Right. >> Thoughts. >> Exactly. >> So I guess to see with the other choice leading ones, each leading one has to be to the right of the ones being one of the last row Vs to the left and some others as well. >> That person's last time somebody said, maybe it was me and my per day that the hormone problem for inaccessible ivory or is anybody verify that they are now? >> And then this morning I noticed that there might be a typo in one of the actual problem is that there's anybody else discovered that. Pulls it off, looking at the right time to check it out. >> Any other votes? >> So last night a perfectly valid question to wonder why there's an answer. >> Here's an answer. >> Before they answer though. Resulting layers that are inherited wall, it works for any size. >> But you know, given how much, how intensive computing determinants in three by three kick some that a fair amount of arithmetic. >> And then if you try to do a four by four, you have to be 33 by three or 43 by threes potentially. >> So really it's, it's practical for two dimension, but it's not very practical beyond that, unless there's some theoretical structure can take advantage of that for numbers really just makes sense that you might use, let's talk show at work. >> So a x is equal to a, if you'd like to remember from now on, all major seasons where x and b are for legend overflow, given a and B, you're supposed to buy some fears formula. >> I don't know. Like little bit helpful role and pretty much I don't know why people call formulas pools that they're just born. >> Okay? >> So it's the ratio of two determinants. That denominator is the determinant of a itself. Okay? And then the numerators are different for X1 and X2. >> That seems good, the determinant over so for x, y, and we take a weird place, its first column for x1, we replace called them a one with b. >> So in other words, instead of writing a here and then the other columns left alone from n, So a row one, column two. >> And then for the same thing we started with a, but since we're in component number two, we replace column number two. >> High heat >> B and then column number one comes from a. It's essentially just automates the whole row elimination process. >> This is what is obviously trying to divide by 0. >> You'll see that the important reason momentarily, but we can say there is no unique solution. >> So remember all the way back to the first day, if something is true, then you have a unique solution. >> Otherwise you have no solution or infinitely many. >> So this is something, if this, there's a unique solution. Otherwise, I'm sorry, this is nonzero, there's a unique solution. Otherwise you got to just go back to whoever off. >> So for example, let's say we have a system of two plus 13 positve stuff here. We don't care about names a variable. >> It's just that there are two variables and the equations are linear. >> In those variables. >> There's no use square times v cosine. >> So we can let x1, y2, x2 back scattered mutation. So a is the coefficient matrix. And the new order that I'd find that Maria, you've already so minus 131 by J. And B is the stuff on the side. And now it's plug and chug type. So I guess since we need to renew it twice, we can just do that separately. So determinant >> Just have one minus three x one turn and has an inverse exactly. And to start with a and replace the second column by column alone, alone, one minus 21 word or two. Alright, the formula that central mythological time, just a couple days left in linear algebra before we're ready to move on. >> Differential. Great. >> So one of the things he did worse. So if we are trying to solve a x B, then we can just write a little bit about the good. >> I want you to go. No, no, no, no, no, no, we don't. >> Alright. >> Last time >> It's sorted, but we have to be more precise. >> So for numbers or scalars, right? We have the reciprocal, or like a multiplicative inverse, a to the negative one power. >> We don't usually write it that way. >> Usually we write one over, obviously, a has to be non-zero. >> And what we mean is reciprocal of a day. >> Alright? >> So if we had a, B and steel or world, it is not 0 and multiply both sides by its reciprocal. And then you have associativity of multiplication. So this is just one. >> And so this all histology and reciprocal of eight and I realized probably were not to challenge. >> Okay? But the point is that everything up to that very last bit hold true for matrices, everything but the last bit writing another fraction, and this bit I for one major planning, becomes your identity. >> So remember I is a diagonal matrix, and I use that word before, but it just means that you only non-zeros are those on diagonal running from top left. And those values all I, I times a is always a, a kind eyes. Alright, so now we have a definition a. >> And there is some Cs, which is also n by n, So that C times a is the identity. It looks like the reciprocal of a, but of course a matrix, real matrix, where all we have no communitivity. >> So we also have to say a times b, given that same thing that we say C inverse of a, and we use that same minus one epsilon. And you also say that a Edinburgh parable means hasn't had a few quick facts without any justification. >> First of all, if a inverse exists, it's normally want. So mathematicians have a very lawyerly street. >> Great. >> There's a difference between Zamzee is the inverse and inverse. Technically, you should start by saying there is an inverse and then prove better can be only one thing I just stimulus. >> That's because