I do get is yes
Point point to me lawyers Yeah This is OK so and under 2 major multiplication size take the 12 clear on this side of the 12 thousand 125 plus 8 times . 5 22 Then 12 times that Pelosi said Yet it is me me to be serially had earlier multiplication matrices yet universes et de determinants very things minimally done is pretty much in this problem like that Japan Adieu right now is organist solve every system of linear equations with And variables and equations The same number we have key the same number of variables as equations where assault on the universe that had unique solutions and do a little bit series some notation and stuff but at the end of it all get computation OK so start with giving me system a mn linear equations so this is how the combination of everything that we've done we started class by steadying systems of linear equations then we sat there and we we practice role operations then we went from there we just 18 matrices by themselves now are coming back and doing systems of linear equations in putting it all together so Kramer's rule does so I'll get end linear equations so saw a picture of bunch of a list of equations and then I had to tell you how many variables with n variables so these are the same as is a special case that and equations and variables somewhat Pegasus is a square matrix so that he can do Kramer's shooting at a square meter rained down What would you do if I had as if I had a list of any equations with and variables what I would do is I would grab all the coefficients input matrix And then I would grab all the variables and I'd put down a little Victor and I call that ethics and on the right side of the equation However list of numbers recall that he was not very satisfying because it just like I've said before his leg and I can't do that Soviet DoubleClick middle order stepped inside this is a very compact way of writing it but it's true a matrix times a list of variables equals constitutes a list of commands Coca-Cola that's expanded little will double click on that if I double-click on it would I get budget coefficients on a amount could be careful about this 1st one they won 1 the next 1 is a 1 2 then I had a 1 and follow it I guess you could rate that that I go all the way down end throw 1st column and throw 2nd column and row and And X that listed and variable also 1 through X those variables and found right side got T-1 through Pte But If but what was saw the equation
And The power of that is that would be solving all infinitely many Linear equations with the same number of variables in the universe of the same time getting nice formula Critical overwork the book at 1st thing I wanna do is I wanna make alleged that Sony so you get a that's the coefficient matrix make new matrix from it and what I would do is Asa J. That is why would I college-age is thinking a mile or else use the letter J is Jay's described how I think I'll die of AIDS Jason the of so many something with the G-8 column that's what that notation implies that with the G-8 although that would be a good guess what the He I related to this issue Matrix obtained from the bank that's that's why we use that name notation having a year wiser comes from the original than said Jay the Matrix obtained from a by placing the G. Collins would be for that stuff on the rights said would take that put it in the 8th column that there's any of these I can make I can make a wanted me to dollar day and every time I just take this and I just couldn't place of the the date car was still a picture of it case said J. that's going to be a 1 1 day 1 Jew and then and I have been 1 through P N A 1 end the day goes always down a and won a N Q and and the Jafar call That just remove the gave column and put in place it it whatever I have over here but just for fun let's go Calculate a school near 1 of a raft of speed 81 If that should be the easiest way to get what we're doing right now you know you've got had of experiences quarters to know that Swanson a while ago officer during this period the eventually arrived at the formula that Cheney I attribute this is when I was in school I saw their saying at the end the at a formula so don't necessarily have to write all this stuff down it's just maybe just let it wash over you and you'll get the formula is what this really is is this is just an opportunity get a little mathematical maturity and deal with this so if you just sit there and you just copy it chances are you have to get much out of it in this moment and then we laundry three-year notes when it comes to this sat there and watched in copied it and India much added menu tend to make up for all you don't return so maybe just watch it A little bit like a mental mechanical bull rider So when I started here get the ball gone please see how long you can hang on at some point
