Math for Economists  Lecture 3
Formal Metadata
Title 
Math for Economists  Lecture 3

Title of Series  
Part Number 
3

Number of Parts 
15

Author 

License 
CC Attribution  ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
Identifiers 

Publisher 

Release Date 
2013

Language 
English

Content Metadata
Subject Area 
00:00
Matching (graph theory)
Observational study
Multiplication sign
Algebraic structure
Variable (mathematics)
Rectangle
Numerical analysis
Dimensional analysis
Connected space
Mathematics
Mathematics
Term (mathematics)
Different (Kate Ryan album)
Operator (mathematics)
Matrix (mathematics)
Vertex (graph theory)
Configuration space
Nichtlineares Gleichungssystem
Physical system
Social class
03:32
Matching (graph theory)
Positional notation
Lecture/Conference
Equaliser (mathematics)
Order (biology)
Rectangle
Dimensional analysis
05:55
Area
System of linear equations
Focus (optics)
Sequel
Insertion loss
Variable (mathematics)
Numerical analysis
Power (physics)
Explosion
Expected value
Lecture/Conference
Matrix (mathematics)
Nichtlineares Gleichungssystem
Physical system
08:58
Metre
Diagonal
Multiplication sign
PoissonKlammer
1 (number)
Lattice (order)
Variable (mathematics)
Rectangle
Numerical analysis
Connected space
Latent heat
Arithmetic mean
Film editing
Vector space
Lecture/Conference
Natural number
Order (biology)
Matrix (mathematics)
Square number
Cuboid
Nichtlineares Gleichungssystem
Game theory
Diagonal matrix
Set theory
18:15
Metre
Group action
Divisor
Sequel
Diagonal
Multiplication sign
Real number
1 (number)
Sheaf (mathematics)
Food energy
Subset
Mathematics
Positional notation
Lecture/Conference
Natural number
Different (Kate Ryan album)
Operator (mathematics)
Matrix (mathematics)
Square number
Energy level
Integer
Nichtlineares Gleichungssystem
Series (mathematics)
Diagonal matrix
Algebra
Social class
Physical system
Multiplication
Focus (optics)
Equals sign
Linear algebra
Rectangle
Leak
Numerical analysis
Connected space
Category of being
Order (biology)
Right angle
Identical particles
Sinc function
26:18
Addition
Multiplication
Matching (graph theory)
Division (mathematics)
Rectangle
Numerical analysis
Dimensional analysis
Finite element method
Differenz <Mathematik>
Lecture/Conference
Operator (mathematics)
Order (biology)
Matrix (mathematics)
28:44
Randomization
Matching (graph theory)
Positional notation
Lecture/Conference
Natural number
Matrix (mathematics)
3 (number)
Mortality rate
5 (number)
Element (mathematics)
31:17
Multiplication
Matching (graph theory)
Lecture/Conference
Meeting/Interview
Scalar field
Numerical analysis
32:42
Rotation
Area
Building
Multiplication
Scaling (geometry)
Divisor
Direction (geometry)
Real number
Model theory
Dimensional analysis
Numerical analysis
Lecture/Conference
Term (mathematics)
Natural number
Universe (mathematics)
Matrix (mathematics)
Musical ensemble
35:49
Geometry
Multiplication
Mathematics
Film editing
Positional notation
Lecture/Conference
Interior (topology)
Multiplication sign
Matrix (mathematics)
Game theory
39:50
Multiplication
Matching (graph theory)
Lecture/Conference
Euler angles
Multiplication sign
Negative number
Physicalism
Right angle
Element (mathematics)
Social class
Product (business)
44:27
Area
Trail
Multiplication
Matching (graph theory)
Multiplication sign
Equaliser (mathematics)
1 (number)
3 (number)
Rectangle
Numerical analysis
Element (mathematics)
Lecture/Conference
Natural number
Different (Kate Ryan album)
Right angle
Associative property
Resultant
51:02
Greatest element
Matching (graph theory)
Lecture/Conference
Multiplication sign
Negative number
Numerical analysis
52:35
Geometry
Area
Arithmetic mean
Multiplication
Matching (graph theory)
Lecture/Conference
Multiplication sign
Drop (liquid)
55:14
Matching (graph theory)
Lecture/Conference
Multiplication sign
Right angle
Mereology
Resultant
58:29
Addition
Multiplication
Process (computing)
Euler angles
Concentric
Real number
Multiplication sign
Division (mathematics)
Numerical analysis
Category of being
Mathematics
Lecture/Conference
Natural number
Operator (mathematics)
Matrix (mathematics)
Extension (kinesiology)
1:01:12
Point (geometry)
Geometry
Threedimensional space
Multiplication sign
Real number
Student's ttest
Graph coloring
Food energy
Dimensional analysis
Element (mathematics)
Power (physics)
Latent heat
Mathematics
Goodness of fit
Positional notation
Manysorted logic
Term (mathematics)
Natural number
Matrix (mathematics)
Square number
Negative number
Genetic programming
Position operator
Area
Multiplication
Theory of relativity
Matching (graph theory)
Mortality rate
Numerical analysis
Vector space
Ring (mathematics)
Order (biology)
Right angle
Resultant
Annihilator (ring theory)
1:16:34
Covering space
Trail
Free group
Tournament (medieval)
Exponentiation
Sheaf (mathematics)
Water vapor
Heat transfer
Grothendieck topology
Dimensional analysis
Numerical analysis
Mathematics
Positional notation
Lecture/Conference
Term (mathematics)
Network topology
Order (biology)
Matrix (mathematics)
Genetic programming
Right angle
Position operator
1:23:09
Category of being
Multiplication
Positional notation
Lecture/Conference
Term (mathematics)
Matrix (mathematics)
Genetic programming
Open set
Measurement
Wave packet
1:26:58
Beat (acoustics)
Lecture/Conference
Closed set
Heat transfer
Resultant
1:29:02
Proof theory
