Bestand wählen
Merken

# Math for Economists - Lecture 3

Embed Code
DVD bestellen

#### Automatisierte Medienanalyse

Beta
Erkannte Entitäten
Sprachtranskript
So what we've done so far in this classes we we looked at systems of equations And then we realized that awaited solves systems equations is to use these matrices and then his practice with that and basically when you these role operations in the matrix the same is just doing the same operations on the equations just that you eliminate the variables when you focus on just the Matrix now what we're gonna do it a study the mathematics of matrices just As a separate issue eventual Bring the 2 concepts back together again and say they hot Had a major cities after we studied mathematically out is that relayed back 2 systems of equations Tora Bora think of what we're doing right now is a little bit of a detour From where we've been and we will come back around and make a connection will start with Bill example here is how much of an example but we just wanna have a couple matrices up there to work with A standard matrix a chided keep the numbers wanted people Rice exonerate Candace that's easy for you to copy here he does glanced at it you know what it is Here's another 1 here 1 2 3 4 4 5 next In a different configuration the last time we learned about how you how we identify these matrices in terms of dimensions and it's really just about the rows of the columns of this guy here this is 4 0 1 4 0 2 just doing the anatomy of a major going and see where all the structures fit together This is role 1 wrote to go 3 and we have our columns a so we have 3 different columns there Columns 1 2 and 3 over here I have common 1 and then colleges to even know AT&T had the same entries you might say they're there there like each other in the sense that they both at 1 2 3 4 6 in them but they're very different matrices in terms of their dimensions this guy A a at world stew by 3 and this guy he is a 3 by Stewart's 3 rows 2 columns to the Rose and then the columns at the convention that we adopted so really have the same entries their different matrices and we can say it's it's obvious gotta be careful when he said Obviously the map glass is not always obvious to everybody in this case I think it's obvious these matrices and not the same because this guy is more of a flat rectangle was a vertical rectangle attorney match rent so it's obvious that place obvious That gave his so in this example let's clearer now we won't make mathematically at accurate exactly what it means for 2
Of equal or not call I will miss a major cities are
Equal and yet I have to things up 1 they have that this same order Syria is the rectangle has the same as if erecting the afternoon match at a Soviet differ rectangles urges not equal dislike that case but when you do have the
Same rectangles and you have additional thing that they have the same exact entries as well so will illustrate that them that they have The same that was the word we used for those dimensions there Dubai 3 let's see order they and order of these 3 by to say that the same order And then Entries are not only say that that's I'd say in English the entries are equal but I wanna be more specific as you might think all those at equal injuries but they have to be in the exact same location so mathematically the way we could describe that we we can say high J is equal to be That's says that the entry in the Shiite throw the G-8 common is the same as the entry is the I throw in the So for here we say a is em by hand and he is also an empire and that And at the same order and what it means for entries to be the same as in use these subscripts again This is what you need in order to solve problems his intuition needed to build the do it but as we doing now we want to his gains a mathematical maturity and so if you can understand why is that that notation means This A statement in English China and if you don't feel like doing it OK at least as understood and budgetary gap to be equal pocket
Has little example where that make it be there's a little problem similar to 1 in the book may yet but something like this on the mid-term too So this is a way to incorporate this new concept of matrices with
This systems of equations that we've been doing I have 3 Linus accident 3 Z 9 X Y Will be so that's a major expense got variables and the entries now and you wanna think of this as if it helps to make a little grid there I'm thinking of this is to buy 2 matrix and I'm saying the dot matrix is equal matrix Next Y X bluesy negative 1 too And the command would be solved or X Y and Z This is just a way of disguising a system of equations its linear This is a linear system of equations wiser bioassay linear just as I see only power the 1 that can attach that word linear anytime I see the variables only having powers of 1 OK what welcome we extract from here I wanna I wanted to find out what the variables are and there's basically for equations got explosives equals 3 I've got busy minus X is negative 1 he just pause for a 2nd look at it you can notice that you have you have to equations here