Math for Economists - Lecture 7

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Math for Economists - Lecture 7
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Multiplication Lecture/Conference
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OK we needed finish up section a 10 . 1 that we need to do Section 10 . 2 is 10 . 2 can put everything together that's the final culmination of every Section 10 . 2 By 10 . 3 years is a little addendum and it's actually it takes about 15 minutes to cover 10 . 3 10 . use a big deal OK facility Finished 10 . 1 4 weaving get their introduced this notion call they have rank of a matrix makes it sound delegates smells bad but it's a different version of the word rain it's more like a ranking system that's that's the word rank you wanna think of OK so changer thinking it would so far we've had it Major matrices been 5 in this class we see him a couple different forms 1st of all we use them as a tool to study systems of equations system integrators we could take the coefficient of aluminum matrix that as a tool to get the job done then we stopped and we studied matrices as a mathematical object which could add them we could multiply them we could find the inverse we could do about 2 different things with the majors Manipulate them low operations so for now what I wanna do is I wanna tell you that matrix is actually a function to use The functions being at the events equals X squared somebody like that believe will study that in detail but in the 2nd part of the cost to the calculus but right now we wanted This is the culmination we finally said OK we we studied these matrices and what's really going on behind the scenes as geezer function but we could talk about them being functions because they could before because I couldn't say whether what they eat with their domain in range were so I had to stop and go and talk about vector spaces because those are the domains in Rangers so let let's set started changes thinking a little bit of matrices choose to view them that way they it as functions between vector space there's they're not just functions their diet while they can be linear transformations that the word a linear transformation but we don't really
He could spend the whole course talking about linear transformations it's a big topic Just a simple Violet to say there Their functions for our purposes but they are linear that's the thing to think OK so it's our little example of this if he already little bit abstract so
A bite matrix a there there's no I'm telling you that this is a function right now it's just a major but for now we choose to look at it in the form word is a function of how is that well it is gonna be a function Oneida have Things that I put in the Function like Apple take Exit put in that The exit break that that is the input so what are the inputs of this function and a little nectarous because they come from the vector spaces also set all the up-to-the-minute when I get a few pieces of it up on the board OK so there's a vector and now what's gonna happen is through multiplication that this sector to another so let's look look at what happens if I do a movie Let's just multiplication and 2 matrices was formed as a matrix but we call vector is matrix multiplication set up that's nothing new But when I do this multiplication the first one I get that put the 1 2 1 on its side there I get mad at 0 for the 1st component and then I get 1 plus 2 plus 0 3 So But let's visualize what this guy started off where it's got 3 components of it very often are 3 so the domain of this It is 3 Canright like this a is a function from Part 3 and where plans for whatever I do the multiplication I always end up with only 2 components so it can land there are 3 it lands in order to do a little picture of this video So here's 5 years are 3 and I just took a vector in 3 a while the compiled what are the components of the little The Exorcist X Y Z case here's the x coordinate that's 1 and then the why courses to so just down and that's why plane got this little This little bit and Amarillo up 1 that takes me about 1 location and that's my neck so that 1 to 1 looks like North 3 and this function comes along and hits it and it takes it over announced an alliance here plans in order to And what was the result of 0 3 0 0 3 looks like this and that that was the result That's not very enlightening to do this because OK that's is 1 that How I get a feel for what the whole graphic Well It's really hard to do because we're going from three-dimensional space to two-dimensional space above a bunch of stuff happens along the way and so you don't really know who a unanimous all this example is Is here for illustrate how complicated scrapped sold in attracted do is just get a little glimpse into what's going on with high clearly do talk about just doing intuition behind When account dimensions as will do it Now Look at that I want you to notice a couple of things 1st of all a always says 0 0 Think about that if I if I put 0 0 0 here So that's the 0 victory in this space here if I put 0 0 0 here and multiply by 0 0 0 and I'm 1 Miami at 0 0 so I have a bunch of zeroes here the only thing I get over here 0 so that Gives us a landmark case we had a from or 3 So hard to you but at least we know 1 spot to get fixed the the three-dimensional version there's the origin here And haggle over here A 2nd That is 0 here but other sectors don't get 0 necessarily this 1 got sent from there They're so targeting and on the overall Graf releases 1 thing gives you a little landmark linear transformations 0 0 these matrices half bucket summers villagers series of examples show you the possibilities that But said that they now I'm going from a large 2 to order to so Just picture that clears the symbolism and if it if you cover over it with your mouse ready you could see this picture and there's the origin of wrote that with your mouse to see that picture in a years going from here so what are the possibility
The 1st committee across the So the 1st injured The discount mention this to hear that that represents the number of dimensions in the sectors of this is two-dimensional Is to do his well
But what were interested in it Will take this whole space and then the result over here or what it would have a was just 0 matrix than what would be the all the things that are grabbing