Standard error
Focus (optics)
Multiplication
Lecture/Conference
Order (biology)
1 (number)
Musical ensemble
Extension (kinesiology)
Social class
1:32:16
Geometry
Point (geometry)
Divisor
Multiplication sign
Sheaf (mathematics)
Heat transfer
Rule of inference
Latent heat
Mathematics
Lecture/Conference
Meeting/Interview
Square number
Matrix (mathematics)
Energy level
Genetic programming
Series (mathematics)
Nichtlineares Gleichungssystem
Set theory
Commutator
Line (geometry)
Numerical analysis
Category of being
Logic
Large eddy simulation
Combinatory logic
Order (biology)
Right angle
Pressure
Arithmetic progression
Resultant
MaÃŸ <Mathematik>
1:40:29
Multiplication sign
Sheaf (mathematics)
Grothendieck topology
Element (mathematics)
Positional notation
Lecture/Conference
Meeting/Interview
Wellformed formula
Matrix (mathematics)
Square number
Negative number
Nichtlineares Gleichungssystem
Diagonal matrix
Set theory
Computability
Numerical analysis
Residual (numerical analysis)
Voting
Network topology
Configuration space
Cycle (graph theory)
Game theory
Identical particles
Boiling point
MaÃŸ <Mathematik>
00:06
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
03:34
Of equal or not call I will miss a major cities are
03:46
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
04:10
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 G8 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
05:56
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 midterm too So this is a way to incorporate this new concept of matrices with
06:11
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
09:22
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
09:52
Information so here and get information so we could write it this way when last time we talked about column that role vector so I can organize my variables in a matrix that saved 3 by 1 major stoic that a column vector we'll call matrix A special name and then I can say that if those that's the column vector X Y Z than that must be equal to 2 1 the organizers information In a matrix Next minister cut Talking about special matrices I have to introduce you to the World of matrices at the show you there's some special ones and then of course there special audit have names attached answer them when I refer to those named Jurgen Oneida Avenue image pop in your mind of those specific matrices so the 1st set of special major cities is where the nature of the game The name implies that were not dealing with just the general a rectangle but it's now specific a square matrix of I wanna say with the order of all square matrices I will work in order to get a square Oneida have the rows be the same as a cop that's what that's the way it Think of it The the same number of those columns of a would be in and by then Regions And by then think they refuse them by M because it's you know in general it's an em by hand matrix but that's usually what we call square OK so what's happened just a couple simple examples like pursued by 2 Dubai to that's a square meters wilderness and a lot of time with 2 by 2 major cities is if you think about it 2 by 2 matrix is the 1st case the 1st that square matrix where you can actually get some interesting stuff happening sold but it's it's not as complicated as 3 batteries who spend a lot of time learning the world of matrices I could have a 3 by 3 News or warning introduced for the 1st time all 1 by 1 matrix I was elected go back to the The simplest fundamental Example if I can and sometimes when you do that it becomes ridiculous let's take a look at that but what if I have 1 by 1 major that means if it's 1 by 1 minutes 1 Rowan 1 column letters 1 number This is a 1 by 1 but you're already familiar with 1 by 1 major season that you dealt with your whole life these ages numbers that's that we so it's Ridiculous that brackets around and no 1 had back its is just so 1 by 1 major cities literally right brackets around that there is no numbers with 1 by 1 they receive but what that is is is an opportunity for you to learn learn a little bit of abstract so we're used to the numbers are ready and now this new concept in history we're doing is making a connection between something you learned a long time ago in a really familiar with with the news that the new thing being nature sees this as a 1 by 1 matrix OK So the villagers basic square matrices now within square matrices we have some other special matrices first one is diagonal major so think about the language there will be a diagonal world why would they be called a diagonal matrix if you can even if you can answer that question will help you remember it if you try to imagine what the answers before I tell you that you Maginot OK you may get it wrong I tell you that it'll stick That very raw or are about to stranded download the information little trick we can do to help that out valuable OK Dad make of 1st of all it's going be a square is within the square matrices This is not a diagonal matrix this is not a bag of tricks this facility will be Dagmar matrix whatsoever but says some examples here so I give you that that's a diagonal matrix so now ask yourself why we use the word Dad Well here's the bag know that I'm talking about it's not zeros visits the the 1 going left to right from top to bottom and we could have a larger 1 Is 1 0 0 0 0 2 2 0 0 0 3 that's also