that involved X and Z and a number of losses focus on that and we just get those 2 areas so the 1st thing I see years is I have Z minus X equals negative 1 that's from here and here And then I also have X plus Z calls 3 I will have to go back in user matrix method you can just do this the old fashioned way like you wouldn't School 3rd add those 2 I choosy close to that tells me this is unique because 1 and then because I know that equals wanna go back and find out what Exs so it acts plus equals that implies that X plus 1 compulsory which tells me that seek to dialect 2 of the men I got 1 that got a couple more equations to work with actually got 3 minus X equals why I could use that because I know it Exs or I could use plus vehicles to try to find out why is there no Woods E is now so let's just use this secret agent I must have a wide Z To that implies life plus 1 equal So now I have wiII go so my solution is sequels to what he called 1 of these calls this is actually an over determined system because at 40 Guidetti even write down the 4th equation that can shows the over determined there said that so this is an example of an ordered we actually do get a solution but if you don't usually
At 40 equations of only 3 variables that It Hi or what Wanna get used to writing Louisiana tears in matrix because after all nature A awaited think of it is if they a add vessel of information it's a litter box but each little entries
But for key Our right now Bush's on creating here's a little Russian dolls and other things of the biggest group is the square matrices than within the square meters is special ones called back matrices and now within the dad who make few announced a special matrices Todd and energy majors
When you hear not the way when he 1st learned edition And subtraction multiplication those called Operation loser operations between numbers but you don't really focus on that were you just focus on addition and subtraction multiplication of the translation from through this word is plus or minus multiplication and division those of the operations that we do that
So want do all those things we did them so successfully in nearly life with 1 by 1 major cities now we'd like to do all those things with 2 late matrices and then 3 by 3 and 4 by 4 2 very familiar with that of multiply and add and subtract there but it's a little more complicated when we get Higher dimensions Ocatillo star off with additional border take a look at what does it mean to add 2 matrices for 1st of all it only makes sense A man have doctors gets in your mind which is that they have the same order they gotta do the same rectangle I can't match a rectangles that don't have the same image Cell They have the same order this debt and by then I am now we're in the that The general world here or not We don't have to be square matrices ridges at that match so for instance by it's easier to see where it doesn't work this doesn't make sense that's the 1st thing they have to match up as rectangles of this 1st Show
But OK so if they do have that both a
Or by So if we do have that situation than what what should be if the threat can a plus be OK that's going be here is made in here be they get a match up so I can draw with similar size is there and what should we do what with what should be the case on what's going to make a lot of sense I take entry here So what is a random entry in the middle of this nature That's going to be A high said Jay where this week I throw and this G-8 column undertaking any random injury rate may be the 3rd row 5th column whatever it is and then What should I do Russian match it up with a corresponding element over here This is by J in a match those of the Vivian new matrix this 1 was em By the end of the song was and then the resulting matrix is also the relief and by then and what should we have only I throw and G-8 column while I should have A kind plants BIG Here Mr. that I described mathematically is it It's not that hard but what would it is it's it's an opportunity to practice with notation To so set an example We see it only many examples of this it's it's them Intuitively straightforward in fact I usually kit that if I came up to you on the street stick of teller enters this well what else would you do when you if you know anything he has saved her life Tracy yet again so I think I would guess a the match those up and hope for the
We had a lot this all said just Match entries and Adam like you would if they were won by 1 or the numbers of 7 post its and the negative about negative 7 1 and 1 is 2 0 0 4 4 1 and 2 3 0 4 or so as the obvious thing is nothing else you could think of to do
1st If a John matrix world We have 2 kinds of multiplication we have multiplication of major cities and then we also have what's called scalar multiplication will start with scalar multiplication it's also got back