OK with 0 matrix just pretend that it has all zeros here That has all zeros here then Any vector I take is giving 0 back more building away from 0 so that 1 possibility is I'm taking this whole space which was two-dimensional and 1 possibility is the outcome is just a single point 0 the is takes the whole station just matches it down the 1 . 3 single those is OK so this is not very interesting because it The only outcome 0 but this is the 1st step in understanding what could happen with these maps this 1 here winner say this 1 has reigned 0 the 1st your 1st exposure that term right and the way to think of ranks 0 is image Of this graph has only 0 dimensional space a point In the next possibility is 1 up from 0 dimension that be one-dimensional next dimension up and what is 1 dimension in this context it's a line has gotta go through the origin because remember the graph always takes 0 0 so no matter what it's Kennedy 0 2 this 0 but it might Take basis to just a single 1 And this will call break 1 As you build up you you just get a higher and higher rank this is like a rank in the military or a ranking system The a hierarchy going on here
But if Q and 3rd brass possibility might as well as to the label these just Here's quarter of the 1st on the 2nd 1 and then the the last 1 would be where I take this guy and giving you the whole space that's have rain 0 1 and 2 and that's the largest I could ever have because it's a two-dimensional space that it's arriving So here I think also face and I arrived the doesn't mean it doesn't do other things too it I want you to think that this just isn't exactly the same way that's why draw these 5 1 way and I didn't do it We should drop like this you don't think that it's autumn and might rotated to be rotated the origin could still say 0 that might be years new transformations that rotates This is what we call rate to the idea here 1st exposure to the rank is it hits the dimension of the image of this function
And in this case if we if we had our was from Arturo to only 3 possibilities you could have Dimensions 0 Dimension wonder dimension to marry I can't had mentioned 3 of the space of arriving at doesn't even have 3 OK so that first one let's to a couple more these deceased To see the possibilities that ultimately will remain wanna do is learn how to calculate this rank
Add algebraically I give you a matrix I say What's the rank then you go calculated they would we have a was from 2 dimensions image of three-dimensional what are the possibilities so here I am taking my two-dimensional space are too and I'm going into three-dimensional space it's always and the origins of the origin of the the possibilities again we have 3 possibilities that I have this guy gets sent 3 options so we have we have the same possibilities as we have over there except that each 1 of these Hester landed in 3 states so in have ranks 0 example ranks 0 it's everything your images just a single point which is 0 Then we could have possibility where just like over a year They could the two-dimensional space in wine but now or line is in this three-dimensional space drug that show everything here gets knocked down the one-line 1 before I write up there wanted you ask yourself what will agree to be in this context if you can make that jump rate to over here is a ticket two-dimensional space and then my image is a two-dimensional space here start with a 2 dimensional space and but I still have the land in the 3 space
But if it's going to be rainy to these images going out in addition to its order that looked like a sucker plane within the 3 and again I put there that the origin as into the origins of really does take this Thing here and it just puts it in 3 space has a plane that's right to play games with this whole life sit there and said the dimensions of the different pictures Ms. tuition for break really
That it's the dimension of The image that you end up with a these trees the go backwards Let's do a is now going from 3 states and the truce what the possibilities here well Yeah I'm starting off 3 space and I'm going in and landing in 2 This is like the original example I gave you What are the possibilities elected only had 2 dimensions over here so there's only it at the point of the long-running levels that's again those 3 possibilities this guy can go over to just a single point that ranks 0 go over to a single line of those in yellow Hawaii illustrate its distinct from its a rank 1 and then we could have the whole space get mad over to the whole to mention too No 1 of the things I wanted to develop For use with intuition see this as 3 dimensions and Thailand in 2 dimensions of what has happened to the last dimension wears ago something happen here I've got I've got 3 dimensions here In the I'd map everything over to mention just ask yourself in your mind with what happened what happens only loser dimension where does it go Well the answer is 1 of those dimensions goes To the origin its smashed down your there's always wonder whatever the extra dimension is passing it center 0 that's the way it works is a big subject so you could take the whole class on China do little sketch of what's going on here Hello if have a picture in your mind when you're looking at that stuff and you 1 more let's do the possibilities were we have a three-dimensional going in the three-day event
I had a large area of our dream than what we're talking about here is three-dimensional is going to three-dimensional so I've got a book that get an extra possibility here for talks it out the ranks 0 rank 1 ring to it now ranks 3 because the potential for images three-dimensional that biggest rank you could add 3 9 0 1 2 and 3 the possibility they once again it always takes you urgent little rigidity so I could have won since the holds
Based in New York If you like hadn't been doing that I make yellow that anyway So ranks 0 and then I could have ranked 1 that would be alright I get a line there in 3 stages And they would have a ready to go watch Frank to that means a dimension of the image has be two-dimensional so that That's a plane again it takes origin to the origin of the grain to generate 3 will be just takes the hall 3 space the three-dimensional space into a three-dimensional space it might do such things along the way but result the three-dimensional finish If you want a quicker filled out a were talking about the whole thing and then The result is nonsense 0 0 It might make rotate things around her use of things it's not just sending it exactly 2 itself those the 4 possibilities for that 1 You can really do for romances can draw stop here get tuition but the same idea holds You know something from are saddened Are 3 you have all these different possibility so we want Avila do is from the Matrix detect what's going on here which 1 of these things were in which 1 of the possibilities of algebraically just from looking at the Matrix without having to grab anything I do anything like that so that comes down to who I'm looking at the Matrix and finding out