a diagonal matrix and there's the diagonal analysts at once A goal 5 at the bag no majors is a ridiculous 1 it's not really interesting but as long as were looking at look at the most trivial examples is Raul bag no major season where they have in common Ms. here What they all have in common is this I can say it in English and we have to write it down with mathematically so what's clear here is that I've got stuff that's nonzero on the diagonal and zeros everywhere else in the have language to describe its called this The main diagonal because there is other dead and then here you might even think maybe that bag no matter then so there are other things that could potentially be called diagonals but we're talking about only this 1 here which is called the meeting Auction A stroke by 1 describe this in English their square With only non 0 centuries being on the Clinton years this new vocabulary the main diagonal At age No 1 emphasize the language here And that is very precise no words their wasted a peso There's something specific cases with only nonzero entries being on the main diagonal so sometimes would you might think especially when you only look at those 2 examples of those 3 examples you might think that you can have 0 on that you can it's just that the only place you have nonzero things is on the back so that 1 more example I have there's a 4 by 4 I can actually have to use the bag nor I can actually have a 0 the road doesn't that doesn't make it a nonbank matrix but still it adheres to the definition That The only known Nonzero things around This is still there will a scanner come up later on because they guilty leaking diagonal matrices durability making them sometimes it is 0 there so that they can cover that you don't worry about having a 0 meaning they
18:35
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
19:17
The special notation for use by her identity by and this is good and bad and matrix village swear And here here is an example of well with slits add some stuff here so can buy and matrix and then it's diagonal but then it's a special kind of bag not where you have only 1 The main not so as some examples of this is the general 1 Hisar minutes and then banned matrix go down to the basic level about diet of 1 of us Now 1 by 1 they tricks 1 by 1 as example break it but let's get other way So it's a 1 by 1 matrix and it's got out 1 in the main diagonal reentry moving so there's only That is the main diagonal so there is Let's leak identity by the 1 by 1 This place ask yourself abuses 1 special In the number world the special What's special about it What OK We help you if I take to multiplied by 188 to great effect take 5 and multiplied by 5 back so his Iranian a number that does that where You multiply take the number the mystery number and multiply but something I always get back the same thing again the 1 is that unique properties so 1 has the property that 1 time saying he called in the in the real numbers are aware every 1 of these integers has the special property cases unifying a the that property exists in all these and by matrices This is not a connection between the original number world the use of familiar with And this new world of square matrices so said to the thing that's gonna be a 2 2 matrix and had gotta have gotta be diagonal so I can have anything 0 outside the main diagonal and it's got out only 1 in the main deck not there's I said to him then I 3 These are the ones remain abusing so well get those other and down it's a diagonal matrix arising 0 upset the main diagonal and I only have 1 of the main now I can show you that this property is true yet because we don't how do matrix multiplication that's coming in a few minutes but since I've got this series make a connection between your familiar setting the special case of 1 where 1 times anything equals the anything it's also true in a matrix so that by the end of a sequel to a tenant On the other side to order matters but multiplied on the right side by 1 right here It's also true that a price that is equal You don't know a matrix multiplication is this is just 1 Hatick insisted here and yet more questions what did Additional service Yet while personal or just dealing with square matrices in their systems of equations that little extra column represented the other side equal sign No factor that's where saying it be here is that what we're doing right now removing ourselves from the systems of equally just steadying matrices but the man behind me to the subject matter would be considered linear algebra which I was still linear algebra we're even before but now or just looking at pure matrices and what we can do with them the algebra Eventually will come back and will make a connection with Hey Those systems of equations But But The focus of now would a Doolittle abstract show If you pay attention to the 1st lecture and mentioned that the goal of this class 1 again mathematical maturity What we do is we abstracted that that's a skill that you learn in college and a lot of different subject matter so what would I just stand here I've got The whole world of major seeds for that nature she wanna think of rectangular Oregon square anything but then within this world I got square matrices and and Here I could and that's a bigger world within this the square matrices exist then the within the square matrices I've got back and then within diagonal matrices I've got identity and a lotta times 0 We do in mathematics as we say OK there's a general thing and then there's little subset that's interesting and then new within the subset there's another subset that's interesting ammonium with Amabel subset theirs but the thing is is that if I could say something is true about this it's also true about all these automatically that's another way of thinking of this nested set so that we just just identified some special matrices we we start off appears rectangular them we said OK square the that the identity you should be familiar with what all of those The next thing here is So that's all Section 8 . 1 2 with matrix Operation what operations while use that word will if you remember