But Guy with the words scale areas scaled about a scale model who is a scale model it might take If I had the City of New York and I wanna make a scale model of the remedy Every dimension of every building in every street and multiplied by 1 over 10 thousand and the reeling down the my little model so you multiply Everything that's was scaling is multiplied everything you see by the same value that's why we use this language You're skill OK so what's an example of that if I have to see a notice the rotation here I'm using a small see this indicates that
We were talking about a real numbers here and a is a matrix sake and stand by N & M by in-house OK solos What is this scene we can write down what the portrait done every mn nature of the universe where there are 1st century A 1 1 minute all again this is the 1st rd 1st column this is the 1st row And columns and I have been M throw although it down M. row 1st column and and corrupt and I write that to bat and so what should this be if it's if we use or scalar we have scale models of Manning's multiple reading by the same number than this should be the matrix 1 1 seat 81 on him And 1 seemed a M and half so I just did every existed the scalar multiplication of every major so the universe great their done So that's what's nice about matters we get these general formulas and then we can start to do some specific examples of the specific examples of 5 . 1 2 3 4 Well that's just 5 10 20 I just scale that a Scaled everything and then similarly wanna add to the board here If I had let's just looks to another example if I have 9 12 6 A negative 3 you could notice that every single term has a 3 so the you to factor out 3 He wanted for a layup you'll see it but to hear I see 3 sows factor that out of the 3 outside And then I have 3 4 4 4 like 2 negative 1 so that's the opposite direction factors
She P 5th should if But scalar multiplication now we begin Daddy actually multiply to readers will do this by example
Just watch me do 1 and then will try to extract down tease out some of the And from some of the method used naked generalization of that solid start off with a game beat to matrix multiplication matrices multiplied by each other the first one is to 1 negative 1 3 and the 2nd 1 1 2 FOR negative not financing game I say stick them not tell me what the interiors then you're intuition would be well and go to times 1 1 tends to the native wonders for 310 negative once and you would not live Fast that encounter from its intuition doesn't work here now will will discover later is that what we're trying to do with this matrix multiplication is trying to preserve some stuff inside these matrices and we we haven't talked about him yet but there is there's geometry inside these issues is a hidden picture for each of these matrices from only multiply and were trying to preserve that geometry and when you don't know what the geometry is it's hard to imagine not any time with the yet because I wanted to the matrix multiplication that's why we have this Funny animators Multiplication is to preserve this stuff that's going on behind the scenes and will learn about that later OK so how we do this well 1st of all a tell you that it again tell you what type of major treaty it back from a multiply these to really get another matrix but right now if I said what kind of a matrix would you get you would know unless I tell you unless you that's some experience so I'm telling you right now when you take it you buy 2 matrix and take another 2 but matrix you get to me Morgan after learned all the particulars of that the to want is now if this is going to a tutor 2 by 2 matrix which I'm telling you it is that going at entries there's the 1st entry 1st drove 1st column 1st 2nd column 2nd row 1st column 2nd row 2nd call the entries now I have to tell you have no idea each 1 of those I think I'll start with a clean board But critic of far-right so for the 1st 20 year I wanted to show you Cut Picture quarter so this is going be C 1 1 how we obtain C 1 1 does not think about it we got this notation here this is mathematics we see 1 1 that mean some centimeters random numbers
So the way entertain the element that goes in the 1st row in the 1st column is by combining the 1st euro here in the 1st column that the combined those 2 things and that's going to create a 1 1 tie
The entry of that appear that's how I create C 1 1 1 minute duel is and take the 1 for hair and turned on its side match those like this if you put on 1 4 here I'm now due 1 times 2 plus 4 times 1 talent that was not there scene got product or something in a physics class or something like that some of you may be in high school doubt that We're doing here is the dot product between the book doesn't call that price This that see OK so C 1 1 is going to be late 1 times but 2 plus 4 times 1 leg is 6 of the 1st