the number of rows littered linearly independent that's what's that Is that the method it would be a rate that there will be an algebraic way to determine the rank of a major so what the rate is just sort of intuitively is its dimensions loved it image the Matrix a flat doesn't quite Tilley added to it yet but once we know that then the way we find that it is it's the number love linearly independent Bush rose and technically it's called But since we do everything with roads each with a stick to the road and just to do everything consistent with what we've been doing OK so you're scratching your head OK will great How we find the number of linearly independent rose what there is something that we don't know what will we do when they take the nature and ruin a reduced it to route to relax on the reduced it to row Echelon form than we can just look at it sailed the the number rose independence are the number Roser Still there and then we can say exactly what the rank so this rank idea that discovering what the rank of a matrix is basically does boils down to reducing it to roll Echelon form which is something wearing just adding this new concept it's a new application of the road reduction of tying the rain today This is a recipe for Reduced Daily to grow a sure way of faith and count the number of nonzero rose that's easier prey that That's an old trick that he knew already Reducing matrices so you could reduce said and anything can find them Breaking even if you don't know all this other stuff is going on still computed get theaters next week it will be a little tuition and you could do it algebraic pockets think that original nature church that we had let's can't find it shrank 1 0 negative 1 1 1 0 day school find it rank Well was There Reduce aid to rely on Form aware that I need a leading 1 and then it's
The leading 1 year I needed have only zeros in its call on both get that I'm gonna reduced this by doing Roach you This replaced by negative role 1 plus 2 pound 0 negative on the 1st change his 2nd row becomes 0 1 negative or positive mocked by this negative added 0 vote by that by negative editor you want that by negative added 1 and that is role reduced form got this Echelon formation and all I need add as many Debbie Echelon formation together with the leading ones have only zeros and its called you see here leading 1 only zeros in its column of the years a leading 1 only zeros the the rest is so this is an issue you guys have when you're new to this is When you use that had euro Echelon form so that's what I just said you can just make sure that the leading ones only have zeros Elsewhere in the column but also if you were to try to make these zeros a that's another way to stop I wanna get rid of this man those mineral and that 0 You're done this as far as you go OK so once battery row Echelon for count the number of nonzero rose users to means the rank of a mixture And the picture that you now we know the rank of a to that means it's taking three-dimensional went to dimension The that's the case for the rate to do But Is anybody remember what the determined that matrix was I did it in glasses at the reason I think you might remembers the scanner Venezia matrix to remember you might remember that we get 0 for that chips and determined So The determinant plays a role on the lake issue and spent a whole lotta time focusing on but the ideas of the determinant 0 That means you lost a dimension along the way There's another way to think of it determined 0 you lose a dimension but least 1 ship are excellence that's calculate the rank of a Well this a little bit Worse than last 1 because I have more numbers and no zeros to play with Mehmet Take that upper left-hand corner headed to start making some 0 and the 1st thing I'm As I meant to replace wrote to with negative for rolled 1 plus and Leonard replaced through 3 with Negative 7 Pro 1 last is doing a reduction in the top row doesn't change that this reproduce it and I know that these again become 0 those that was by design now what happened was to the 2nd row so I have a 2nd rower multiplying negative 4 times a negative to that makes negative 8 and added a negative 3 here than multiply by negative for Harriet negative 12 and added name 6 Than most to row 3 multiplying by negative 7 adding it natives 7 is a negative 14 had that a negative 6 Multiply this by negative 7 that's negative 21 at 9 idea 12 That's out your 1st step and then we can order shiny Rover Road that Echelon forms so what I needed this is my first one I need this to be a 1 do that Eminem multiply But it you choose steps here The multiply row to buy negative 1 one-third The seminal multiply rose 3 by negative wants it I want to 3 0 1 1 0 2 1 2 and then 0 Pursuit After Now we're almost there on that at that see visas senior common due to steps Shearman replaced growth with A negative growth to broke through and I'm also getting rid of the studio As attitude you with leading once this is gonna be my next leading wanted to make sure it doesn't have any zeros and its call the OK summer due negative replaced 1 with negative 2 wrote to Warned 0 1 2 0 0 0 At last from becomes all zeros And now the Top Perot becomes 1 0 multiply that by negative 2 and adding it again OK then what is I'm done that's produced row Echelon form it's got the Echelon structured each leading 1 only has zeros in its columns and any rose 0 is the bottom right The book itself by saying that got to 9 0 so the rank of Where would we
Yeah give you wonder practice yourself wanted to the purity of the is this by 3 This is a good 1 to practice for the editor is so that way is 3 days of practice reducing that
But Cook who OK warrant would do little theory now and this is Yes how have to frame there's a while Like shop Because it turned out to be a rose 0 when I when I went to my whole goal was to just get rid of this 1 here but it turned out I also that so there was no 1 to be had that's what cause it have a lower rank if we got a 1 year there add rate 3 so in this step we just discovered along the the way doesn't always happen it is possible that you 1 that it would be 3 This may be The special cases If you look at Pier that's a 3 by 4 matrix So And A tree by more than you can say that what that with the mapping a horse for our race had a figure that out who will in order to do the multiplication I'm gonna multiplied by something right but in order that to work when I do that multiplication that lineups I've got have 4 spots there and then when I do that multiplication what's gonna happen only do it 3 times during its 3 little slots for the answer to that Yeah Image space is being are 3 high so special cases when you have a square matrix like in the last example that 3 bags 3 So when a focus on that a little bit of a and matrix which means behind the scenes if you choose to look at it that way it's a function from March 3 or 3 Talk R and R But 3 is 1 of the examples phenomena right down some facts of useful about that's so let's suppose that the ranks of day equals and
