Merken
Math for Economists  Lecture 8
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Erkannte Entitäten
Sprachtranskript
00:06
OK so last summer I wanna do there were put up this Agon value us situation again in order to expand little but there's so there's some interesting facts that come along with begging values that I didn't get a chance to to cover because I was trying to do a whole problem for you last last time we had a matrix a was the matrix and what we did was So we started looking matrices as functions and this takes the plane hard too Into the plane too but when you look at this But may Churchill really know exactly how it does this so the study of ideas values it basically finds that taxis that this operates on what we found was we found the hike in values to be I think it was those there's this was you this We already down to you was the egg in value 0 1 1 and that corresponded to an ID value land equals 3 and then we had a lame equals 1 and he was 1 negative What they really are that taxis on which this operates Get it chooses to view the plane like this and then if you will Choose to look at it that way to you can see what's going on it basically takes Anything along this axis and multiplies it by 3 the land equals Street and than this 1 just keep keeps multiplied by 1 of startup looking at it as grid like this and then moves over and multiple is a song by 3 and stretches that then we also formed an orthogonal matrix cue from these vectors Analysts rate those down ahead 1 1 1 negative 1 of Hugh We had to divide by the likes of those with the likes of those guys if you like we can write he pulled with 1 or 2 out like that and then this is orthogonal we learned about orthogonal matrix was last time and what that meant was a few times transposed is equal to the identity and really what that meant was that the transfer was was equal to these numbers so that what we did was we multiplied AI neither side by QMQ Trans whom we will just I told you to do this this is what's called the process of diagonalization so we took our original matrix a multiplied on the left by Transco's which by the way is also a human errors then multiplied on the right and what we got was a diagonal matrix And the sky had the eigenvalues that's where we use this Capital landed to describe the matrix because inside of that nature to gut the little lamb those that all we can do a little bit more of this this is actually this is the part that the economy is actually just when we were when we're studying matrices rural we did what we start off saying OK welcome we Adam yet we learned that out of there we secular multiply and yet we learned that a multiply matrices what we didn't learn how to do it is by will you sort of learn Ataturk powers of matrices like a that would be a kinder times it
03:57
What if I said I data the 17th out we would really wanna 17 multiplications of matrices deal OTTAWA this allows us to get powers of matrices really easily only show you that and also just Morgan ahead toward says giving some think about it years thing what will be the square root of a matrix With all the things that we did with numbers was starting to do with major cities add numbers that choose multiply numbers multiply matrices square roots and numbers we take square roots in major cities I could take a number and I could do it to the 100 power We do that to matrix into it easily so that's what had words here on a rewrite this formula race here as CQ poses really queuing 1st and then Armitage was on a multiply both sides by Cuomo left and few adverse over here and that's going leave us just a supplies of array due over here on the final hole manipulation by multiply the left but you hear on the right by embarrassed all in order to keep the equation equal had to do the same on the left side so have to you there but you see or hear I get the identity of the area Researchers exposes a probe a is equal to queue Lambert you anniversary remember Q is that specific matrix for this A and land is that specific church sometimes saving time and writing these matrices over and over again by just using the symbols Well let's check it out of what is a software for a squared is this Times itself right so that's fuel Lander inverse times land Hunan writers and by doing that said lined that the Q and a Q and Ursula canceled planned with you The square Human so that's got a curious maybe that continues on what I would I do a cube will In Cuba be Guy and then I would just put on extra Q and a Q Anderson accused cancel yet so we continue on this process but what we learned pretty quickly is that a man any power of a square matrix if I could break it up into this Factorization using the eigenvalues and eigenvectors then the power of a statistical to And that matrix with the aid values and human and majesty power of that guy and this is a diagonal matrix so this is easy to take powers Lamberton straight down auditors Slammed to the end is equal to 5 in this particular case of 3 0 0 1 at the end our which means is multiplied by itself and time with all its Internet happenings end multiplying the 3 2 at 1 so truly easy when you have diagonal matrices to you powers So what's happened here here's what's right down a minaret That is as matrices field is This guy 1 over ripped you want over Route du then I have 3 0 0 1 and then the transposed laboratory to talk to negative chew on her So I gave you start off with a basic matrix and what happened to rolled through all this calculation always essentially Dennis taking taking a interceded with a sledgehammer only broken up into its 3 main components as is like a factorization maybe you could think of it like prime factorization nature broken up into its essential parts this middle part is that Agon values this tells you how much again multiplied to those vectors by so this was multiplied by 3 was variable Led by 1 that's that information and over here I've got those actual vectors put in the matrix so when you combine them like this you get back a and then writing it like this allows us to take powers of really quickly please see we can also see what happens when you take hours a day A takes this Already here will then what happens if I multiplied by a again will take that lengthen multiplied by 3 against Alaska and and that's what we got here if square 3 squares 9 then the scale stay 1 of us is Kownacki keep stretching back more power they do it is keep stretching that rectangle out further Back that square root issue where about the square root of the scattered disk We're de look at By the way in on a date for the it's so nice to see so many unfamiliar faces welcomed back to the collapse of the odious Guerra a great thing a welcome yell back with a guilt but hadn't had a pop quiz then calculate squared But are at what let's say this is some majors example always socalled will it if it's the square root of a then it must be the case that be squared has right That's what square root that it's that being said that when he squared But you just tell you would rather than trying to calculate it we can just get shouldn't it be that aid to the onehalf Askew Lambert onehalf human research should be it's harder calculated using this method here because we're going the other direction but what we can do is you can do is get yes but say he is equal to 2 0 5 Land The 1 half due there's