That's OK you to just understand that part the craft off OK let's just get 1 little example heroes go a long exhibiting gay 1 at all that the Wednesday 1 its I take a N I replaced the 1st car bomb would be 1
To be in the 1st column and they are set to stay the same history a This is the 1st call 2nd their 1st Roe 2nd column 1st Oh column and throw 2nd columns and rows There is a war we did I replaced the 1st row with but also Get the determinant could the skit the determined And when I do this determinant and expand along the 1st call so is acting give me it's believe he 1 times cofactor C 1 1 plus 2 times the cofactor will them when I go down here to see to 1 because that would be the 2nd row 1st column and so on holiday on the last 1 end time Seif and 1 that would be the expanded along that 1st column I 101 point out here is that these cofactor say these are the same ones from it because notice when I got a calculator all these I'm always across the seller and until all the numbers that are left overall from it So the use of the Co factors is the same Co factors how The city could believe that they are all the numbers that is difficult to calculate those factors are they just all the numbers from the beast orally show they show that here Not 1 calculated these OK still on my mental mechanical bull how are mostly Howard rose organist volatility Italy's this that is Taylor's at the solid everyone theaters a X Goolsby Well let's assume a inverters exists that that has happened that I have a situation where the inverse exists because then I consult at the injures exist other X is equal a and B we see that Some multiplier both sides they But we know the formula for was right that I would say inverse that 1 over the determinant of times How much to overriding here 5 times the cofactor major it but that's right what the formula 1st this is times the adjoint today All of a universe that's just this artery That's 1 over A times OK let's write down what this said joint is its C 1 1 C 1 and C N 1 tenant Dollar co factors and transposed times be 1 To be Truth So what we have here His X look at what happens if I take The 2 the transpose now looks to us worn over a delay due transpose for almost at the end here so just hanging on it what happens when do the transfer all this stuff becomes the columns The exceed 1 1 although it down to see 1 and that's different than this 1 in half Rice's sea water too And this is seemed to win 1 of them will succeed but were ready to go so here's what I want Let's just calculated X 1 had yet X 1 on the multiply out the top row here with beat see that That's how I would get X 1 I would multiply the up Barry And I'll give you the very 1st location of my new maker You see that village right out what X wanted X 1 is equal to 1 over the determined today times of him will be 1 times this plus 2 The next 1 times this to 1 . . plus 10 times the last 1 c Believe me He sees that anywhere else on the board But a letter
It is The determinant of that special matrix 81 So I could rewrite this another way that I could say this is that 1 is equal to the determinant of 81 You see this thing right here is the same as that of their
OK so now if Missouri special let x 1 x 5 we think that would be Have X 1 N A 1 not a hard lead to realize it x 5 x ended nearly a over the war in general I write down what called Kramer's role in that X