That means it's it's the the highest rank possible but that's that's way of looking at that in this case here here's a 3 by 3 and we did not get a rank of 3 of the lower ranks So if you have a rank of ended in other words full rank that's actually returned
What is that we a look back at how we calculated the resent that means we reduced it and we didn't get any roses 0 we reduce this will only gotta rose 0 so that a little bit less of a rank as 1 count the roads this 1 doesn't count toward the right So if we got a full rate here then that means that the reduced to a Has no rows of 0 to reduce it all again we don't get any rows of zeros not think about that thing I've got my reduced nature it doesn't have any rows of zeros was determined is there the terminal 0 turned that will look at this example of what she was mentioning if we got an extra 1 here than that the determinant 1 they hear the determines 0 So when you're when you have a full rank when it went Nunavut with no rows of zeros than that means that determined It's an implication you Excel and is making a connection between there's a rank They it's the full rank then determinant is 0 And and then that means I 8 if I take a vector I can't solve Equation a Equals 0 The problem but 1 more thing them so as to determine its 9 0 is determined tell you tell you whether there's an inversion So that was a member success tho half what they really need to cut his Herceg because I would extend this damn thing will it The interest exists and I wanna solve this equation since the universe exists and that means that he is equal to a universe of 0 0 so the only possibility for this equation The Eagles 0 so on the user all equivalent thinks rank It is full rank determinants not 0 universe exists and then this equation here only has 1 solution namely 0 2 We have Tell you Larry worker with a described this as nonce
So the word nonsingular for a matrix implies all of these things OK sometimes you need 1 sometimes you just need be Anderson
Sometimes you just need the determinant doesn't it 0 Sometimes you won't find the rank depends on which backed it's useful for you but this word implies all of these OK well if there's nonsingular than unity singular and singular is the opposite of all right all those Ms. if OK so if the rank of and it's not equal then is less than It's the other parts of the rank of a cut It is less than half of that this case here where the This is a 3 by 3 the rank was to wasn't 3 Well and that means that when I reduce it If it's lesson they had get rose 0 soldiers the reduced has a role of zeros at least 1 road 0 And if it has at least 1 row 0 doesn't mean the determinant 0 It's the opposite If it didn't have any rows of zeros terms not there it does ever 0 0 then the So The rank is slightly less determined is the determined that 0 so then that means what that means injures does exist
It's the opposite of the existing right that of the determinant determines 40 whether or not the interest is high hollow again and this last piece here a equals 0 OK here V was 0 vectors the only solution and now this is good and being led by a lead equal 0 stress this There is a not equal to 0 so that The word that we apply to that is singular
Really this is serving as partially review to remind you the connection between the inverse of the tournament on but also this last back here will use the next thing we do this fact that if it's got rank less than and then there's this scene so that 8 people 0 what that means is it's going on the dimension of its full rank it doesn't lose any dimensions and if it's less than full rank them something at to recent 0 either liner planar some things had loosened tensions along the way Much of the Boca So vocabulary for instance on exam I might give you a matrix and I to say is its singular simple questions what would you do tested you could do anything you want you could that unifying inverse the matrix the new Party founded
Then the inverse exists and say nonsingular Brittany volleys to protest the basic mistake that determine the determinant tells you whether it's singular 0 its singular that's not 0 that's nonsingular her Q. alright pack so Now it and put it all together section said 10 point to hang in value
Brawl The problem and of having almost everything that we in this class and you got solving systems of equations you got multiplying matrices all kinds of stuff OK board are getting quite get this year with the dual background stuff from the 1st thing is A minute value And orthogonal matrix says faces Added to another Special matrix gotta tell you what it is not born knowing what that means the amount before I tell you what which it it would've orthogonal matrix the word orthogonal means right-angle parade so does it mean for a matrix be orthogonal military vectors orthogonal means it means they're perpendicular but what about matrix four-member within a matrix each little column is elected So what this is going to mean is that the sectors in the column or and means something election 2 I'll tell you OK so that columns about an orthogonal matrix chewed Robert the first one is that the the columns er orthogonal that's where the word comes from each column is orthogonal to all others will have an example on a 2nd group and the others feature is that each column has length 1 0 0 2 features a Louisville example of a orthogonally if if About it They got some nice features but they also usually has some of the features of you this is our Diagonal matrix All-Star Dubai by Poqet have about 1 use 1 Avery Fisher 1 order you negative 1 of which is 1
That's an orthogonal those check and see that as these properties but call let's called this year's columns
What's called IU B Phillips check the you India perpendicular Trans Posey that's equal to 1 over route to 104 2 times negative 1 or 2 1 to yet when I do this job multiplication I get that negative half plus a half 0 so they are thought the 1st conditional
The setting condition is that each column has like 1 village checked the length of yield better we do that take the square root of the squares of the components of a square 1 that 1 Overridge you square-rigger half plus a half in the same with me
That's going to be here have negative attitudes of matter is the squared anyway negative that have squared plus none have square suicide Group 1 2 that seek one-half half of what I have So it satisfies both those conditions there's a North body major she wanted a little bit