so that peace squared that'll be Hulu On 1st time you plan that Anderson and those cancel and get back to you plan that embarrass which is a is that a that's the factorization focus that's my guess ballots go with land at the 1 onehalf the square root of land and our land as our land over there was 3 and once on the square history and Were So is equal to 5 Q managers to swear 3 You winters need multiply those 3 are give you the square root of Agon values that allows you to do this kind of a just a little word here that I have a little bit of a mathematical missing piece we were doing all these accused that we always get in this class The enemy orthogonal because we're starting off with a symmetric matrix and the reason we focus on symmetric matrices and discusses because in economics often the matrices cannot be nature There's is a will there's usually no difference between good wanting good to you when you combine the matter with you had good wanted to agree to a good 1 so that That symmetry always I always that often leads to symmetric matrices and so if you a furious steady this again another math class you'll find this is much more complicated and these were were picking Arbour hand picking really easy matrices to deal with
13:00
But when you do it again is much more complicated if you do it real math class the kind of focuses on this theory OK Some so of that target powers A couple of their Saks about I value
13:23
We had A decided convinced you early tuition or Or provide evidence that these ideas values really are the essence of the nature They have they contain all the information on a lot of important information well by way of review we could calculate that traces of a what's the trace a river with that is that means I had a bad moments the elements of the main diagonal politically too close to where they would be calculated determinants of day the determinant today is for minus 1 3 but that we could go over here we could say more about the agony so that I can values were land equals 1 and land equals 3 and build lamb and created this this slammed a With the ideal values in a notice that the traces the debt and the determined the same for this trace of this guy It is equal to 4 is same as this and the chairman of the sky is also equaled 3 that so you see if can it That tying things out of the way we did before if I look at these Columns here And grab them than the determinant represents the area of the rectangle so that the the area this rectangles exactly the same as this 1 except this 1 actually is A rectangle that one's apparel agree so you see we've done is eat you take this matrix an inside that there's this information here and this is the essence of you could take a 3 0 hero 1 of that area's 3 new graf about 1 minute series 3 but that information is preserved when you do this kind of computation interesting if it did it fascinates you dig math class size and that's about all Oregon and you want thing we have left to do Would be 3 3 matrices market put that on the midterm soldiers focused on 2 by 2 and then enter the course come back and review for the final you'll get a 3 by 3 of the final C or you will be only get away from it but that won't you won't have to learn that by by Friday Yes About Yet The injury but is well actually the determinants is always the product of the the alien values and b Tracy is the sound so So you see at a fight that pleased me another wait for a quick way getting the idea that I can say that lambda 1 plus landed people for that trace and then I could say land the 1 time was limited to that's equal to the determinant because that's what it is here that Lam want plan to label like that if you like so when I do the determinant Harry Atlanta 1 landed 2 that's gotta be the same as tournament over there 3 So if you like you can solve this way that lies that and you could just look at this 1 plus serious foreign 3 times 1 history that would solve it I switched here but anyway The group of careful anyway But spoke Keep track of the assistants is kind interesting facts seeking users if you want to solve years hike in value problem on the Aegean Any questions about that busses kind cleaning up Value it is really interesting for me From my perspective this is this is the most interesting stuff there's always really cool stuff behind the scenes of a matrix but we had to spend a lot of time getting to that point we finally get their nite test you so Soviet Yet If you're fine or little mini midterm on it so that the class divides nicely into 2 parts is a linear out for the calculus for So we're finishing the linear algebra Parwane starter calculus part when we get to the final you will had 2 cards and so you'll have a little meaning through this summer with linear algebra pardon the little mini Minter that's on these multi variable capitalist simply another section that will Visitors later on the course there is 1 thing that that outlined how they're getting you ready for the midterm there's just 1 type of problem they have to get ready for from the section on over detailed party ready for before you
18:50
Blasted it What really added eightyear steadying OK so this is what called wide erratic this section OK 1st of all disaster somewheres or quadratic mean to you what pops in your mind if I say quadratic maybe quadratic formula but what what does the word quite dramatic me this is quadratic is 1 step from linear quadratic the quadratic formula solves a quadratic equations but those come from parabola so that's 1 step up from Lanier you have You have aligned and if I decide today in the line the 1st thing I do is I make the power instead of having a 1 attitude and that's what creates that occur in in the graph so quadratic forms should remind you of Travel or circles or something like that maybe it doesn't remind me of circles should help you help to be reminded of circles equations like this this Here is a unit circle when you have X where plus Y squared equals a constant that is a circle and in general If I had legs were plus Y squared equals cars that a circle centered at the origin of radius R if I wanna move around I couldn't have modified this to be X minus a or something like that only get into the formulas for circles this is enough for us so what happens This is a basic quadratic form It's a portion of 1 A circle here has equation where the variables have the powers of 2 now What happens if I put coefficient of its weapons of open numbers from 1 of these It shouldn't really change the overall Structure but the Constance what they are doing is if I put a constant here than that might change the ExxonMobil Mike But it can't see you might work a little bit this way if I put the same amount warping are still in a circle but I put a little more over here a little less over there than get is an ellipse closed that says are Things like However we could say That's weird Over like this more where Plus 9 while I swear the without the coefficients Belvia circle the Sooners I had on different fishes and letters shorter works in a different direction and we end up with something like this Now that's kind of a prerequisite for this class however not a Duma members itself as well as the money That is what we're about to do As Of these kinds of poker So what is a quadratic form its Here is an example of what we use queue for quite erratic excellent the tell you would acts as excellent plus 2 x 2 of this part is a little bit familiar even I X and Y over here here 2 variables warns X 1 of the other ones X to this part here I have to explain what this so 1st of all X is vector on taking this and I'm putting it in here and I get out that so 1st of all as a function by the way they used function here This is why we use the word performance that a function because this is not a function right past the vertical line that is not really a function but yet it's a graph so think of the word like function except you know we can do these kinds of graphs here that are technically to replace the word forms graph OK So what is this guy doing this guy is a function of form that goes from all are To because I start up into variables and ends up on so as an example it might Take Here's back exodus X 1 X 2 so here's X 1 here thanks to that little court system there and then Q Over here and it ends up in the real numbers so it only has 1 dimension and it might for instance take this to Points on the real numbers argued example now numbers conceived