aspiring to be professional additions, but it makes the inverse. >> And the other is to say that yes, instead of left to right separately. >> But it turns out they always good. Yep. So C a is equal to i if and only if a times C. >> So as it works out, we don't have to worry. >> Those are kind of just technicalities that way. But the next thing is big thing. >> So month long rebuttal models where non-square matrices, we don't even have an idea. >> That's very interesting, heavy idea, but it may not be. >> The 0 matrix is one example. Because 0 times anything is 0. >> So hopkins 0 times anything, but okay, that's kind of to be expected. We know 0 doesn't have a multiplicative inverse is one non-zero matrix. >> But if we were to try to find an inverse, let's just say any old dangerous comes along and we make it into a inverse. >> Well, we take the inner product or dot product of 00010 and a CD with 000 c times a dot product of this c. >> C. >> We have little problems. >> Begin at z equal to 10. >> If this is practically empty and identity matrix, no matter how you choose a, B, C, and D. >> So the word for these matrices, sometimes people say non-invertible or singular means no invoices? Well, because it has to be here. >> And I don't remember if I put here when I'm thinking about the 0 vector. >> So these are my only choices. >> They all two-by-two matrices look like. >> Yet, no matter what members like, whenever a, B, and C, And I can't my one in his head if he hasn't won. >> So I can't make that product matrix if it were possible, if this matrix has an inverse as possible. So with that being said, genocide a non-zero singular matrices, it turns out that super important and I've domain on one of the main thing that happens when you go from one dimension to morton. However, if a is invertible, then we can repeat the same trivial looking calculation that I did before, right. >> So we started with a X equals b. >> We multiply both sides by a inverse to kick left and right straight both sides on the left, I use matrix multiplication is associative, but it is associative. >> So a inverse times a, by definition, is an identity. >> So well, that's tomorrow. >> So I times X equals a inverse, then I is an identity. It's the multiplicative identity. >> So I axis, and that is a unique solution. >> We said that if a is invertible, the inverses, you need only one choice for a inverse is unique. >> Vein multiplication is, is unique, right? >> And so there's very obviously, if a is invertible, there's a unique solution. >> Less obviously, it flows in the other direction on the back that minutes. >> But unique solution for a also has to be a slight. So your questions, it looks easy. >> Urinary tract. The thing is that it's easy to write. >> The kind of begs the question, OK, solve this linear system. >> How do you find a inverse? >> There is an algorithm based on row in the nation. So let me tell you I feel about this algorithm. >> So in predicate to Danielson and Daniel somebody and said that's a yogi go to Japan and then he sees a poster for another martial arts school. And the school is like dynamically chopping a tree like Hawaii classic correctly, a tree trunk and splitting. >> Asked can you do that? >> It says something very typical. >> Accidents don't know. >> Ever been attacked by three. You can use this algorithm to find a inverse outside of math classes, I've never had to do it. I don't recommend it. You can ask a computer to do it, that you really need one, but you also have to ask, why do you need the inverse? So if our goal with solving linear systems, the problem with this algorithm is that it's equal to solving and linear system. >> By the way, I think you've never had a teacher views karate kid who hear references to consecutive classes. >> I think that would be my last cranky preference. So same as solving linear systems. So if my goal is to solve a linear system, and you're telling me I've got this great method for you. >> All you have to do is all your systems. You might say thank you, I'm gonna keep looking. >> Doesn't make sense, right? So the inverse is actually namely a theoretical tool. Occasionally you might want one if it's larger than two-by-two. Ok. Thank you. >> That's my advice to you. >> I will not ask you to do a three by three Universe. Omni guys try to force me to do it. >> I will, we'll kick and scream like a father doesn't Birth. >> There is one thing that's rather today's, Well, there's two things that are relatively simple to do. >> So one of those is that shed her singularity fairly easily, or at least without trying to find a way to show that a matrix is invertible is to actually produce a difference. >> But that's hard to say. >> There's a shortcut goes right back to the term in the determinant of a matrix is non-zero, it is invertible. >> And sometimes that's enough to that. >> There are certainly situations where if you just want to know whether the matrix is singular and there's a unique solution even without being able to produce. So this is another effect that would be a lot my agenda. >> So that kind of mace that kind of, a little bit more later on. >> Cramer's rule, right. >> In Cramer's rule by the terminal a, the denominator. And now we know that if we're trying to divide by 0, that means the singular inversely. >> So what should we expect? >> A little mayhem. >> The other thing that we can do pretty easily with inverses is finding inverse of a two-by-two. >> So you just have to make a formula for the inverse one over a minus b a, which you'll notice is German. >> Okay? And