But it would too Dude More to be done on matrix of I'm showing it to the show some of features By take QNA take its trailers Bozena multiply the vote each further you get the identity
There's a really special matrix because What that says is that the Trans poses actually offenders that usually not case chance poses switching the rows and columns turns out With this kind of a matrix egregious which the rows and columns with Sanders doesn't a matter which order you do the same for this state here we can verify and we will honor a 2nd but this state here that implies another way of saying that it is that you Transpose is able to use that really special matrix can make it makes finding nurse really easier I just write it down you to do any calculation so orthogonal matrices have this property that the transfer is equal We can check that with ours there are so few Transposed Q effect to the trans posed by the negative 1 over routine goes on the lower left-hand corner on order to negative 1 over route to worn over route to 1 or 2 Which I'll let you verify that gives you again and you can see you just doing this when you end up with a half that 1 the year native hapless the bus apart About 0 negative or positive 0 the New positive at such verifying it this is a key factor this high Now new these transformations is matrices functions a road that would try to get a handle on what's going on inside With all this stuff has gone on behind the scenes inside the nature of this really will revealing here it's like a hearing of his blob of numbers but if you look at it the right way you can see this the geometry that's going on inside and out Trying to get the training see similar geometry focus ready for example of what will develop this theory of agony values and eigenvectors just using this example men next time I'll do some more examples Just this example that you would take the attitude with this that if you can't reproduce this example and you have no business doing never seen before they became even reproduced when you have seen before Why should you be trying when you haven't seen before so there's a lot to do with this 1 so I recommend being able to take a clean piece of paper and reproduces example before you even among others OK will look to write this down as a function a series of sergeant from too I immediately to see that as I know this is 2 by 2 major 202 a 202 But she was fired at with this graph does So 1 way to see it is let's let's take a look at the simplest basis so that we can that we've looked at the 1 where you have even 1 and he She where this guy You know This is 1 of this 1 so anyone is equal to the detector 1 0 and too Is that 0 1 This concept and basis so easily won think it is these are the building blocks for this whole vector space and if I use those as building blocks the creates is this grid So I can get anywhere in that sector space using these 2 guys And this escape on this little grid word Every square has side equal 1 school revered Stewart does that will aid due to the Well A bus to a B 1 that's equal to 2 1 1 2 1 0 That's equal to 5 2 1 a He won That's to 1 And that looks but calculate the length of this to put the length of a new war that's equal to Square root of 2 square plus ponds clearly which is So right away Mr. Siegel of what happened this 1 sector It started off so nicely or perpendicular to the Y axis and I hit it with this nature and it rotates it moves it from being here 2 years they think of it as a rotation But that it didn't just do that because it had just rotator would keep the white race but if it rotated and multiply it by 5 1 from handling 1 5 so that transformation just looking at that 1 vector Yet Rotates it a little bit and it stretches it out all of us take a look at the other had a year to the next June 1 1 2 Times 0 . that can be 1 to So here's this is a B 1 and then we get Want to say here that aid to It'll make those yellow I So what is it done to that nice perpendicular agreed that we at it's given us this grid so this is a lie near misses a making connect here A year that or so that's not what happened to our nice perpendicular grid under this matter if it takes you imagine it just skews everything so even like a parabola here if I had if I had this Little curve rain here he gets sent over this here stretches that The reason I show you that is as this is complicated case out of the question is are there any where it doesn't mess it up like that's what's also calculates the kind of sorcery so what happened here it is right now What happened is not only did I wrote to if also multi those vectors got longer they multiplied by Route 5 We see Gave road tapes and stretches So there is the question you see what's happening is there is a set sees it is operating on and we don't know about it
We don't know there is a set of perpendicular rectangles that is operating on and if we can't find that we can see with a relief to see this is not helpful because it takes over used to an excuse in such a way that is not going give us a good picture what's going on so the question is are there and this goes back the batteries this is the agony value problems are there any
That vectors In order shoe That a doesn't see it that rotation this really messed that that's which skews it too early Molale floor connected to get stretched out over knocked an all-out effort to be rotated this is what I want I wanna find the answer to to this question and it'll be yes always in in our cases the lawyers is nothing but the question is that you find that this OK so let
That's right now what mean this picture a pictorial pierced ears are 2 And a does its magic so what I'm saying is are there any vectors there it doesn't rotate so if it doesn't rotated the there's going to be this vendor and I get it doesn't rotated so I get the same angle but it might be a little longer than Make it stretched out but it didn't rotates words it's the same vector just have a different lengths that's in the same direction Another Think of this as this sector sitting on a wine and this sector stays on that line OK now bye confined to these victories than what that victory is is a set of axioms through this nature We try
Need to start attacking us But the but the uh My goal gave him a matrix a majors in this case it's always a square matrix
We do it that this problem is an applied and square matrices so give any day And they can find denied equal to 0 and Lander So that what has happened algebraically when I like this and taking I multiplied by a surrogate AT nasty so I'm really trying to solve this equation a V equals land that's that's the agony value problems are very vectors where I