24:23
So the matrix form formed with that could be it so the reason use this this terminology matrix forms I wanna write this in terms of a product of a luncheon each fund that I have a matrix terrace where user capital
24:58
And that on either side of the assignment X on 1 side and text transpose on So when I write it that way there's more to be said Ray look at that New Tokyo is ideal for me it's just math notation at this point ripped unravel that Cancel desolate Saturday but that just take the simplest 1 with income of what the other than 0 pursues the identity matrix was the identity matrix give us what we do Exercise is your list of variables that's 1 next to the Nikon write down what Mrs. X transposed is this guy a is the identity and an X 1 2 so that the unraveling this notation now palaces matrix multiplication Is that is what happens when I do this multiplication here what I get is just X 1 next to multiply X 1 plus 0 0 plus 6 2 And then when I multiply this out I get squared plus to square Now that's the quadratic form we had originally so I can write this in 2 different ways I can say queue of X is good where plus 6 2 squared For I can say that that's equal to X 1 X 2 times this matrix No Zurich will state it's not a question of our unity after years I give you this form right here can you right in the matrix that's a little attitude Can also do it 3 variables were limited to 3 that's goal And it's actually pretty easy to do want somebody shows you drink bullets this study this 1 per 2nd 3rd duo of attacks equals X. want squares plus 6 with let's let's see what this kind of graph gives us school fueled above want 1 So let's say all points where L. equals 0 The Telegraph on once said that she's still here Here's sex blind and here's X 2 and then years In a V Culex Askew wants plug it in is in a number it is 0 2 So wanna find out all points with few Mexico's 0 so that means that X 1 squared clothes and shoes where people 0 Not think about it If X one's anything other than 0 This is positive so them Muckadilla subtract anything from it to cancel that the only solution for this is X 1 equals X. 2 equals at the point where QMX 0 I just get a single point Now let's look at all points where you'd better want so that period here's ears to acute x equals 1 assigned . 4 Culex equals 1 that means the excellent square plus next to squared equals 1 possess Q and them weary learn what that isn't a circle Is this a circle of radius 1 very seen that that's unit circle so what that means is if we go up to where 2 of X is 1 All the points they give you that our inner circle We could think of this coming from and here but plugin all those points and you X is 1 at every single 1 of those that graft there is the unit circle And I Continue this on affected Q equal for the next 1 squared plus back to squared 4 and so I will appear more and there is another circle of radius too And so the squad erratic form where is a graphic threedimensional is it's actually would call a parabola and that coincides with the the word quite yet breakers quadratic was parabola as 1 in your old man now we get big boy math and quadratic means something in 3 dimensions that's that's This is called a parabola of gravel a paraboloid threedimensional psychic prevalent spinning it will study that's quite a bit After the midterm only certain calculus
30:39
1st exposure to it over the whole and now and now we find these uh matrices from the quadratic form Start with an example here next ones where does for next 1 used blurs is a little bit more complicated once it it's like the 1 before but now I get this added they've already got a little excimer X X 1 X 2
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Next to And I wanna right has
31:35
Human next was transposed decks for this part artery done I know at this part is excellent next to a paint and then the next 1 next to those 2 parts are done those adjust the variables to relay the hitters theatrics that we won't find anemic and just put it in the court to have to work real hard to get that will see this is now ready so all that Is gonna make sense is that if we use these little indices minister put things in a matrix The indices again needs of what we do have a major here This is a 1 1 spot in this is If they choose but this is a once used by And this is paid to 1 spot so those little indices there correspond to the indices in in these equations this guy here This is X 1 X 1 so the number inferno that McGowan a 1 1 5 and this is a 1 2 but he sees theirs There's 1 2 into 1 of so here's a delivery rate X 1 X 1 X 1 plus 2 X 1 X 2 plus 2 x 2 x 1 that We see this in the same thing because even I which these I know order put everything there is the 1 1 spot that have the 1 no 1 wants that I have 1 to put it to their have to 1 with that in the 2 1 spot Had Deutch you put 1 that's the injures Checketts Aizoaceae we believe that if if that isn't that the matrix that goes in there and want multiply that I should get back to the original equated to you can always check to work for that want to what we did when we do this I'll be careful when you multiply that set its X 1 plus 2 and then I have to X 1 2 of them and multiply this out its X 1 this plus x 210 that that's all of it is a major modification that Some variables things and I wanna mulled whether I get excellent squared plus duets 1 Plus 2 6 1 6 2 square which is exactly how I had unraveled And the put altogether if 1 square of 4 for and 1 execute these 2 are the same as the matter with X 1 2 Woodward hinders Americans which set that to square Focus list do a quick as you all the nearest about that will have a form of I had fueled is equal to A ones where floods B X 1 X 2 plus CD costumes Thank you write down the matrix that few of people Export and What do they had at work I need it We ever had here But that number in the upperleft in court Would regret here But that in the lower righthand corner because this was 1 1 2 2 and then this guy would I do with that But it at his half of it along with 2 1 and half 2 This has been you over to met exactly what we did Did that that for became too great only have there's er formula for 2 2 and since that was so successful 1 only do 3 by 3 see how that works but it but we do 3 bite 3 would keep in