J. determinants of J But with the face of given any system liquid is On was the exists to reimburse exist I mean the determination is 0 so it's OK to divide This gives me every variable only by 1 x 5 ago determine 85 rural terminal 1 x 3 a determinant of 83 over It's a nice formula after all OK So that's it that's the combination of everything new fewer sleeping you should wake up now kilometers of examples along those lines but start with how about uh well about 2 excellent books at a system of equations so X that's the only variable so that should be Equal to the determinant of a 1 over the determinant play Well OK What is this This is this is a serious is day and was beat this case a assisted Chubais to have 1 by 1 matrix will do the budget but determined and then what they want and they take bidet replaced the 1st column of air with 3 the there's only 1 column to begin with so that it can be very very hands rockets silly to do it that way obviously going to do this in your head but the Kramer's rule still works even for this case we have won by what we have a single equation with a single variable still works there gives us hope that mathematics works with every define it it it doesn't work a certain places not called the ballots do a they have So let's do That's why text Minus why would want rosary they'll tell you the solution solution is pools are too negative practice Iran if you want If
Chg Will Correa Kramer's role here say Texas X is like X 1 here so I'm gonna do undertake the determinant of the Matrix where replaced the 1st column with the right side of the divided by the coefficient illustrate that at least
X am considering that the 1st variables like X someone if you like you are replaced with X 1 next OK so X had I get that that's going to be the determinants over determinant this guy here is just the coefficients was 1 1 1 negative 1 that's the determinant of the determinant goes here is a list football 1 3 in the 1st column the 1 3 year and then I keep 1 negative that's equal to a negative 1 minus 3 over negative 1 minus 1 negative negative too which is why that's equal to appears be a determinant over determinant the bottom determines can be the same as just the coefficient that's always the what changes now is now take the 1 3 instead of putting it in the 1st column put in the 2nd column may be arrested that 3 minus 1 over 1 minus Of nearly of warm minus 1 so that's too will negative Their solutions That's not to say that we could have done 1 more easily just using a high school math is were just practicing it's easier to buy 2 editor 1 by 1 their due to buy 2 and then finally will do a 3 by Really do 4 by 4 speakers again we had to do likely could be a lot of 4 by 4 determinants close to a 3 3 in Colyton X 1 plus X minus 2 eggs 3 negative 1 2 eggs 1 6 2 3 4 4 . 6 would 2 x 3 1 have the solution you could work toward yourself and make sure carpeted rate will get a disaster here Ontario But at a computer for terms that wouldn't I need a is on the bottom of everyone of papers determinant of that's going to be the determinant of 1 1 negative due to Avoid negative Now I do this I could reduce that I was gonna go on the top row in each case his duties 3 three-base by brute force remember this is a plus location that the minus location and This guy is equal to its 1 The Times Mirror raid at sea That's 1 times the Matrix leftover which is why it would made it then minus 1 that's this minor straight here via a little tricky but time the determinants of 2 negative 1 5 The 9 minus Times a determinant of 2 negative 1 negative votes That almost Siemens You know what they do there I copy that went wrong This 1 here For looking at this rate February believe that Acreage I believe it Kosovo symbol negative 3 minus 9 minus Q times Street which has gained if you believe that That's the first one because we believe that 1 computer or the other It's not that 3 more eager high over 1st 81 81 is determinant booking I'm taking that original matrix but I'm putting the rights Saggion negative 1 4 1 and that's going in the 1st half and the rest of it's the same 1 negative 1 2 2 2 2 2