He said And I get it back again but it might just be modified so you know Think of this as being a 3 2 hours The plan a real number OK so let's try to solve this we have a little bit of experience with this if I want assault Something like this maybe that means a TV Minus landed a equals 0 This said 0 and then on a little extra step here is what painting big bucks Those are the same and justification for this step is I was really equally high the identity it acts like 1 of the major reason they did that we've seen this little trick before the reason it is as the factors things out factors via don't do that this implies that I had a minor slammed a half time lead If Africa
Here it is that with all the news Ozal faxed that I was talking about with break here is The only place to use any means that those facts available to or at this stage here And we know that he is 9 0 if you look back in men in the previous earlier in the class was talking about singular matrices 1 of the features lawyers
They have a matrix times vector equals 0 And that vectors not 0 that means this is singular Look back at what singular men you'll see that that was equivalent to saying that the that that it had a vector a nonzero vector matrix that victory equals 0 1 of the conditions and the other conditions was that the term is reminding you that Petition and this rate here is will ruin news To find would call the Haganah value Ursula me that The I am values of a of the Islamists I'm going to solve the determinant of a minor slammed height go 0 for land that's my 1st goal eventually move back and find out what is will talk about that the 2nd so this is enough radiolysis go get the ACGIH values of a matrix by solving this equation
Each so a that matrix Then I wanna solve the tournament modes of a miners I equals 0 and I'll tell you what that landed here this is not a now
So it was a couple things going on here for so we a was rate that them the numbers involved Thursday a about this case here Lambda we know eyes the identity matrix of this is A minus Lander times the identity matrix And then the vertical lines here that means I'm gonna take the determinant of that once a figure out what the Matrix actually is minimalist said that equals 0 A notice the only variable involved volunteers land lobby sulfur Rolando careless to a couple steps years this is the same as 2 1 1 2 minus land 0 0 lambda may not Durant Ali step The 1st time through Mariehamn all out Eventually which he realizes all you're doing is attracting land from the back when I Jack saying it to minus slander On the upper left-hand corner than a 1 minus 0 assesses the 1 their male 1 minus 0 is 1 2 minus slammed show When you good it this which he realizes that to go from here you go from here directly to this because all you're doing is you just subtracting land from each share that with GM Doing but when I'm I'm writing it out for the 1st time this to see all steps in all their glory OK so we can take this determine what's the determinant here it's a B D 9 BC to minus landed square minus 1 square feet below 0 that's that's multiply that are you consolidate every 1 of them The sources said 2 minus slammed the squared equals 1 so to minus Landers equal plus minus 1 so I have to possibilities to Linus lenders want and to minus landing negative here Landers equal 1 in here is eagerly 3 of those er what you call it a day and value of it Before we move on us go back and see what we trying to remind ourselves more in the goal was to find these vectors so that be Equals landed it at all we did was we found the land so far we still have to These vectors of 2 things a unifying unified were recalled the agony values which we just did that the complete solution got the idea value now that we have value so now we can begin checked this equation and say OK will land that he called want the little 1 here The little 1 here is For 1 for everything and then I can solve Let's find out would be its rebound ahead of jagged vectors Each value of slander Find that soda a equals limited So looks Just duel over the That implies that a B minus landed 0 Which means that a minus And I equals 0 And we wanna now solve this week now we know land 2 cases landed equals 1 nonetheless solve this with plenty of 1 a minus 1 I and equal
Lawyers that amount to A minus 1 eye that means subtracting 1 from What threaded a is 2 1 1 2 minus land higher slander a limit 0 0 But there's buddy This is my goal to solve for the horrified at what the spectators so that no 1 is a need to know What about here Lamnidae equals sources put here Now subtract those I get 1 1 1 1 1 2 equals 0 0 So it Now boils down to a little system of equations new defined as too little variables you want me to be her Reproduced were my last step is there unless that is 1 1 1 1 1 1 1 1 8 1 8 2 equals 0 0 It on agenda solve this system that was put in an augmented nature The same is and I wanna try to solve this system and give you the values we wanted to What we do here Well Let's do to replace wrote To with negative wrote 1 plus Get back and by the way if you see that the 2nd row just canceled when you're doing this in practice you just crossed back out of undoing it your notes but sometimes you get pretty sick a rating matrices over against Warner developed a few short OK So if you about this this year says V 1 plus a V 2 0 0 which says that equally negative So what we do with this takes us back to An old situated near the course we had headhunter determined systems What essentially what happened as we lost the equation So what What happens is there's no infinitely many solution was right Downer solution I'm interested in these equal want to 1 of I found out about V-1 and V-2 where I found out well I found out that I ve choose Eagle a minus V 1 Ray Solis if that is what I have a good be 1 and then this is minus V 1 That's equal to be 1 1 tie in you can consider this year agree that if you like gloomy just do that step away from the sky the the fastest way to do it now But to be consistent with how I did these problems with 4 Ever so I had he to go negative wonder what we could do is say said we want see use parameters that accusing the negative TV the movie is equal to be 1 of 2 that equal to see negative which is team talent 1 that's the same vector I got out again that would probably the I can do at this stage there's lots of little tricks to get the eigenvectors 2 ways to do it Another way do this is say said people 1 He said global warming you did it 1 negative 1 me but the reason I do this is illustrators actually infinitely many of them it's all why we just wanna get 1 of the there's there's not a good news even understood complicated the 1st time there's really complicated but we get to repeat his every fairy