mind as we move from this portion of the class to the next course of last we're increasingly involved with higher dimensions of this is to dimensions for 3 dimensions looks do we did with the circles of threedimensional private X squared plus Y squared equals 1 that was a circle the bad 3rd variable which should that was the threedimensional version of the circle gets a sphere had to drive dimensions there that's the sphere radius 1 by changes are square here that it's a spear Radius r And if I chipped in effect change the coefficients a little bit slightly differently for each than you begin ellipsoid instead of we'll have a last year later circle aid but we go from all that Still the so know from parabola 2 parabola circle year OK U.S. assault background will get much more in this later when I wanna I give you acquired erratic form with 3 variables what that's going to be 3 variables its guarantee that strands a X him outside a latex that has got 3 variables now this multiplication X 1 X X 3 men entered matrix but in order for the matrix multiplication even makes sense at other 3 by 3 matrix time that what So what if I just put a simple thing in here like at 1 3 and then multiply that out multiplying this just used X 1 2 x 2 3 x 3 assists slowdown a little bit where multiplicity X 1 plus 0 plus 0 than a good period 2 x 2 Plus 0 plus 0 0 plus 0 plus 3 x 3 When I get that as a result when I'm not by this that I get X 1 squared plus Q X stew square plus 3 3 square As an example of a quadratic form with 3 variable so Khadija language again acquired Radick for ministry variables so that were quadratic implies powers it 2 Powers of 2 are the highest powers it to get even I have 3 variables Still rely allow maximum hours to that happens from that more
39:41
That It so this is an example Rowley the squared terms
40:06
Just like over here I could have the variable smashed up and without powers of to here's a here's an example I have 3 x 1 squared plus 2 Thanks to square Minus 3 squares and then I can mix up all these variables Plus 2 x 1 next to minus 4 1 3 6 6 so that's a more general quadratic form it's kind intimidating breakers all this stuff going on you have no idea what it looks like By the way a general way of looking at this What that thing does is it basically rotates futures curious that's what these things do they wrote to focus on how we needed The matrix Zadeh that's all we could do it the same way we did over here with the pay into To indices so there's the X 1 1 term that's gonna go here in the 1 1 slot And then use it to to If you want this is like a 1 1 2 . 3 0 here's that 3 years ago the the 1 1 but to those in the 2 to start and this negative 1 here goes in the 3 3 stuff And then how about this this is the 1 to do so would we do over here uses as an example when I had a 1 2 year Addis Split up between The 1 Jews but In the 2 1 spots whatever this was divided into it put it in Location is where are the 1 to put a little box around that where's the onetwo in the 2 1 that's here here Undertake take that to an ominous polluted at those 2 locations because this is this is a once used by this is the 2 1 8 years 1 in 3 But also fool around that wears 1 and 3 in the matrix years 1 3 years 3 1 8 1 3 Split up and the 2 parts of goes and then again the 2 win the 3 there's always couple spots left a cloud Are So you see that 2 3 spy And these 3 to stop A divide that 6 and 2 matches so we know the X I write that in matrix form irate out as X 1 X 2 X 3 times that Matrix 3 it 1 negative tuned One 2 3 . 2 3 1 and that's what So you can check you multiply that out see if you get back to Mr. Norway's check it but that's the trick Pages look at that you look at the indices for the variables and then you put them in the proper locations in the matrix of course it makes sense that matter that this is where we store information 5 0 2 2 0 put used to spot 1 to put him in the 2 locations that involve 1 to any questions about that but do 1 more for Travel I'm not explaining others quickly That the Yes exactly so if this was just missing Heritage's wasn't I wouldn't put 0 paralytic key to clue you in that but this was missing than I would have 0 here 0 there OK so they do the 1 from your Cedarbaum intern right now an hour away from that public you go raw here for a which last question here a section the levee is give you a big quite erratic form and 3 variables sources say was 1 of the At 1st it's intimidating launching know the decoder ring he just use go Few of them 1 text The exterior the butter 3 by 3 matrix here on a products 1 6 2 6 3 on that side and which is read off these are the diagonal stares at 4 1 5 negative to the dad 1 5 negative and then here's 1 2 so that's going to hear twoyear there's 1 of 3 negative 1 1 and 2 and 3 3 That's how we want to know Do it then you hope I put on is 3 points But if you've never seen this a news show up and United figured out any questions about that The now review
46:03
Done all the new material remember here during the summer they before the midterm and now the days before before the midterm because each classes like to lectures So now it's the day before the Nikon Not sure The ahead World 1st See The For a long time for the 1st portion of the class who spent time doing systems of equation and whether we realize it or not the 1st couple weeks was devoted to that we got sidetracked Lucy nature seasons of a capital that is we ended up with 3 methods for solving systems of equations linear equations you can't a drop Woodward when everything seems to be the but today we quadratics that signing put that were back in linear equations so I had M. equation And and variables that would end up with I can write the equation like this is just an extension of what you were doing when you were younger where you have a little way And what is a a is big block a stuff and it has M rose and it has And the columns in the 1st hour That is a 1 1 and Alaska 81 and 1st getting here's a M 1 of M and then would I do attacks well There's any of them so I can rent amount that can be It is just a list of numbers So our goal was to solve this big complex thing so the 1st thing we did was to learn how to do this you slowly if they learned me to these things to move them very specific situation where I had the exact same number of equations is very who spent a lot of time where this thing turned out to be a square matrix in the most general case it's a rectangle of wants a rectangle you have things available to you like finding the universal Nature or finding using Kramer's role because Kramer's role requires that Determinant and that requires a square matrix so this is the most general form if you're in that situation the only method that we learned was Rovers method 1 was reduction to size grow echelons songs and you should go back and you should look at what I talked about this there were 3 there basically 3 types of examples there's a warm breeze get a normal solution used get X 1 legal something next 2 of them there was over determined and 100 tournaments Stop start In the next case was that we spent a lotta time stating square matrices When restating square matrices the implication is that we have the same number equations as variables so now when when I go to end my hand case it's not this general cases more specific when I had and and we have this equation eh That EU which is equivalent to that it's a square matrix we could solve exercising a conveyors be as long as the determinant today was 90 The reason determined may ask to be zeros Skelos were 11 Anderson 1st remove the way we find the Anderson's 1 over the value the formula minute but it's it's 1 over the 8 times the adjoint so it's the determined 0 were dividing by 0 to start so doesn't work so we could solve this equation by finding the universe and the 3rd method that we had using Kramer's that took us all the way through Chapter 9 that's how long we spent studying systems equations even know we can't get detour and studied at nature sees the last met was Kramer