So I just told you how we how we do see 1 1 c to 1 his B B 2nd row with the 1st column the 2nd row matched up with 1st column of that same thing that we just did So I'm here I have a legal right to use it to help wanted for their and negative 1 3 who was 1 of 4 so when I do this same multiplication I turned on its side match them up dear wonderfully for now I'm gonna have 1 instead of 81 negative 1 phenomena have for 3 years less That's the next entry With 2 more days We are a long mallets to see want to that correspond to the 1st euro 2nd column 1st row here at 2 1 in the 2nd column 2nd column is To what to negative 1 I do the same thing where take this mentally Lesley Turner said attitude you in your mind you can you can just the match them up like that guy at 2 times 2 plus a negative 1 by 1 That's for minus 1 history The last 1 to know it Established a pattern We go along more quickly to chew that's going to be the 2nd row here with the 2nd column And here negative 1 3 so take that and you're in your mind in Machynlleth like that to times negative 1 plus a negative 1 of her positive 1 positive times through the war So now I consider what may be That 6 to 1 that 2nd row 1st cost down here and and see 1 2 6 3 and Nancy to 2 That's it Yes Today write it incorrectly Oh yeah whatever this is it should be that guy get diverted closets wife thinks that's a negative so this is a negative so that this is a negative Chu Plus negative 3 5 0 this would please please do that we're going to see a mistake counter don't be afraid to tell is soon to be quick
Fixes the sooner I you on thanks I do my my best to make perfect lectures but I don't have a long track record of being perfect stuff That we expected that still try her So now it's extract out what happened there was still a little bit From Ultimately would like to build a right Down in general what it means to multiply to make from this We get that from this 1 example alright so if I had
A times B equal Sieverson what had happened notice That is what I just did there would no more the columns of the Rose didn't match up right if they have the same number of elements so it convenient chose it but That 1 for matched up with 2 2 1 by have the same number of elements in those cases they would match up so that's the 1st thing if I can say that this guy is in and by And matrix And then this His by Are nature said be welcome we say so what I just wrote here it describes as a phenomenon that the number of columns here has equal the number of rows The was macho OK and then when we get as it were really it as a result would immediately outside Associates M Are tricks and become more clear we do some rectangular ones in different values So 1st thing here treated gleaned from this is that these must match so area starting ILS someone's in the book that you ask you do matrix multiplication you can't do it says were a trick question we have to make sure that the inside numbers match and that represents the number of columns here has equal the number OK and then the result The result is an are so the outer numbers give you the resulting nature you can really see this happen in the 2 by 2 case that start to back a teacher easiest 1 their things it to their C can you know These matchup in memos also match That's why you get to buy 2 because the outer numbers not because the pocket so that the the general said at her multiplication MHC 1st he have the habit that situation than you can then you have to learn I executed to practice executing a little What if So Tsub simulator Macia sees here she 12 Work OK so that a and multiply this guy here he is able to provide free and revived to so if I were to do a D than that's going and I I know I can do it you because the threes match up and I know that the result of doing that is to buy 2 The threat out more steps to have your notes to buy 3 Time 83 by to those match and they result The outer OK so let's get started so if if this is gonna be a CD that can be to again a matter seem want 1 he wants to see you want to do that as of the end Tuesday at defined as for entries but do exactly the same way we did now we have a larger major studio little bit more but to do so 1 1 that's going to correspond to again if you like that helped with the picture that can be the 1st row With the 1st column them are correct the numbers in the notice so the image of its 1st 1st call OK so when I do that would I would I wanna do I want to take this row over this column here return on its side of the match it negative 1 to make 3 we get better it this year after actually write it down but at 1st if if you're not comfortable a better right That today OK so C 1 1 is going to be on a multiply these 2 guys was gonna do the multiplication case that's negative to mind to clients 9 and check my arithmetic But export
OK see 2 2 Their seat now when you use when users 1st call on go ahead and use it again on the 2nd 1 so that's why I always do I do see 1 1 and then seek to 1 that where I used the 1st column in both cases proceed to 1 that scan and be a 2nd car star in the 1st column again with the 2nd row
So we see match them up there it's negative 1 2 2 2 0 phenomena go negative 1 the bottom 1 that's negative 1 and 2 times 0 0 and negatives 3 times native to its 6 his 5 please ask a few not sure on getting these numbers seminary produce a here just ahead Thereafter Paulette bore down again OK So now or movement on related to 1 so that correspond to These 2nd column With the 1st row and I'm their match
So now I take the 2nd call their 0 1 negative to and I matches up Brady here with these rose 0 1 negative too