then, okay, one thing, remember the x and the determinant of this diagonal, diagonal terms slot, the anti-diagonal terms stay in place yet. >> And so just to put numbers on it, hopefully not breakdown singular matrix. So one over three to four. And as long as the compiler a little bit, I get negative one negative back into the majors. Everybody nice thing about finding an inverse of any science really is that while it might be a pain to get the inverse, it's easy to check that you have time. >> You can just check all multiply these two matrices together. >> So four negative one dot product here, eight minus three is five. >> But I did like advice. >> I'll give one there for negative y. And first row, second column, negative four plus four is 0. Brave second row, first column, negative six plus six, and then second row, first column is three. >> Second row, remember zone, this native Muslims bomb. >> You can always check your work more easily than do, but I'm going to say still less emission and Cramer's role. >> It was good to have both of those things. >> Okay. >> Yeah. >> Yeah. Yeah. >> So the key to determinant happiness is to look for zeros. >> Take advantage of zeros early. >> Often though, easiest thing to do is to expand along into the second column or the second row. >> And the first term doesn't matter, the last term doesn't matter. >> The middle term is c times the determinant of this one. >> Wildlife. >> Let's just do it anyway. Why didn't you should get another thing? So if I'm at 1000x there, and let's say B is then there is an INV. >> I could say x is time. >> Anytime that ball, again, MATLAB is not always using exact numbers, so you may only get 14 or 15 digits of precision, so that's 0 to prevent it. But this is still the same thing applies in MATLAB as it does to you. >> This is not the best way to solve a linear system. >> And there's, there's a speed reason. >> There's another reason as well, which is called stability. >> Best use a backslash, look like the same result. >> You probably, even if I go to more digits, in this case, it won't be any difference directory, nobody cares everything. >> He's faster than you can type. >> But if it was a big matrix, it would be a big, And I guess maybe now a logic behind this notation. >> So remember the first thing that I had to write scribble out was b slash a fraction. Well, that's not going to work because order matters. So the a inverse has to be on the left and a B has to be done that way. >> So these flash, it makes no sense. >> So if I write a first, they'd be next. >> If I write this the other way, a is still in the denominator. That's why nations to do it that way. >> And it's frustrated a lot of people give he's ever since because not everybody is good at the difference between slash backslash. >> But just remember that that's healthy way tells you which one is supposed to be different. >> That's mathematically the same as inverse of a times b. >> Question about Jack map is a contraction of matrix. >> I'm putting out map. >> So it really is supposed to be all about. >> So every, every course dedicated linear ends up with this long theorem that they build throughout the whole semester rather than just adding to it. It's weird because nobody seems to real with the name of it is I find the fundamental theorem of linear algebra >> But why is it not theoretical? >> But I think you should see these things at least once. >> Or do you ever use an advanced material? >> Might have a chance. >> If I want to talk about suppose a basis vector, each of the following statements with all of you will know why linear system has a unique solution. >> Linear system has no free variable. >> Or to use terminology jerk jargon, you might, you might have used the same way, but where he got it, we're not gonna do it by just putting it in there. >> The row reduced form of a, the augmented matrix a itself is a burden. Columns of a are bases, n-dimensional space. >> Remember, basis is the smallest set of vectors that you can use to describe. >> Is that a space like that, like on the dimension of the null space, 0. >> Remember, we say that happens when the null space is just the vector. >> 01 >> Determinant of a is not 0. >> And the last one I'm putting here in anticipation of x yet, but just the leopard, a non 0 as in the afternoon. >> So again, he did some things, seemed like the same things over and over again with different words. >> And here's where all that comes there now in the opposite gates, right? >> So if the determinant is 0, a is that singular form of a something else and has a free variable, all that. And then might has your resolutions that amazing machines you need to look further to know which. >> All right, okay, let's pop it out. >> We're gonna have one particular use for eigenvalues, which is solutions of differential equations future by eigenvalues or one of the very biggest things about matrices. >> So by one count, there are at least ten Nobel Prizes in physics that are related to the eigenvalues in some way in certain point of view, all of quantum mechanics is the study by Google's original algorithm was based on finding eigenvalue and a huge matrix, which was which web pages point to the other web pages just as a big picture. >> All right, so let's just jump in and invite the other lambda and a non-zero vector. Such that a, b, then we say lambda is an eigenvalue a with associated eigenvector ideas. Just a German prefix, dependent special, particular glue that hold this word. >> So course 11 interpretation is that for an eigenvalue and eigenvector