52:37
And Kramer's role a X he's still the system of equations there's a expert on through equaled be year and then Kramer's role us to do is to just hone in on 1 variable while but said I just wanna know what Well X I was equal to The determinant of AI divided by the determinant today this being determined that minute to tell you with a AIA is a where I take less than foot in the I can't think what I did it before I use J. syllabus which affected GA represents columns truck with feed Want to Be a Millionaire and then leave everything else along than I would take the determinant of that appear to take chairman of all matrix and that tells you what that individual variables So on an exam I forgive you work for variable system and disable find me x 3 basket Final Four By disabled funny X 2 X 3 than you can just horn in on that 1 variable values Even those those ideas came from different parts of the book They were all part of the same thing or off solving systems of equations now 1 of the back in a while some the details like At you know what is the inverse of what is determinants of stuff because we had to learn most things along the way that's essentially what we were doing Time although through Chapter 10 Chapter 9 a lot of the stuff that we just studied with pure matrices sometimes is just notation you do you know a transpose by put it see there you know what to do that if I if I put a little negative 1 there you know what to do with a lot of his just notation but you have to know to do the basic operations operations petitioned a beat when you doing a plus B which matrices they have to be the same ordered a both that Begin by and you can't mix matrices but different when you're adding and then when you multiply the them you also you have to be even more careful I do a Times beat you can do in by Annan by Annan unless these little numbers match 5 of them twice and And die or are these must be the same and that the resulting matrix CD that's the outside numbers sources said Quick example for your notes by have 3 by 4 And multiplied by bike to I can expect that to be a 3 bags to hot Scalar multiplication let's see times matrix that means multiply everything in there that set this is an example of that I have 5 times 1 3 and 10 5 by the No don't forget is you're not just more buying 1 thing 1 row over a vault level we had the idea that transposed the notation was a to the seat power Now as say it may be that it's just notation that tells you switch the rows of the call Whatever the just a little pictures for that I have a matrix here and I had this road and this road in Israel in a trance turns on those guys In the column If Daiwa steady for the test I take these concepts and I go find looks to the notes its water that Useful facts that are associated it on doing this for something you could do the terms of the heyday of the chairman of a transpose we use this in 1 of the proofs of identified
58:35
Yourself spirit 9 . 1 if you like former members as a fact and we also had a chance to vote Said that does itself is useful facts about the transposed The removed our way to the concept of determinate peso there's the notation like absolute value of the Matrix If I write down Kazan and I ends you can even talk about determinant it's a square matrix to begin with straight that ain't as Annan and has got it 1 1 came on and a N 1 and that's the most general thing do and then what is determined it's quite a complicated thing to describe skull is little pieces too Ghostwriter the determinant is What's called cofactor picks Bianchi and prolonged any here Besides the need rope or call will Since we got this 1 here muzzle just do it must use the 1st so if by if I go along the 1st row here tomorrow I do with this cofactor expansion I take the elements of the Ichiro a 1 1 in the sense that it wants to know if and 1 and those of the elements in the 1st row and multiply them by their corresponding cofactor than I had I have to tell you what all these little Things a little cofactor so each of see Jay's that's equal to negative 1 I J Times em I J and now I have to tell you what am I doing this Wesco complicated formula them I J is the determinants Iowa my unit do a minute take A here's Thursday and taking the determinants us why have vertical bars here to indicate determinant and then I'm gonna removed Jake Columns and Let compact way of describing what to determine if you practice at a bunch of value to see somebody do it you'll really reminded all those things are but that's the whole world Mathematical formula for What does have a quick example go with make sense of all that notation example I like to users it's this 1 here and what I recommend we doing his duty to get really fast at this just picked different rose due to the determine along let column along the that road stood on different rows and columns the answer 0 So you can always check conceded just see people get past that The short cut further over this Part of the call factories just put the pluses and minuses of the upper left in quarters always a plus the scientist alternates from them so when I do this determinant like used the That is used this that road there so I see that that the plusminus plus Like Star office 7 times a determinant of minus time determinant of something plus 9 times the Journal of something and then put those determinants of what I'm doing the 7 I get the 2 3 5 6 Where do the I get 1 3 4 6 When I do the 9th I get 1 2 4 Dodgers reduces down to a budget to buy 2 of the city 0 save us some time But that She should do to practice and then Andrew have to do a 4 by 4 of them yet I You 1 Divorce Now you should get yet what he's saying is if you have a matrix like this 1 2 3 0 4 0 5 0 6 and you see that that's triangular then the determinant is 1 times for 6 and even if he didn't you just use this column in you go 1 time determinant of 4 5 0 6 and then that's the onetime So you can just say that in your head go for
1:04:43
Don't forget these new role operations in order to modify the 1st you won't necessarily 1 that would certainly to buy 2 and 3 by 3 maybe that your preference I personally goal for the 3 trees and the other 4 by 4 and the time reduce the matrix 1st Florida terms
1:05:07
The inverse today would this guy represents like dividing by matrix here we write won over a week Israeli inverse that but it's like 1 over at the reciprocal of a matrix and the implication is that it may take gave you multiply it by inverse them with his vanity and vise versa you on that side that the idea the inverse is it's it's a reciprocal and identity behaves like the 1 in the number system the formula for a vs. gone however the determinant of that incident hinders doesn't exist unless the determinant is nonzero