And now and then go 0 times too 0 2 1 negative 1 that's negative 1 to minimize negative 2 tons 3 that's negative 6 years The last 1 is this week due to bat in the course onto the 2nd column in the 2nd row At Now all match up 0 With the 1 0 and 1 with 0 0 negative too with negative to positive for us now I can say that a is equal to Negative 13 5 negative 7 when it 1st learning it might be helpful do these things with you guys cannot appreciate is a little tedious see want China drop that he also wanted
Drop due in this every time that you need to do whatever it takes to get it right so That helps to do that Until you're more comfortable but eventually just look at it and you can't do that arithmetic in your head that I'll say a lot of time off any questions about how I get summers entries yet Wrap Yet worry that they do actually have meaning that was alluding to their earlier that theirs There's some geometry going on behind you there's there's a parallel attorney and areas of parallel to we need spell more times this rudimentary stuff eventually or started show you some of that stuff Are still But turned aside said now a more advanced 1 but I'm not gonna ride out all the different steps because it just gets a little ridiculous especially when if I switched to order here and we now have to be a case of a that's now than a be is is a 3 by 2 and a visit to about 3 so OK I can do that multiplication because they match
The result is idiot by 3 that's get 9 entries their right out all the 9 entries so hopefully we watched it A little bit so can see the method malpractice here with a larger major gotta have 9 entries here if you like we can write man C 1 1 1 2 1 3 2 1 2 2 2 2 C 3 1 3 2 3 3 Want Yet there to get a little bit of a better method this 1 after rent out each 1 of those 9 filets That's right amount next to each other so be it is a negative and 0 2 1 negative 3 negative Chu and then a this to 3 1 0 2 2 OK so what was our method our method was Take a column And then turned on its side and then will reduce take that hit each 1 of those that give us their 1st 3 entries because that'll be see all these Column 1 race so when I Yukon once get seen 1 1 to 1 each to those 3 entries come from what wanted 2 times negative 1 that's negative to plus 0 that's the 1st entry negative 9 and move on to the next row 2 times to his 4 plus 1 times 1 is 5 London extra 2 times negative 3 Is negative 6 plus 1 native to the negative 6 minus 2 His who stared That for a 2nd if you believe it See if you wonder yet Yeah we're done everything we can with the 1st column move on the next car next column as negative 1 0 And do exactly the same thing match those up that's 1 plus 0 that is my 1st entry here 1 Then I do negative 2 plus 0 negative Thank you Part this is positive because native 110 negative to positive 3 plus 0 That gives me the 2nd call
Anybody readied a riot in the streets Yeah I think that we don't think we can further setting comets the 3rd
It's very methodical you don't really I think the main thing you learn here is concentration is The weighted develop your concentration because it takes a lot of already distracted look of butterflies I that it's hard uses a bunch of numbers of it so 3 negative to go that's 3 times native 1 and then made to 10 0 so that is negative and got 3 times to his 6 plus negative to the And got 3 times negative 3 is negative 9 plus 4 made it but that's B a now there's a richness to the Multiplication of matrices it you don't get when you multiply 1 by 1 natures To use numbers so noticed if you have 2 times that people the refinanced so with you when you distillate numbers you used to is multiply want but look Here we have a B Is not evil to be here so What When you move to visit The matrix world from number world it's a little bit of an extension in the sense that we can do a lot of the same operations of new addition and multiplication William learn attitude division But some of them Properties that happened in numbers Don't translate into nature's namely the multiplication is not community red that the word we use a phony is a fancy math word to describe this this is not a non community where is the real numbers a community doesn't matter how you multiply and you get used to this so this is a weird thing not only are they not equal I mean they're not even closer it was too late to of the 3 3 2 different and complete that's a little bit interesting play around with that a little bit but Strider right down the process of multiplication of matrices in general
Villagers in special cases and more move on What I write down the process so that in your mind chair think about what how would you write down this process how would you describe it the alien comes down and says he I would watch you from above 2 this matrix multiplication What's the process
Needed to support the book's 1st see now how Cook