26:25
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
26:54
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
28:46
But OK so if they do have that both a
29:00
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 G8 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 G8 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
31:21
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
31:51
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
32:42
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
33:28
We were talking about a real numbers here and a is a matrix sake and stand by N & M by inhouse 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
35:52
She P 5th should if But scalar multiplication now we begin Daddy actually multiply to readers will do this by example
36:53
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 farright 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
39:59
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
40:14
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
41:10
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
44:36
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
45:28
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
51:03
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
51:41
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
52:36
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
52:49
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
54:01
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
55:25
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
58:18
Anybody readied a riot in the streets Yeah I think that we don't think we can further setting comets the 3rd
58:32
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
1:00:56
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
1:01:40
Can you describe In general Q will 1st always have something in describes the orders because we had never specific thing happen with the orders in or even do multiplication Hugo petitioned a is equal to see a them by men and beat equals and by all are but that's so this is just notation here were aware really saying we get this great little thing I ve i.e. before I is another way of saying another way of saying The Another way of saying this is the number of Columns of war too The number Of relatives and beef that their starting point you cannot Multiplan each season less that happens could you that little multiplication won't lying at the latter an email that symbol of the drawdown Estrada so here's here's A man hears beauty and earlier see sissy here So this is this particular element in your seat J. it's in and I grow and the G8 thought that weird is so how I get that back well I'm how we do it right we got couple examples here now if I had 1 2 I took 1st and 2nd column so you here I need through l But some of them will this year to guide us X will die a little square So there's me I have not specified in the city's some role in their like And then over here and that The column of many years The column see on doing this in general are not his undoing or matrices interior simultaneously 3 powerful Book is and will remain do Madam up put entries here so hard I get CIJ J is equal to all its extra acts plus that time was . plus Square Square close right that out in English it's easy see the power of the brain drain likes visual Things if I tell you that The method you could follow that with your mind and you get really quickly now That's just words and let's compared to see how your brain deals with describing it as a CEI today Formed as it for me Spread out were made to amid a multiply the elements of the ice with the elements of the column and that that's will do Marek that down formed by multiplying elements of life brought 0 but is not as any I've throws the 8th wrote a and then a duet multiply that would be J. calling beat And then I had the results won't be good to be good for you to sit there and stare at those 2 things and see if you can read Those words you happens is if you just scan your eyes serious like a novel The register anything you go AST ideas for find elements of the I throw it over Wednesday Colombia other results got it now has about the way the brain or so if you really want understand math when you're reading it you really have to dissect it carefully you have to go through each little word and really understand each word otherwise you can't just read the 2nd novel but this year describes that and that's pretty clear you consume opening and then you also want some examples