1:06:08
At times the adjoint today in the of course and you tell you if adjoint is it's the cofactor matrix transport for every enough at the upper lefthand corner they won 1 you go you see 1 1 foot in the upper lefthand corner for every little entry through 1 of those ideas
1:06:40
9 factors that will make that matrix take the transpose would called the agile it's useful to just have the member said the formula for the 2 by 2 memorized foods have that right now
1:07:03
That's a little bit of a cumbersome formula would come too late use you can just memorize what it is like to a seat 1 2 3 4 a embarrassed and 2 by 2 case it's it's 1 over for minus 6 the determinant times well What we do we switch these 2 areas for 1 and make these 2 negative Take it do although the universe saying it's all those guys then a universe is equal to 1 over The determinant of Which is Latest PC times a minus be enough as long as A minus Germany is not If hoof
1:08:24
But That was all the Chapter 7 8 is that any ideas that Chapter 10 starts from the abstract things like vector spaces in the history of eigenvectors nite about lot stuff Chapter 10 Only chided narrow it down for you certain things you have to know about vector spaces so we had to talk a lot about it so I explain what it was but then there were just a few computations you have to worry about 1st of all what is the definition of a vector space
1:09:21
You and we pardon That But at these 2 guys and that means the some of them is and if I have any old in there that means a constant times the sector In the sectors so examples Artist unfamiliar setting the real numbers Always go 0 at the 4th 3 The vector spaces is being examples that will have to deal with it and then not vector spaces are portions of these you can have a portion of this India And be vector space so 1 of the examples of the exercises is the upper half point the reason this fails It is because if I have a vector you hear Sevey then minuses Daniel and minuses not in the vector space that's why fails to be elected not ending class I did the the 3rd quadrant of the 1st quarter of that if I'm questioning is that a vector space the way show that it's not a vector space as they take a victory in a week and thereby multiplied by negatively hits outside the vector space of fails the 2nd of the 2nd part of these Both are not vectors exist And then there's another example of the exercise Rice's shows the unit circle the vectors of filled it's not that means it's gonna fail 1 those 2 condition 0 A length of that debt The agent for that that double line so as not to terminate its double 1 the length of a vector if he is equal to Do you want to The new It is the square at me once where plus to where You just take the components to square foot the square is like a general version of the Pythagorean there 2 On the of again ask you to compute 1 of test Giuseppe recognize this Rotation that means take like that whatever you get in said the inner product that's another simple computations Hideo Nomo before July the United overdo it But you can be our end to transpose be that will you be safe if chance but point on Tsai become the horizontal Row vector se You want be 1 plus bodily damaging you if you 1 3 0 4 5 6 then in a project series here even have to ride out like this you just you just go in and that matching up the 1st CSU 1 and 1 and then you to indeed to matching those of multiplying eminent adding the results of 20 times for the last 2 times 5 plus 3 Times 6 which was 14 Pelosi if a
1:14:18
It'll probably be the case that I'll give you a couple of vectors and I'll say Are they were thought the piled sectors will say which ones are You'll see an example like that on the sample The key is EDG his where comes after the inner product theater product is tells you whether things areas are orthogonal not Inner product is 0 than their perpendicular are thought to have all you need so if you're To determine the orthogonality of 2 vectors it's basically like saying Pierre product and tell me whether I get 0 OK so now I've got that I want to put a little at a time into this linear independent but only that the twoyear Vonnegut the Aggies values and make sure that we get another chance to practice is so I gave it recipe for a marketer repeat the recipe gave it to you last time as time so I wanna do it I want to be a hike in value from a sample views that as waiter review the number 1 just giving a matrix and ask you finally agony values and the eigenvectors physicists like Ghana hiding had a trick during So there's a matrix and the recipe is if I wanna find the hike in values what I'm gonna do it is I'm gonna solve this equation the determination of a minor slammed high school 0 that's what I As sulfur lamp that inning The ACGIH so what is that this is a the determinant of a major And what am I doing here on duty subtracting landed from because this Islam NYTimes I which just that land is in the bag also that's all I'm doing is subtracting 1 minus Lander 2 2 former slander and make regions said that he goes 0 and sulfur lamp Opec 1 I multiply that out but I determinant they get 1 miners land time for my missed land mines will ripple 0 and then multiply that up further Notice earlier for here a minus for it so aminic Atlanta squared minus 4 land mine slander mind 5 land equals So that landed final slammed mind 0 . 0 0 course It was like to argue that as the 1st step get gagging values that for each of those values a movie vector associated filmmaker down the process the 2nd step is to get hike sectors and we do that it's for each Linda salt a minor slammed high school 0 for the vector of Peso so 1 land equals 0 That means 0 here on market is distracting anything from a The dreaded after a minus 0 5 0 that gives me a year
1:19:10
That a marketer subjecting promises beginning at any time he wants to equal 0 0 at my goal is to find V1 and V2 the way we do this is weak put it it augmented matrix
1:19:34
Sommelier rid these by multiplying by Wrote to his replaced by negative to rolled 1 cluster of when you do in these Things you don't have to rent out these steps you know what you do when you can get back you can do is go this is crossed that offers you know that's what you're going to do the next day And then you just go right for this says that he wanted us to be 2 equals 0 so he wanted to be The school over here our Arab So let us take stock of what we have so far more interested in that sector we will believe sequel to leave 1 To what we found that We found out that you want this to too 5 gay thanks thanks negative Now backed up to and there's a ragged The Now you can't do this again with another Lambda that landed equals 5 for land equals 5 A minus 5 5 W Equals 0 the but a different sectors they're just so not having the same letters For that same as feared by subtract 511 negative 2 to 1 I'll make W W 2 0 0 this funk quickly for 2 to see have permit circuits theirs What is so when I am at this stage Series C I could get rid of this firms going across not just go right to the says that too W 1 minus W 2 equal 0 W 2 people to W 1 and the veteran interested in W W wash W 2 That's equal to W 1 2 Go to water I want to The issue The What we do it the eigenvectors now Willis's collective question Yet so that's a minus 5 I don't take these days there and subtracting 