The OK So that was that was matrix operations in our answers this section 8 . 3 8 training for short get through those pretty quickly Section 8 . 3 this is about not tricks trans position in the English-language if I take a word and a switch to letters what's it called I take the word him as telling him not necessarily had said that they could recover the example or use which froze out of what's what's it called a research 2 letters on a word or even a number to switch switched to the scholar trans position that colony looked at opening in the dictionary transposition is just switching its moving the position they treat you think OK so now But you know what that is in English the trans position is switching to letters what in the matrix could we switch What would we have available the switch yet arose McCollum just like I did their here you switch the rows and columns and and also You can think of that in terms of the subscript AIJ his been which has become a JI that's That's the trainers position I'll show you that all right off the bat but an example 1st just see what's going on here if I have a is 1 2 3 4 5 6 then the notation we use free transfers diesel exponent transpose that's that The notation so a transfer most of what I said gonna switch the rows of the columns OK so I see this row that's 1 2 3 that's now and become the 1st call them and I see this that's now organ becomes the 2nd column the way we say this we don't the you could say it's a transposition of aid we are sure that we call it a transport stated in English a tree and sister notation You don't work with math notation it's always confusing radio if you take some of this stuff and show grandmother view show this year's tournament impressive Mathers scholars fancy notation nobody has any idea what it what it means when I look at it looks really impressive we wanna think of math notation is too short because I don't feel like writing out if I if I wanna tell you I wanna switch the rows over combat or after right out OK now take the role and make the column how wanders have a symbol that means that and that's the origin of math notation is the easy that's all Soledad so We wanna think of his This is like a high common your death if you double-click got all kinds of information that pops up by hover over it says a transpose cover over it it says switch the rows and columns are by double-click on have an example where something so notation is really a convenience you shouldn't be intimidated by But it's hard not to be but it always is always think of did double-clicking on it would meet at site could start to deal with the teacher it's just an icon guess it's that extras meaning behind it Sir OK so we're guy here's the trail boast that implies that the Rose a Collins of Patriots that's the easiest way to think about and we can have a little visual if I had a year's the columns of it then A transpose will be this That's patrons and them we consider this a little bit more mathematically safe OK with stray The right the order of the day it is Em by hand with Entries water entries a veteran them simultaneously that's a major that the high end Jared no I goes from 1 them J. goes from 1 end Then a transpose which had a chance because they could really see that the word transpose here To see if it's done by a man in a trance poses And Manley transposed the dimensions and you we can also see with entries with entries a JI I I which A switch If Chris
Don't know A year Open to happens it happens we defined something in then we stirred thinkin about it
Realize lot the things we can say about it there's properties a transposed We noticed that he had voted to the transpose Bozo something and then do it again for what should happen just look at it in terms of this if I do the transpose because an abandonment and do it again goes back them by hand direct back to the same You can write that down and symbols like that This is just an opportunity at practice with the symbology getting comfortable with looking at notation so there's your 1st we can have an example with 1 2 3 4 5 6 days I do the transport of that adds 1 4 2 5 3 6 And if due the Transco's again is a transposed measure transposed well on the first one this They do trains goes again Back tourist urgently They pose transcripts But if But They also covered I had to make add 2 major season and take the transpose Madam add these to get a matrix that take the transpose that's the same as if I took the transpose individual majors market into an example that he practiced them on When you do the same when they're ANB in earnest on here a play with so multiplication just think of it as you can distribute It's not exactly that you could think of it that way to distribute the so that if our of followed by 1 up here said this is true what's this to come up here this is what the then your natural instinct say A CPT make you be wrong at but what matters interesting it's not always predictable this is equal to be transport The listen and deceit says hello players wanted to wounding 1 3 0 and he is really of want 3 0 2 1 2 2 chances to some quick multiplication here a B transposed 1st that what that notation means that double-click on this er hover over a little window shows that that says multiplying 1st then take the chances are excellent do that this multiplication Manigat 8 I can tell right now nominated to light to matrix hitters do this multiplication and that negative 1 plus 0