of Newfoundland You can download it This practice on some fancier once got some trick would always throws people and yet there so that's what I get book soso Utah Jazz yourself A now will recall that we got a row vector it's a major The specific under matrix of color Road vector threedimensional and that he could be a column vector And now announced to Just like we did before it's a little harder A deal this because don't have clear rows and columns they only have 1 entry so let's let's see if we can follow our method On this will not be good practice our star after they'd be whose go slowly so 8 times dealers take care the lower which we expect a busy 1 by 3 And is 83 by 1 so I'm expecting a 1 by 1 and was the 1 I 1 is just a number and so I'm expecting a real number as a result of that has to be surprised should be the 1 multiply those 2 that I get a real number that the way the matrix multiplication works Gaylordsville it Lowell ideal It will a slowdown earlier we
1:08:55
I know the revealed 1 by 1 so there's only 1 entry rate C 1 1 and take Single column over here and a single there are no match em up let's take this on said just like we did before 2 1 negative 1 match at those 2 numbers will be 2 times through already out again to see if there is 1 kind of 1 negative 1 intends to all that up at 6 plus 1 7 minus 2 5 3 treated you know which call for a just a splitsecond ring column nature so 1 by 1 but that's not how we write Spot It's OK the other way about believe day Well In this case is a 83 but 1 in a easily won by 3 it's finds a multiply them because of their numbers match but the result is revised 3 got giant matrix that I'm gonna get got 9 entries will do that but there are others will face serious A conciliator our So 90 Debbie on the Left side day have A on the right I know in advance it's going be a 3 by 3 get 9 entries are at the pay attention what the method would have made the C 1 1 minute do the 1st column see here the columns are very interesting is it's a single number and Acadian amid turned on its side Over here match that there's only 1 thing matter so it's actually pretty easy do it 3 times to nothing that rates 3 times to plus nothing else it 6 3 times 1 history 3 times negative with his native 3 done with that column and move on to the 1 a onetime students to 1 times 1 1 110 negative when his native 1 than area can look back on MoveOn 3rd column 2 times 2 4 2 times 1 to 2 terms Cry what uh for But June Sir There's with me or against me on irony Annie questions for Q OK Just a quick fingers to remind you this the fact that the multiplication isn't community not another little example the practice time if it were a little more familiar with that here's a British No question how would I get from it but said that I know how I did it So there's a there's be sort of I'd done to get from I'll tell you don't have to answer to happen Where at It in English The Rose became the call rate the rows of He came The columns that's going be a move that we do with matrices in a minute I'll tell you about the 2nd but there's a danger 1st exposure to What I want to hear 1st multiple prickly faces a major show you how I would think about that how quickly I do this in my mind may take the 1 2 and I just match it with the 1 2 SIC that's 1 plus 4 5 and take the 1 tune enacted appears in my mind the negative 1 plus 2 1 and now and then area making without 1st column energy the 2nd column and turn it in my mind a match that negative 1 up with the 1 and the 1 with the chill so that's negative 1 plus 2 is also won and then not done with the 2nd column I'll take that match it up with the name native won its match with the native won that 1 and then 1 warned that Maybe they have RBA The thing is about these 2 by 2 major is no matter what you get to buy 2 major so it's possible that you might think that aII these people the BA in this particular case because the rectangular case it's obvious onetime you get 1 by 1 the other way you get a 3 by 3 that's clear that they're not equal here but see that they're still not equally even when we have the dimensions OK a good time practice VAT can you can you put a on the right side And see that Solana do what I do this Take this column and a match Their prey here at BA so already out put your 1st I want Negative 1 case I'm taking composed good practice for your brain Trend not do that see if you can still do do the multiplication look so undertaken at 1 negative 24 inside at 1 plus 1 is too are already there so gradually to continuing other people we will continue to practice faith to minus 1 1 done with that column move on this column And it 2 minus 1 1 and then to tend to his 4 plus 1 I really recommend you practice enough for you can't do that in your mind Rather than after radio before you get get comparable with the vast added So that what this is meant to illustrate is that they be added however As was mentioned by 1 of you and also by myself their stuff going on inside the Matrix that is preserved so what'll happen is the geometry and said this 1 will be the same as geometry and said that is that the preserved only multiply and it doesn't matter what we we do it however were not in a position
1:16:37
Needed to support the book's 1st see now how Cook
1:17:05