5 along with that of 1 minus 5 is negative for the former 1st lady would focus on that landed even 0 I guess we vehicle to put here on tho wears is 2 1 a negative In the end effort and 5 had that W. Gould but what Now I said this earlier than expected is Solar absorbers said the way this matrix is designed it's a it's a symmetric matrix and so What's gonna happen is a way for you to check with you get these vectors right these vector should be perpendicular they should be fervent so you can check with got right of their perpendicular fight to the death that In product alienated 2 plus 2 0 so that's like a check to make sure you get the matrix that I give you will make sure that the eigenvectors Perpendicular OK now on make this Q. accused the orthogonal matrix that puts the eigenvectors in there but the 1 thing I have to do is add to divide the likes of over the length of B & W over the length of W so length that he is equal to revive the lake W is also illegal to reach 5 so I get negative to over 2 5 1 over Route 5 1 on Route 5 and make their positive too That's Q and then I am left with the resulting diagonal matrix that comes from future and pose a few we get this right that you don't have to do this multiplication could you know what it is in advance if I put this Eigenvectors firstyear than Italian value shows of 1st But put it in the 2nd eigenvectors hits by value shows There's the those of the ACGIH values in the back of the resulting debt Questions about that take those 2 example of their reproduced on you'll quick at this there's not a whole lot that I can do that any different than the 2 to by 2 there's not that many that I can give you the volcanic turnout saying so you practice these 2 Should be ready for the 1 on the using to their daily class this 1 the 1 on Monday gets yet so see these are the bag values It's a diagonal matrix I know these tours 0 but then these 2 values LPI values and specifically effect would be 1st year and a half with 0 1st by switch the order but this 1 here and that there the gets which do not reach economic that
1:26:36
But you know I just ask you for the Lambda Nu give me that you have to do the multiplication but if you get there Order Another issue that comes up sometimes about whether I at this stage wouldn't doing the vector Where's my eigenvectors There's 1 there But So sometimes you might get a different sectors we that say you had to negative what you might be wondering is that OK you might get into negative 1 of its fine I'm aware of the fact that you might get different sectors of its OK if you if you if you do this in you don't get negative to 1 of city 2 negative 1 just the the way you're algebra worked out that's OK as long as at the end of their perpendicular you make amount length 1 of Any other questions about the But I know this will be the result of doing that you could not by battery could just say no I know the answer and that is what is gonna be CSA yourself the the time plus Haiti tumult don't get that for you and I feel Mrs. Clinton cost you a lot of time in 1st edition Nuno I But what I don't have to do is do this multiplication of 3 matrices because I know the result is gonna be this anyway 2nd is ready and we have to write down you yes I a Was Where As saying myself sometimes what I could do as I could do this that I could do replaced world 1 with Road Plus room 1 I could do that and then You want And so it since I know that's what I'm going to do I can just crossed that often rewrite the nature his savings time at their orders Yes that's true if you mauled by the top half of monstrous avoid the fraction yeah but perhaps Get rid of the bottom Yes the case so in general if I had A just a bunch of stuff and I get and I get the idea values so this is just a general 1 The state ABCD and I go get the idea values are what they are land wantonly to enable the vectors D 1 and then the lander nature so it's going to be banned on land that it's a bag majors right put the agonizing so since I found my agonized 0 5 I just created drag no majors was 0 5 in the bag The remainder of Yeah yeah But say it is you might get a slightly different sectors mind you're vector always be multiplied by constant so you might so that they deserve to extracted the need to find out That eigenvectors what's really going on is you're choosing a value from 2 if I pick 1 2 than I get that back But if you take negative 1 thing meaning it what you said and you could also picked to their immediate negative for 2 that's also 1 that different people different sectors it's OK I look for that immigrated legally here's the general formula Q is equal to 1 over the the length of more than 2 over to And so this this 1 has to go with this 1 this 1 They have to keep that would gay a couple more adjustor miscellaneous things from us We're very did quadratic for but I just wanna remind me of a few other things that we did with matrices vocabulary and That Vote This and if it's not just a squared its 8 ahead of idle The way things In its as you multiply by itself doesn't WAR Q again and then the U.S. traces of a matrix recovered that big so a big 1 On 1 hand and 1 that's the traces the sum of the dagger The identity matrix That's the identity of the 2 by 2 matrix and 3 3 and so on and not for you by doing more of that them than the other concept that I haven't covered today that I will be on the test so it's pictures that remind you that is rank of a matrix But there will see
1:33:35
You're close to a deal This I did find the ranks over what would it was you basically reduce this down to row Echelon form the new count the number of nonzero that tell you computeraided sister mechanical way to do it what's really going on is this year's figuring out what the dimension of the images of this man let's That's kind of complicated leaders will find it Mr. reducing it OK had I wanna make that set this up 1 0 0 so I wanna replaced 3 with A negative report to row 1 plus row 3 Multiplying negative 2 years positive to Now you can see these 2 are the same Phillips go to here a minute due replace rose 3 With negative road to plugs broke 3 just to get rid of that 1 1 0 1 2 0 0 0 and it's in reduced form This is a leading 1 everything in its common 0 this is a leading 1 area common and I can't do any more of this is ranked At the end of it all I count of the number of nonzero rose so this doesn't count 2 nonzero rose that essentially says that this to linear independent arose and then the rank is to compete Talk 70 questions remain The GAO go over the sample midterm as much as you want for the next hour of Artie email the tombs had a chance to look at it and it's our posted on the website for maybe look at it for the next 20 minutes earlier had requested that you have I guess I severity has electronic devices with them yet Yes So Well it's actually hidden a um I just Extracted at the algebra for example so that I will do that example with so that would be like Number 10 When I did that he Yukon example within it we had to do some matrix algebra and a so that I just extracted that from example on question on the midterm from that Show us all this equation tell you with a A 1 made it 1 0 and D 0 4 0 Solve this equation for X Find a X That's when you get there it is sell by the way we do this is easy if this was just say 2 X plus exit polls wonder something I would just combined