That Mr. Wright these things out the saves time and again I'm gone along the 1st row so that way in the plus-minus is always the same last month that minus is that sector minus the men of course if you believe me while I'm doing it that the cofactor expansion along the 1st row And that's equal to 3 . 21 miners 2 times Witches A 3rd as books said 2 renowned for a tour taking that that beat and putting it in the 2nd call for everything else stays the same 1 2 negative 1 and related to the world If And that's difficult to foreign negative 1 of 5 going again along here as saying that plus 2 negative ones native 1 5 minus 2 to 4 1 of his 21 plus 9 minus 2 terms 6 positively city believe that He and 83 now I'm taking That these their negative for 1 negative won 4 1 And putting it in the 3rd column The rest is the same On the 1st role again and do this I guess Negative 1 4 2 1 of mine To We don't would this guy is negative 1 might is negative then Q My name negative for that positive 6 but that had a minor set for a 6 and then for minus 1 is 3 with a negative home free those 4 matrices eatery get the 4 variables are the 3 best now 1 that 81 over a negative 36 negative 18 just to next to The determinant of 82 over a that workers Teen over negative 1 x 3 8 3 Look for a solution comes down the 4 major not tell you that the EU may prefer it is a differ away so that the same answer that's fine market that Put you in a position where I say you must use Kramer who all colleges give system and you can have the choice however If I decide to make the answers fractions Awami use Kramer's because imagine that the answer was 14 over 27 that Sysorex 1 said 14 over 27 Well I would amend the visitors 20 7 and this was 40 and that didn't require any fraction is a discount of 14 for that and I a 27 for that the answers a fraction of the valor the nasty to trade you reduction so it does have its advantages also making program computers program But in 10 variables and equations tell the computer duties determined and that a programmer computer solving these systems of equations so it does have no advantage fractions and things like that also whereby dissent here's a systematic acquittal of 4 variables go find X 3 I don't need or for might just 1 x 3 moment in that case just to deter criminal does have its advantages and give you a former variable system and a single finally x 3 analogist to force Determined answer was a fraction you can get it through the simply OK that's enough for today yet No no no no of these you're saying switched to I that way but what I'm really doing his undertaking that raid the negative 1 4 1 and I'm putting it in this matrix in the 2nd column That's in the 1st year and putting it in the 3rd column And here I'm putting it in the 1st column that That's how I make 81 pending state thing on the right said the conference on the right side and put them in the 1st column this 1 state constant on said the 2nd of the because that's how we do The Yeah because when you doing the determinant it's just the locations Yes The same location yes Because that's the role I'm using to determine Get the determine if I could I did the same ones to keep it kills 3 simple because I want have the same plus-minus plus minus So have to deal with that
Matrizenrechnung
Matrizenring
Determinante
Einheit <Mathematik>
Gebäude <Mathematik>
Mathematisierung
Zahlenbereich
Vorlesung/Konferenz
Faktor <Algebra>
Element <Mathematik>
Wärmeausdehnung
Matrizenrechnung
Sterbeziffer
Zahlenbereich
Fortsetzung <Mathematik>
Unrundheit
Element <Mathematik>
Euler-Winkel
Äquivalenzklasse
Übergang
Eins
Zahlensystem
Weg <Topologie>
Negative Zahl
Poisson-Klammer
Äußere Algebra eines Moduls
Vorlesung/Konferenz
Konfigurationsraum
Auswahlaxiom
Vervollständigung <Mathematik>
Determinante
Amenable Gruppe
Güte der Anpassung
Frequenz
Teilbarkeit
Arithmetisches Mittel
Rechter Winkel
Wärmeausdehnung
Boolesche Algebra
Nichtlinearer Operator
Matrizenrechnung
Negative Zahl
Matrizenring
Mathematik
Determinante
Rechter Winkel
Theorem
Mereologie
Gruppenoperation
Vorlesung/Konferenz
Wärmeausdehnung
Euler-Winkel
Ordnungsreduktion
Eins
Nichtlinearer Operator
Matrizenrechnung
Determinante
Natürliche Zahl
Gruppenkeim
Primideal
Eliminationsverfahren
Teilbarkeit
Ordnungsreduktion
Physikalische Theorie
Topologie
Negative Zahl
Differenzkern
Rechter Winkel
Meter
Vorlesung/Konferenz
Dualitätstheorie
Turnier <Mathematik>
Matrizenrechnung
Sterbeziffer
Natürliche Zahl
Besprechung/Interview
Zahlenbereich
Mathematische Logik
Term
Eins
Knotenmenge
Variable
Reelle Zahl
Gruppe <Mathematik>
Theorem
Endogene Variable
Radikal <Mathematik>
Vorlesung/Konferenz
Gerade
Determinante
sinc-Funktion
Reihe
Flächeninhalt
Differenzkern
Rechter Winkel
Sortierte Logik
Beweistheorie
Ordnung <Mathematik>
Resultante
Turnier <Mathematik>
Matrizenrechnung
Wellenpaket
Punkt
Matrizenring
Determinante
Hausdorff-Dimension
Zahlenbereich
Kartesische Koordinaten
Wärmeübergang
Biprodukt