AB parallel. >> So our go-to for finding eigenvalues is how is that? >> Where'd that come from? >> Well, a times d lambda, I mean, certainly subtract lambda v from both sides and equal to I times the matrix. >> Distributive Law tells me that I knew A minus lambda. >> Now, if lambda V bar all of hiking here, then this equation is true and he is not a 0 vector, is not visible. >> That was buried in my definition has to be, because this is trivially true if it's true for every McGuffin, every lab. >> So what's the point of parity? >> Has to be non-zero. >> So I had a way, I have this linear system, the homogeneous linear system, and say it has a solution other than 0 as a solution other than, of course, 0 vector has a solution to that system so as to be theorem. >> Since this fails, they all must fit the determinant of this, of this system majors. But AX equals B, that has multiple solutions, multiple solutions. >> So the determinant of that matrix must be singular matrix singular. >> Therefore, as I said, sometimes all you need to know is whether that thing is singular. >> And without knowing what the university is. >> And what makes that so convenient? Walter, one of the things is that instead of having to solve for lambda at the same time, right? >> This equation has to be Atlanta. >> And you see they hold quite generous. >> It's not exactly linear, but you can get rid of them just fine. >> So we use this condition to bind. >> For example, suppose you'd like to neighbors. >> I have a very busy a minus lambda. I, I mean, well, I remember has one live on the diagonal, minus one, minus one or one virus. So that ensure that 2w squared minus one so fans out i square minus two lambda plus one minus eigenvalues. That is by Euro quota that determinant 0 ruth quadratic land roles squared minus four times negative three. One plus or minus 16 is wars do yes, we just backed by looking at 31. >> Most of the time, it doesn't matter what order you well, so mathematically, it never matters what order you get. >> Most of the time there's just no, no convention, but what order they should go. >> And by now they've got the eigenvalue. >> We're still not done by the eigenvector. >> So this must be true if V1 is an eigenvector by definition. >> So this matrix a minus one times R three. >> So this matrix is negative two y, what I've been harping on, we have non-zero matrix times a vector equals 0. >> It doesn't mean a vector has to equal 0 because this is a singular matrix. That was the whole point of finding the eigenvalues to make that thing. >> You row elimination >> Two times the second row. >> Well, it's a singular matrix, so it's, it's our arm. Yeah, can't be the identity. That means some row is going to be lacking a leading one. >> And the stuff over here zeros and we're always going to get is 01. >> Not a problem with that. >> That's a good. That means liquid is going to save me. But this power is that he is free as a negative x or negative two. So all eigenvectors this fall, which I can also write as S times 1.5, right? >> So this is an eigenvector paul S set S equals S is equal to 0, then v1 is 00. >> So normally the problem might stay on an eigenvector. >> So one answer would be 1.51. >> Another answer, the fraction theta as equal to 12, or for some reason SVA employees 2014, why are there all right. >> Well, so the next step, I guess to make this truly are aligned, then you have to write, so this would be my mean, one millisecond. >> So that's always going to happen. So there's some ambiguity and the answer is, we don't want to break into Baghdad. Let me finish this thought that evolved for the first time that I have another one which is negative 12. >> Why it is lambda to be equal to 0. So do I always looks the same way. >> Another diagonal, so a just subtracting them too long. >> And I add so one minus one here, four here mines, which is the augmented matrix Q_1. >> Okay? >> Negative two times the first row added to the second row. >> I'm just going to turn it to 0. What are you going to write it out has to be 0 because we engineer this David, that matrix isn't anybody already something wrong before. >> So minus two to 0 here. >> So the second component is going to be a free variable, as the first component is Q times the first component equals negative s. >> So I guess I get negative one over omega S S times one. So this would be, as you know, avoid shortcuts energy due to their wonderful. >> Let me just introduce this one now there. >> But good one year when I did a minus lambda I this major, right? >> I did the row reductions that, but of course the preferred candidate, I just look at the first row, I switch the order and negate one of them, I will get 12. >> And that's because I didn't get any rise. >> And I know I hear switch the order and negate one of them, negative 12, which is just a multiple. >> Awesome. Now, so what do I do? >> It all you gotta do. >> Form this matrix. Take the first two edges in the first row, swap them. >> That's always been because, right, we can kind of see it, right? If I put negative 12 as my solution while I'm adding a negative two plus two zeros ocean solution. So that's always going to use that idea to be using in similar. >> Any questions about that? Let's start with how I got it. >> Well, I mean, if you're faced with solving this linear system before X1 and x2, now you reduce it to this divide through by negative one. >> So there's your y1, x2 is x1 is equal to 1.5
MATH 305 2020/02/20
From Tobin Driscoll February 20, 2020
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