they can really combined But what I wanted to find Exs or isolated so I've got into different spots the way we isolated by factors of the key step is to rewrite the ads that because I was identity like that there And I times Exs just equaled X east through the same so I haven't changed anything but what I've done is I'll enabled myself a factor of the ethics of the vise a X B That nite sulfur X X books a plus 5 B so now I do Go find a plus I plus ideas I had 1 teacher there dashing also to negative 1 1 1 adding 1 teachers There are also people and a plus Inverse out the inverse of the 2 by 2 well that's can be As the determinant eschew plus 1 plus 1 1 over the determinant that a switch these and then make these 2 negative So there is a inverse a plus I burst and now X is equal to a plus I understand be to the multiplication 1 there in this 1st 0 1 0 plus 1 0 plus 2 Then a negative 1 plus 0 and then positive 1 For act which Yes What I think that's where the deluge yet They replace fiesta deal you will have any calm problem when your finally to the all being that I get a couple proves on the sample the ones that did in class and she's practice those find those in the know near notes from last day off on Friday
00:00
Beobachtungsstudie
Ebene
Eigenwertproblem
Lineares Funktional
Matrizenrechnung
Oval
Prozess <Physik>
Matrizenring
Große Vereinheitlichung
Sterbeziffer
Natürliche Zahl
Orthogonale Funktionen
Kartesische Koordinaten
Wärmeübergang
Vektorraum
Stichprobenfehler
Diagonalform
Multiplikation
Rechter Winkel
Nichtunterscheidbarkeit
Mereologie
Ordnung <Mathematik>
Diagonale <Geometrie>
Leistung <Physik>
03:27
Matrizenrechnung
Prozess <Physik>
Natürliche Zahl
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Symmetrische Matrix
Richtung
Ausdruck <Logik>
Faktorzerlegung
Multiplikation
Symmetrie
Eigenwert
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Zusammenhängender Graph
Wurzel <Mathematik>
Leistung <Physik>
Zentrische Streckung
Matrizenring
Mathematik
Streuung
Orthogonale Funktionen
Vektorraum
Rechnen
Fokalpunkt
Teilbarkeit
Diagonalform
Quadratzahl
Flächeninhalt
Würfel
Tourenplanung
Mereologie
Körper <Physik>
Ordnung <Mathematik>
12:48
Matrizenrechnung
Turnier <Mathematik>
Kalkül
Punkt
Momentenproblem
Natürliche Zahl
Klasse <Mathematik>
Rechteck
Auflösung <Mathematik>
Element <Mathematik>
Physikalische Theorie
Weg <Topologie>
Multiplikation
Variable
Exakter Test
Reelle Zahl
Perspektive
Lineare Geometrie
Vorlesung/Konferenz
Leistung <Physik>
Matrizenring
Determinante
Mathematik
Reihe
Biprodukt
Arithmetisches Mittel
Flächeninhalt
Mereologie
Garbentheorie
Diagonale <Geometrie>
18:40
Matrizenrechnung
Subtraktion
Punkt
HausdorffDimension
Quadratische Gleichung
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Ungerichteter Graph
Bilinearform
EulerWinkel
Term
Eins
Ausdruck <Logik>
Richtung
Variable
Algebraische Struktur
Reelle Zahl
Einheitskreis
Vorlesung/Konferenz
Warteschlange
Gerade
Leistung <Physik>
Lineares Funktional
Radius
Kreisfläche
Graph
Vektorraum
Physikalisches System
Biprodukt
Linearisierung
Arithmetisches Mittel
Quadratischer Raum
Rechter Winkel
Offene Menge
Koeffizient
Mereologie
Ellipse
Parabel <Mathematik>
Garbentheorie
24:38
Matrizenrechnung
Subtraktion
Punkt
Kalkül
HausdorffDimension
Quadratische Gleichung
Zahlenbereich
Bilinearform
EulerWinkel
Eins
Multiplikation
Variable
Zahlensystem
Nichtunterscheidbarkeit
Einheitskreis
Vorlesung/Konferenz
Warteschlange
Beobachtungsstudie
Radius
Matrizenring
Kreisfläche
Mathematik
Graph
Frequenz
Arithmetisches Mittel
Quadratischer Raum
Quadratzahl
Differenzkern
Rechter Winkel
Parabel <Mathematik>
31:10
Resultante
Matrizenrechnung
Multiplikationsoperator
Sterbeziffer
Extrempunkt
HausdorffDimension
Klasse <Mathematik>
Gruppenoperation
Besprechung/Interview
Zahlenbereich
Gleichungssystem
Bilinearform
Ausdruck <Logik>
Eins
Multiplikation
Variable
Kugel
Vorlesung/Konferenz
Indexberechnung
Leistung <Physik>
Radius
Kreisfläche
Mathematik
Frequenz
Fokalpunkt
Quadratischer Raum
Quadratzahl
Rechter Winkel
Koeffizient
Mereologie
Dimension 3
Parabel <Mathematik>
Ordnung <Mathematik>
39:35
Matrizenrechnung
Matching <Graphentheorie>
Quader
Bilinearform
Biprodukt
Term
Division
Gerichteter Graph
Variable
Quadratischer Raum
Quadratzahl
Unterring
HeegaardZerlegung
Mereologie
Vorlesung/Konferenz
Garbentheorie
Indexberechnung
Streuungsdiagramm
Leistung <Physik>
46:02
Erweiterung
Matrizenring
Determinante
Natürliche Zahl
Klasse <Mathematik>
Rechteck
Zahlenbereich
Gleichungssystem
Lineare Gleichung
pBlock
Bilinearform
Physikalisches System
Ordnungsreduktion
Ausdruck <Logik>
Variable
Vorlesung/Konferenz
Tropfen
Grundraum
52:12
Nichtlinearer Operator
Matrizenrechnung
Matrizenring
Determinante
Wasserdampftafel
Inverse
Zahlenbereich
Gleichungssystem
Störungstheorie
Physikalisches System
Term
Gerichteter Graph
Skalarfeld
Übergang
Multiplikation
Variable
Zahlensystem
Exakter Test
Schwebung
Beweistheorie
Mereologie
Vorlesung/Konferenz
Leistung <Physik>
58:31
Matrizenrechnung
Subtraktion
Matrizenring
Determinante
Mathematik
Element <Mathematik>
Ausdruck <Logik>
Zahlensystem
Einheit <Mathematik>
Betrag <Mathematik>
Kompakter Raum
Mereologie
Vorlesung/Konferenz
Faktor <Algebra>
Wärmeausdehnung
1:04:33
Nichtlinearer Operator
Matrizenrechnung
Determinante
Nichtunterscheidbarkeit
Inverse
Zahlenbereich
Vorlesung/Konferenz
Ordnung <Mathematik>
Inzidenzalgebra
Term
Ausdruck <Logik>
Topologie
1:06:29
Matrizenrechnung
Determinante
Flächeninhalt
Vorlesung/Konferenz
Grundraum
Teilbarkeit
Ausdruck <Logik>
1:08:19
Resultante
Länge
Punkt
Klasse <Mathematik>
Reihe
Kartesische Koordinaten
Vektorraum
Drehung
Skalarproduktraum
Pythagoreisches Zahlentripel
Quadratzahl
Exakter Test
Eigenwert
Reelle Zahl
Konditionszahl
Mereologie
Einheitskreis
Vorlesung/Konferenz
Zusammenhängender Graph
Projektive Ebene
Gerade
1:14:05
Matrizenrechnung
Prozess <Physik>
Physiker
Determinante
Orthogonale Funktionen
Zahlenbereich
Gleichungssystem
Vektorraum
Biprodukt
Skalarproduktraum
Eins
Flächeninhalt
Eigenwert
Stichprobenumfang
Vorlesung/Konferenz
1:19:10
Matrizenrechnung
Länge
Wasserdampftafel
Gruppenoperation
Klasse <Mathematik>
Reihe
Fortsetzung <Mathematik>
Vektorraum
Biprodukt
Symmetrische Matrix
Diagonalform
Multiplikation
Differenzkern
Eigenwert
Rechter Winkel
Tourenplanung
Vorlesung/Konferenz
Ordnung <Mathematik>
1:26:25
Resultante
Matrizenrechnung
Länge
Subtraktion
Abstimmung <Frequenz>
Gewichtete Summe
HausdorffDimension
Natürliche Zahl
Klasse <Mathematik>
Zahlenbereich
Gleichungssystem
Bilinearform
Zählen
Eins
Erwartungswert
Multiplikation
Exakter Test
Rangstatistik
Eigenwert
Stichprobenumfang
Minimum
Nichtunterscheidbarkeit
Lineare Geometrie
Vorlesung/Konferenz
Addition
Bruchrechnung
Matrizenring
Determinante
Inverse
Vektorraum
Teilbarkeit
Divergente Reihe
Ablöseblase
Ordnung <Mathematik>
Luftreibung
Explosion <Stochastik>
MechanismusDesignTheorie
Aggregatzustand
1:40:43
Vorlesung/Konferenz
Metadaten
Formale Metadaten
Titel  Math for Economists  Lecture 8 
Serientitel  Math for Economists 
Teil  8 
Anzahl der Teile  15 
Autor 
Kronewetter, Jason

Lizenz 
CCNamensnennung  Weitergabe unter gleichen Bedingungen 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nichtkommerziellen 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  10.5446/12902 
Herausgeber  University of California Irvine (UCI) 
Erscheinungsjahr  2013 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 