Plus 1 0 And I've got 1 plus 0 plus 0 and I've got 3 Plus 4 plus 2 strippers for close to his A negative 3 plus 6 Nader Super Sixes positive 3 0 and the 94 transposed so that means The Rose Becomes columns or the column become rose aware of there's the result of doing a transposed if I do do transfer was a transpose 1st look just look at but look at age transposed beat transfer that what would that be 8 transposed would be 83 by to and Beat transposed It is a had to buy 3 of this 3 by 3 so this cannot be equal to that the readily see there's something fishy gone if you were to get a chance posed beaches poses this it would work out and get a 3 by 3 in over a year to vacate the real result is that which the order and with the chance
OK so as practice that beach France opposes Bits Flatter like this 1 a negative 1 0 0 1 3 2 2 Thailand's age Transco's at 1 2 1 2 1 3 0 now that'll be it you like to see I have to buy 3 3 bank to hear again to buy 2 kids get equal the focus was to the multiplication of 110 negative 1 is negative 1 plus 0 But 1 his music 3 plus for us to his knife on this column positive 1 plus 0 plus 0 and then have negative 3 plus 6 native Super Sixes positive 3 plus 0 so that's just an example to believe that statement and this is our 1st year and assist They these years to that of the the 1st on that emphasized this is their Corning 4 What is from a next trick if your trip to review a proof everybody's favorite pastime writing if From error walk you through it will be OK And just so you don't think scandalously because a celebrity exam actually put this proof on the mid-term elastomer taught the class so if you think again shown up It will but is not as bad as you think It's much worse if is not and all here do 3 proofs of his class this is the 1st on the other ones will come a little later and the reason I do that is just because I wanted take away the mystery ain't you better actually until the records is the simplest come up with so you could learn so the idea this is 1st of all just to understand what the truth what watch Will be trying to do So here's audited approved the best His equaled seat transposed and fears of personal look back here is just an extension of this this 1 says if I had a indeed and transpose vests switch the Auburn equity and each this 1 that a B and C CDs which the order an equities believable because we have that on
At For all of us What the 1 with Thekla 101 members that a equals 1 that cycle 1 by 1 matrix 1 in there so that's That's a diagonal matrix code could also have to buy 2 were a vehicle to be identity that said that had imposed nature of that a square that totaled 1 0 0 1 2 times what 0 0 0 1
The that on its side give me 1 plus 0 continue this 1 0 plus 0 And this 1 is 0 plus 0 as well And then the last 1 a 0 plus 1 There's another ad impose matrix but isn't the list goes on from there because it is in the when 1 101 cases is simple we could that equation here it's much richer we we can have matrices when you square than they give you themselves back again but they don't look anything like those of 7 others importer We can't counting to tried a finds some items of major find a formula that gives you all the of Casillas check that out at 1st glance you wouldn't expect it to be just got too many things that are kind of bigger numbers so I would expect to get that back again but you do 3 negative three-toed native to time 3 3 to to the take this son put on a side 3 times 3 is 9 minus 6 history And then the 6 miners for his too Now I these er negatives to just to marks on board and I have negative 3 plus 6 negative The last 1 I have negative 6 plus for you so they let put But a trace of a May chooses alas concept from Section 8 . 4 this is an easy thing to compute but nobody tell you it is a no idea what to do so this Tree Sears as awards will be lowered Cherice crying need you the I have a couple things you could trace something where you put it in a transparent piece of paper tray something we can also think of trace elements that's really what's going on here the trace of a matrix is as if you boil it down the residue of the Matrix asserted essence of it is found partly in the trace a miraculous unit for a while but I still tell you what it and that a city congested Ahmed The Tracy said that the notation trees a that that So how would you eat from them That configuration 3 posts 5 years so the trace is used Some of the elements of what what's the vocabulary hits the main backer of has a very easy they view is that you always is viewed voters have 1 more example of the set of that for for matrix and the trace game plus Complicated site has taken a nag and had a those elements we are 27 33 34 and that's an easy thing to compute talk X that's the end of Section 8 . 4 4 start on 9 . 1 on Monday enjoy your weekend
Einfach zusammenhängender Raum
Beobachtungsstudie
Nichtlinearer Operator
Matrizenrechnung
Subtraktion
Matrizenring
Mathematik
Matching <Graphentheorie>
Hausdorff-Dimension
Klasse <Mathematik>
Rechteck
Mathematisierung
Zahlenbereich