Mathematische Logik
Term
Gradient
Arithmetisches Mittel
Menge
Flächeninhalt
Sortierte Logik
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Wärmeausdehnung
Spezifisches Volumen
Matrizenrechnung
Thermodynamisches System
Total <Mathematik>
Mathematik
Determinante
Quader
Natürliche Zahl
Inverse
Gruppenkeim
Zahlenbereich
Wärmeübergang
Fortsetzung <Mathematik>
Rechnen
Eins
Zahlensystem
Schwebung
Rechter Winkel
Beweistheorie
Mereologie
Vorlesung/Konferenz
Garbentheorie
Grundraum
Figurierte Zahl
Matrizenrechnung
Zahlensystem
Sterbeziffer
Determinante
Sortierte Logik
Rechter Winkel
Vorlesung/Konferenz
Ausdruck <Logik>
Matrizenrechnung
Subtraktion
Sterbeziffer
Determinante
Ortsoperator
Natürliche Zahl
Zahlenbereich
Physikalisches System
Ausdruck <Logik>
Eins
Negative Zahl
Wechselsprung
Vorlesung/Konferenz
Wärmeausdehnung
Negative Zahl
Multiplikation
Determinante
Natürliche Zahl
Zahlenbereich
Vorlesung/Konferenz
Faktor <Algebra>
Rechnen
Zählen
Teilbarkeit
Eins
Negative Zahl
Flächeninhalt
Determinante
Rechter Winkel
Natürliche Zahl
Güte der Anpassung
Vorlesung/Konferenz
p-Block
Fokalpunkt
Frequenz
Eins
Ausdruck <Logik>
Matrizenrechnung
Punkt
Sterbeziffer
Wasserdampftafel
Gruppenoperation
Komplex <Algebra>
Übergang
Zahlensystem
Reelle Zahl
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Verschiebungsoperator
Funktion <Mathematik>
Beobachtungsstudie
Addition
Mathematik
Krümmung
Güte der Anpassung
Gebäude <Mathematik>
Inverse
Biprodukt
Fokalpunkt
Teilbarkeit
Menge
Mereologie
Ordnung <Mathematik>
Numerisches Modell
Beobachtungsstudie
Matrizenrechnung
Momentenproblem
Mathematik
Schiefe Wahrscheinlichkeitsverteilung
Wasserdampftafel
Natürliche Zahl
Zwei
Zahlenbereich
Vektorraum
Biprodukt
Variable
Forcing
Menge
Vorzeichen <Mathematik>
Minimum
Bestimmtheitsmaß
Vorlesung/Konferenz
Ordnung <Mathematik>
Analysis
Funktion <Mathematik>
Beobachtungsstudie
Matrizenrechnung
Subtraktion
Punkt
Matrizenring
Determinante
Sterbeziffer
Mathematik
Wasserdampftafel
Zahlenbereich
Fortsetzung <Mathematik>
Vektorraum
Rechnen
Term
Teilbarkeit
Weg <Topologie>
Erwartungswert
Multiplikation
Dezimalsystem
Forcing
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Ordnung <Mathematik>
Numerisches Modell
Lipschitz-Bedingung
Matrizenrechnung
Vorlesung/Konferenz
Matrizenrechnung
Punkt
Natürliche Zahl
Klasse <Mathematik>
Besprechung/Interview
Zahlenbereich
Gleichungssystem
Übergang
Multiplikation
Variable
Zahlensystem
Exakter Test
Meter
Vorlesung/Konferenz
Grundraum
Funktion <Mathematik>
Nichtlinearer Operator
Schießverfahren
Matrizenring
Mathematik
Reihe
Lineare Gleichung
Kombinator
Schlussregel
Paarvergleich
Physikalisches System
Lineares Gleichungssystem
Koeffizient
Ordnung <Mathematik>
Matrizenrechnung
Punkt
Mathematik
Momentenproblem
Zahlenbereich
Lineare Gleichung
Frequenz
Ausdruck <Logik>
Variable
Zahlensystem
Rechter Winkel
Koeffizient
Vorlesung/Konferenz
Grundraum
Mechanismus-Design-Theorie
Leistung <Physik>
Stereometrie
Punkt
Determinante
Inverse
Zahlenbereich
Wärmeübergang
Rechnen
Teilbarkeit
Ausdruck <Logik>
Eins
Rechter Winkel
Mereologie
Vorlesung/Konferenz
Umkehrung <Mathematik>
Grundraum
Expandierender Graph
Matrizenrechnung
Determinante
Mathematik
Gleichungssystem
Kombinator
Schlussregel
Physikalisches System
Flüssiger Zustand
Ausdruck <Logik>
Negative Zahl
Differenzkern
Radikal <Mathematik>
Vorlesung/Konferenz
Matrizenrechnung
Abstimmung <Frequenz>
Variable
Oval
Forcing
Mathematik
Sterbeziffer
Determinante
Rechter Winkel
Koeffizient
Minimum
Vorlesung/Konferenz
Term
Matrizenrechnung
Bruchrechnung
Ortsoperator
Determinante
Momentenproblem
Kondition <Mathematik>
Gleichungssystem
Physikalisches System
Term
Ordnungsreduktion
Eins
Konstante
Variable
Negative Zahl
Menge
Rechter Winkel
Schwebung
Vorlesung/Konferenz
Wärmeausdehnung
Optimierung
Auswahlaxiom
Aggregatzustand

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Titel Math for Economists - Lecture 5
Serientitel Math for Economists
Teil 5
Anzahl der Teile 15
Autor Kronewetter, Jason
Lizenz CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nicht-kommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben.
DOI 10.5446/12914
Herausgeber University of California Irvine (UCI)
Erscheinungsjahr 2013
Sprache Englisch

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Fachgebiet Mathematik

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