Agon value we get a nagging back to pursue this process over again and that'll help us get pretty food with with focus so well I just did as I did all that I found the eigenvectors for land equal of 1 For their use 1 for Kalandek equals 1 we got the idea that is equal to 0 1 negative
Was just check and see if that is in fact the overall goal was to get a Eagles landed Hazlitt see that actually happen so a V that's equal to 2 1 1 2 times 1 negative 1 that we do their multiplication I get to minus 1 has 1 and then won Minus 2 is negative and what isn't that 1 times 1 negative want a His eagle to which is planned doing a fact that is the agony value and the eigenvectors all that original equation if that's go now and do the other Agon value to their peoples
3 air Her coat This recap Start off with planned equals 1 And then I when they found the vector synonymous start off with land hoped to read and I want some vector scholar W. that a double the dealt with That's what I want On the assault a mine 3 W equals 0 that's basically bring the 3 WR factor out the W this When I do this 6 2 1 1 2 minus 3 3 0 0 time W 1 . 2 0 0 0 This overheated gives me a negative 1 1 1 negative Right now we're at the the same stage where were we here related this stage a here here will be this opener augmented matrix than I used the argument make trips to find out what the V-1 and V-2 are prominent in the same year this Hi how
On behalf of the Group F O Right From here so I get 1 native 1 1 1 8 1 0 0 at my augmented matrix and all reduced that to good here
So what I'll do to get rid the bottom row as manager workplace wrote to with a row 1 got good nite negative 1 1 0 0 0 and then this here this that this part of the process is very repetitive the 1st time through doesn't really know what you see in multiple times but stick to pick up at this says that W 1 negative W W 2 0 0 which then the W-2 is equal the W 1 So now the lectern interested in W That's W 1 W too well at another name for W planning is called W 1 major Ikeda structure You can check that if ideal A WTO that's 2 1 1 2 w is 1 1 look what happens when I do this multiplication 3 of the top and then 3 On bottom that 3 times 1 1 which is planned That we have found the eigenvectors associated to that idea about
On the seat That's OK let's cannery Capel we've got here on our goal was to find a special vectors and the special values that go with those that Ursula's recap of started off with a and then we got to hike in values and then through this agony value we got the sector we which was 1 negative ones and we got
The vector for this guy over here which was 1 of the other way It was 1 more These are the idea that So now or ever do is Oregon Amaker matrix says where we do it could do this information will take employers matrix us where we store for me if noticed that day
BMW or perpendicular BMW perpendicular picture will not show you feed us out so here's the picture This is the real way to look at this What this matrix is doing to the party is the way it's viewing it it says I've got these special make special vectors that I that I'm operating on the years the first one 1 1 that's troubling right there's 1 1 and then
The other 1 was 1 negative 1 and soap The way that that matrix is Viewing the space is it use the space like this with his That's the way it used the space and now when you take a new map but over here What is it takes this sector and multiplies it by 3 takes this vector images keeps it multiplied by 1 so all it does When you look at it this way is not so confusing it doesn't totally skew the space if I look at it like this but of other great but look at it like that then what happens this and with this over here this 1 stays at length 1 with this guy So now my new grid is like
The 3 of us takes this stretches out that's a much simpler way of looking at it you can know that from looking at the nature to have to study this theory in and discover it separately and then you go back and say Oh yeah that's what was really going on is that this matrix is based this way then when it does the mapping think it just keeps it in angry
And now we can see what's going on with you when you do it that focus The picture behind-the-scenes ballots 1 last step we had to learn what to do with all this information There would For almost a year here and now it seems like a huge but is a huge undertaking is that all the different things were doing rail systems of equations we have player transformation theory of stuff here determinants So noticed
That we end W or body He printed those W. that Siegel who won the negative 1 times what and that's what they want 0 BMW and I did do the picture to also see that from here The OK let's get that planks of wood that it was the length of wealth the length of POV this screwed 1 Squared plus negative 1 square which is And the length of W 1 square plus 1 square which is also proved too slowly and and if I take a victory and I want this sector has linked through to if I wanted that water like 1 which it idea If I have the number 5 and I wanna make it at length 1 photo ID divide by 5 Right By have 7 and I wanna make it at length 1 divided by 7 of other vector and I wanted to have 1 1 divide by its like so Veto over that is 1 over 2 Times warned maybe it 1 Which is 1 of the route to negative 1 of the and the length of that as length 1 so that is divided vector by its own length I again what's called a unit vector But anyway If you divide that provides like electrically and same with this guy says link RSA years over that if he were then it take those vectors of minute matrix harsh nature if there's been a or father and that that's how we started off this section Utah
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My eigenvectors America would have not been adjusted the original eigenvectors undertake member North part matrix has the columns like 1 Solomon put in here you will may call it The We dealt with those of my 2 columns that be OK would be over its line that 1 over required to minus 1 or 2 And then W. over its length that 1 of group to 1 there's Hugh not killers are thought Which means that if you to transfer resembles a few vendors that's the significance of being thought OK now here's the magic of where thing we're doing here Were going to take our original matrix a going it's like that taking it and putting unit Distillers scholars extra stuff but put it in a distillery melts away all the extra information a giddy just essence of nature would expose the essence of our original matrix 2 1 1 2 By multiplied Hugh Trans pose a Q So what is that can't do that Hugh transpose is 1 over route to 1 over Group 2 Make it 1 2 To a 2 1 1 2 and then Hugh is displayed here want to make it would appear so that's 3 matrices multiply That that's before we just jump right into that but see can make our lives a little easier this is the same as 1 over Route 2 times 1 a negative 1 1 1 2 1 1 2 and then times 1 0 2 1 4 and made it seem to agree with that all I did was just factor at the 1 because it's so ugly isn't it and we all agree that It is an hourly wanted doing multiplication involves that I can avoid it OK so now I will pull this 1 alluded to this 1 a route to together and give me a path Menomonee keep this nature and manages this multiplication by getting it some his done now getting it some of them all