Gleichungssystem
Physikalisches System
Term
Algebraische Struktur
Knotenmenge
Variable
Konfigurationsraum
Zahlensystem
Matching <Graphentheorie>
Differenzkern
Hausdorff-Dimension
Rechteck
Vorlesung/Konferenz
Ordnung <Mathematik>
Matrizenrechnung
Einfügungsdämpfung
Matrizenring
Zahlenbereich
Gleichungssystem
Fortsetzung <Mathematik>
Physikalisches System
Fokalpunkt
Lineares Gleichungssystem
Erwartungswert
Variable
Flächeninhalt
Vorlesung/Konferenz
Explosion <Stochastik>
Leistung <Physik>
Einfach zusammenhängender Raum
Umwandlungsenthalpie
Matrizenrechnung
Matrizenring
Natürliche Zahl
Rechteck
Zahlenbereich
Gleichungssystem
Vektorraum
Eins
Diagonalform
Arithmetisches Mittel
Poisson-Klammer
Variable
Menge
Verbandstheorie
Spieltheorie
Meter
Vorlesung/Konferenz
Ordnung <Mathematik>
Schnitt <Graphentheorie>
Diagonale <Geometrie>
Matrizenrechnung
Subtraktion
Natürliche Zahl
Klasse <Mathematik>
Gruppenoperation
Rechteck
Gruppenkeim
Zahlenbereich
Gleichungssystem
Fortsetzung <Mathematik>
Übergang
Eins
Leck
Zahlensystem
Multiplikation
Reelle Zahl
Nichtunterscheidbarkeit
Meter
Gleichheitszeichen
Lineare Geometrie
Vorlesung/Konferenz
Einfach zusammenhängender Raum
Nichtlinearer Operator
Matrizenring
Mathematik
Kategorie <Mathematik>
sinc-Funktion
Reihe
Physikalisches System
Fokalpunkt
Teilbarkeit
Teilmenge
Diagonalform
Energiedichte
Ganze Zahl
Rechter Winkel
Garbentheorie
Ordnung <Mathematik>
Diagonale <Geometrie>
Nichtlinearer Operator
Multiplikation
Subtraktion
Matrizenring
Matching <Graphentheorie>
Finite-Elemente-Methode
Hausdorff-Dimension
Rechteck
Zahlenbereich
Vorlesung/Konferenz
Ordnung <Mathematik>
Division
Matrizenrechnung
Zahlensystem
Sterbeziffer
Matching <Graphentheorie>
Natürliche Zahl
Randomisierung
Vorlesung/Konferenz
Fünf
Element <Mathematik>
Drei
Multiplikation
Matching <Graphentheorie>
Besprechung/Interview
Zahlenbereich
Vorlesung/Konferenz
Skalarfeld
Zentrische Streckung
Matrizenrechnung
Natürliche Zahl
Hausdorff-Dimension
Gebäude <Mathematik>
Zahlenbereich
Drehung
Term
Teilbarkeit
Richtung
Multiplikation
Flächeninhalt
Reelle Zahl
Gruppe <Mathematik>
Vorlesung/Konferenz
Grundraum
Numerisches Modell
Matrizenrechnung
Zahlensystem
Multiplikation
Matrizenring
Mathematik
Spieltheorie
Vorlesung/Konferenz
Schnitt <Graphentheorie>
Geometrie
Innerer Punkt
Multiplikation
Negative Zahl
Matching <Graphentheorie>
Rechter Winkel
Klasse <Mathematik>
Physikalismus
Vorlesung/Konferenz
Euler-Winkel
Element <Mathematik>
Biprodukt
Resultante
Assoziativgesetz
Subtraktion
Matching <Graphentheorie>
Natürliche Zahl
Rechteck
Zahlenbereich
Element <Mathematik>
Eins
Weg <Topologie>
Multiplikation
Differenzkern
Flächeninhalt
Rechter Winkel
Vorlesung/Konferenz
Drei
Negative Zahl
Matching <Graphentheorie>
Minimum
Zahlenbereich
Vorlesung/Konferenz
Arithmetisches Mittel
Multiplikation
Flächeninhalt
Matching <Graphentheorie>
Vorlesung/Konferenz
Tropfen
Geometrie
Resultante
Matching <Graphentheorie>
Rechter Winkel
Mereologie
Vorlesung/Konferenz
Matrizenrechnung
Nichtlinearer Operator
Erweiterung
Prozess <Physik>
Matrizenring
Mathematik
Kategorie <Mathematik>
Natürliche Zahl
Zahlenbereich
Euler-Winkel
Division
Konzentrizität
Multiplikation
Reelle Zahl
Vorlesung/Konferenz
Resultante
Matrizenrechnung
Punkt
Ortsoperator
Sterbeziffer
Natürliche Zahl
Hausdorff-Dimension
t-Test
Annulator
Zahlenbereich
Element <Mathematik>
Term
Negative Zahl
Zahlensystem
Multiplikation
Unterring
Reelle Zahl
Leistung <Physik>
Umwandlungsenthalpie
Matrizenring
Mathematik
Matching <Graphentheorie>
Güte der Anpassung
Relativitätstheorie
Vektorraum
Energiedichte
Flächeninhalt
Sortierte Logik
Rechter Winkel
Dimension 3
Kantenfärbung
Ordnung <Mathematik>
Geometrie
Turnier <Mathematik>
Matrizenrechnung
Grothendieck-Topologie
Matrizenring
Mathematik
Exponent
Ortsoperator
Wasserdampftafel
Hausdorff-Dimension
Zahlenbereich
Wärmeübergang
Term
Überlagerung <Mathematik>
Topologie
Weg <Topologie>
Zahlensystem
Rechter Winkel
Freie Gruppe
Vorlesung/Konferenz
Garbentheorie
Ordnung <Mathematik>
Matrizenrechnung
Zahlensystem
Multiplikation
Wellenpaket
Kategorie <Mathematik>
Offene Menge
Vorlesung/Konferenz
Term
Einflussgröße
Resultante
Schwebung
Abgeschlossene Menge
Vorlesung/Konferenz
Wärmeübergang
Multiplikation
Erweiterung
Gruppe <Mathematik>
Beweistheorie
Klasse <Mathematik>
Vorlesung/Konferenz
Ordnung <Mathematik>
Fokalpunkt
Stichprobenfehler
Eins
Resultante
Matrizenrechnung
Punkt
Besprechung/Interview
Zahlenbereich
Gleichungssystem
Wärmeübergang
Mathematische Logik
Übergang
Einheit <Mathematik>
Arithmetische Folge
Vorlesung/Konferenz
LES
Umwandlungsenthalpie
Matrizenring
Mathematik
Kategorie <Mathematik>
Reihe
Schlussregel
Kombinator
Kommutator <Quantentheorie>
Teilbarkeit
Druckverlauf
Menge
Rechter Winkel
Garbentheorie
Ordnung <Mathematik>
Geometrie
Matrizenrechnung
Abstimmung <Frequenz>
Matrizenring
Grothendieck-Topologie
Siedepunkt
Besprechung/Interview
Zahlenbereich
Gleichungssystem
Element <Mathematik>
Topologie
Ausdruck <Logik>
Diagonalform
Negative Zahl
Zahlensystem
Einheit <Mathematik>
Menge
Spieltheorie
Residuum
Dreiecksfreier Graph
Nichtunterscheidbarkeit
Vorlesung/Konferenz
Garbentheorie
Konfigurationsraum