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 Englishlanguage 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 doubleclick 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 doubleclick 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 doubleclicking 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
1:23:25
Don't know A year Open to happens it happens we defined something in then we stirred thinkin about it
1:23:57
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 doubleclick 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
1:27:47
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
1:29:18
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 midterm 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
1:32:30
If you believe that was believable OK right Daddy red approved an through such a comedian who gets Ohio rare approved 1st of all you always take a deep breath and you say OK I'm gonna approved now I can't believe I'm about ready What I have to do they have to start with this and I have to have a logical series of progression and the last thing I write Is that OK salutes looks go better start with their fears were approved a twoyear CD trainer because I have to say this whole something to get going and I know this factor know this ceremony use that To get this new result Italy doing now Well This says something about 3 things and the only thing I know something about 2 things so this is the 1st step is why somebody has to show this to you because you don't write proves you won't even know where to start so what's the starting point for starting point since I know something about 2 matrices and managers manipulate this and make it look like to make my 1st that entering a B C and we agree that those are the same from their multiplied 3 matrices among small but those 1st that gives me a matrix and but I see Even now I'm in a setting where I have 1 matrix times another major transposed I can use that there now this is equal to see and most times may a be transfer and when I do this that need to justify justification is around 8 . 4 0 was really gone I feel like to memorize fears in their numbers what's really going on is amusing the fact that ADD transfer equals nutrients was a trade is using that back on that specific said OK can't I do the same thing here that you don't have a fact says He shares those egos beaches Jose Kansas so now I have 1 more step here this is equal to see chance of being change chance again because a transposed because he will have transposed major justified may step and look I started with that I had a logical series of was where justified each step and I arrived at the level last with that's it Start QED maybe river that from geometry of soccer cellular right that proved you just take a clean piece of paper in Satanta you go from the beginning to end looking at between and review column memories I that's fine column memorization I think will happen is below the of a high pressure between memorization edges But it's a hybrid between agent actually learning with that if you're really memorizing and were unit to go OK I start with left open parentheses a capital a capital B capital city right close parentheses this little up above and left France is going to war on wines at 2 hourlong lines Member has it that way but that's ridiculous so chaired you a combination of memorizing understanding or This editions commuter red or yet because all my near that I booked for both there if you don't like to memorize it gave you write down what the theme of the rule is that you However almost done yet Set just this year But the but her OK because section for Section 8 . 4 Is about that was sexually especially those as 1 special majors who wanna talk about club called him not judged musket says that on lowered I IDC poll has had a single item is I or colic impotent so we'll see what I get so These giddy the definition if you have a matrix Gayfer salacity square when I say my and I could say the word square by You're supposed to think of Square in mind when irate symbolism and then it has a special property that squared equals had by an order even multiplied by itself because if I do it and by then this is this 8 times a day so this is then bye And then by hand so I'd had been ordered square had that those 2 equal to could only square A square matrix focuses got this property now But let just look back at the 1 by 1 he is after all a square where the numbers in this case that would be like saying a squared equals A values little a because died about numbers here Not really matrices so we could solve this equation uses a squared minus a equals 0 8 times in minus 1 equals 0 so a equals 0 And a close 1 that makes sense readily only numbers for you she square mute the number back again That's the 1 by 1 case Well and that India that Cisco Chubais for instance just don't really need to do We do 3 back will start to So far 2 by 2 were still have this Case look a is 0 0 0 0 The case where physical today How So that's directly here as well The 1 by 1 matrix was 0 everywhere this that you like to matrix of 0 everywhere this by the way is called the lowly matrix for 0 matrix that side of It's got that property how
1:40:33
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
1:41:16
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 threetoed 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