These 2 guys became the Half the Sky stays the same and I'm gonna do this multiplication here when I do that I want the upper left corner Then I get A negative 1 Yet
2 minus 1 1 Year ahead should do Take where so this is 3 1 2 plus 1 and related to phenomena do these 2 multiply those too I get a 2 The 0 Leggett 6 There is 0 9 8 6 and self When I finish this software again remote But bloodbath These are back in value Broker So what I need to do right now and they need to go through interest Recapture it really did a lot of things and I was explaining along the way What we've essentially done here poses tell you that on a red all that will be done so I start off with it and I multiplied it on either side of it
Of itself with the special matrix victories matrix I use the idea fire multiplied on each side by that and we're like yet I got a diagonal matrix and that diagonal matrix has the agony values of there so this is the essence of aid behaves like this that when you view it in this context if you look at it like this would do multiple is suspected by 3 that's the 3 of them matrix animal but effective war that's what we
Discovered 90 did this cap and now you have a recipe for all 1 a if selected said before An orderly masses you really need take this example and reproduced They a all the other examples are similar to that so that's why this is a good place to start But sir Start off with a major giant all the rest is by you In the goal here is 1st fine The agony and values Said had he do that if you look back in the notes you could find the world was recaptured here's what we do is we solve the determinant of minus 0 can you solve that land That your 1st at finding the values involves solving medically then the next step is defined that I'd been adapted just use vectors go with Each of those land it you found in Part 1 so for each phenomena highlighted that each value of lambda found in Part 1 for each of those notice we did it twice as we had lamb equals Wonderland equals 3 us all a minus the that I will be equal 0 Just highlight here before you know it helps to these 2 things look similar Bashir or solving for me And here were solid for land Here we have vertical lines and take determine here we don't is determined to pay attention to the symbology there to help you remember what if
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3 days just to collect everything we had an arcade and equals 1 and that we got we 1 1 area had landing 3 We had W pulled those a Results Of 1 into that was was it take those vectors and make them have links 1 Don't make that planks 1 alive by their current lengths So it will be provided diet Don't in that case we got 1 out rejuvenated negative 1 2 them we took W and divided by its old white and like it has won over to the next step was we took those once we calculated we took among would put him in a matrix we created this Q See related to I took these new factors that I created would like wanna put of animators and that's the magic matrix that's the 1 that changes a Internet diagonal matrix pesos let's put bank want that Jews From step for 2 Q Hugh If you want it can write amounts lead over length of B W length of Delta and that turned out to be this matrix here up above said So we put those length 1 vectors and we this new major and that matrix is what caused this would call that if we're almost there But On a To them Show we do this modification actually you don't have to to do the multiplication because when you look at what the result see from this example we can see what the result going to be the result is going to be a matrix with this I values and that's why I use this capital land that indicates the ACGIH values unity in the wooded would begin here This turned out to be 1 0 0 3 Where those are the agony and I wanna Also Yet so Q. How when highlight this of Q is equal to this matrix here where was More negative 1 2 negative 1 route to 1 over 2 . 2 this is the 1 from seed and this is the 1 from W. so OK so what was the ACGIH value that 1 would be that should be land that equals 1 and this 1 was landed 3 and so If you have 1st then landed becomes 1st and if you have W 2nd W bag value shows a 2nd over there so reason I'm saying that is because you might you might choose when you make UQU might switch these 2 I can't control that Major said 2 vectors you might have found this 1 1st the new put down in 1st that can cause this to be with the they match this guy goes with The 1st vector goes with the 1st died in huge CAQ Alaska is dead This is this is what's called the diagonalize An essentially it's just that the essence of a behaves like this matrix when you look at that rotated axis Pockets of duties sit there and go through each step appearance 1st finding a values practice that in practice the eigenvectors and want to get those 2 things sort mechanical put it the nature and also want point out I said this before sitting it notice here when I did this multiplication I got this but You can just right because if you know in advance that the thing you get here is the matrix with the eigenvalues than to put a amid year after so don't think you have to do that multiplication if I ask you for the diagonalize you just give me the 1 with the idea so you'll have We will do another full example of this terrible much more quickly next time because I want to explain all the steps of his due major soldiers power through it and you'll see that it's on Wednesday ordered that battle From become better from your sample The quest so you get a chance to see him there do recommend between now Wednesday tried as practices wanted if you could sit there with a clean piece of paper reproducible thing you'd be really ready for Wednesday but if we get some questions from happening Too much debt you have any questions tag at that Ivory questions on Wednesday amid some questions for discussions Took that