classify-the-quadratic-forms-in-exercises-9-18-then-make-a-change-of-variable-mathbfxp-mathbfy-tha-2

Question

Classify the quadratic forms in Exercises $9-18 .$ Then make a change of variable, $\mathbf{x}=P \mathbf{y},$ that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct $P$ using the methods of Section $7.1 .$
$$
2 x_{1}^{2}+6 x_{1} x_{2}-6 x_{2}^{2}
$$

Runpeng Li · Accepted Answer

problem. 10 um, were given a quadratic for that is too sorry to ex Juan X one squared plus six x juan next to minus six x two squared. Okay, so the first thing we need to d&#x27;oh his problem is to write out the matrix a so based on this, or get a form. And because the coefficient of X one squared is too. So that means the first entry of diving off the diagonal is too. And since the coefficient off X two squared iss negative six. So that means the second entry for the diagonal is next six and the product of X y next to has the cooperation. Six. That means, um, the some off the first column, uh, some of first column, a second first colored woman said, Second row and the 1st 1st row, second column. We&#x27;ll have six. So that means we can take the 1st 1st row second column to be three three, and we can take the second roll First column to be Greek so that these two entries have the some six. So this is our make 68 And the next thing is to find find the, uh, I got better. Uh, here we consider the determinant off a miners Lum di, and this will be determined and tough. Two minutes. Number three three negative six minus lambda and determined is zoo. Okay, so we expand this determinant. That will be two minor Lunda times Neck of six, minus Nanda and minus nine. He was zero. So the solution decision off this quadratic equation will be first time No one, We have three. And London, too, is negative. Seven. So since thes to London are not equal, I actually one of the wines positive, and number two is negative. So that means, uh, this it&#x27;s called radical form is as the as across the vacation, which is not definite in a minute. Because in our textbook, the fear of five from our textbooks says, um, Symmetric matrix. If they could do that, if the quadratic form is positive, definite even on leave, the Eiken bellows are all positive. If it&#x27;s an indefinite, that is the even happen. Only if the Matrix has both positive and negative, I get better. So in this case, we have both positive and negative values. So that means it&#x27;s basics as the quadratic of warm that is indefinite and correspondingly we can write out the we&#x27;re going to form with the under. Under the change of variables, I have three white, one squared, minus seven. Why two squared? Okay. And the last thing is to find out the corresponding Eigen Victor&#x27;s car to act number one and number two. So that&#x27;s first. So by our definition of Mike and Values and Aiken Victor&#x27;s, we have this quality we take. So in this case, the yoga value has to be a two dimensional vector. So in this case, we let yeah, get ready to be x one x two. So, based on our first row of it takes a we will have two x one plus three x two ukuleles Well, on the right hand side is just a two off x one. Sorry, home. We have number three. So he&#x27;s three up x one on this gives the relationship between X one and X two. So that is three dfx too is X one and also for lunga too, which is connective. Seven. Yeah, naked seven we have left inside. It&#x27;s still saying, but the right hand side is connected. Seven next one so this gives. Say, um, this is three X two equals connective nine x one. So does x two. You causing neck too. Three x one. So we get right out. Uh, I can vector the first I can vector corresponding to loved a one will be one and three and we have to normalize it. So that gives one over square root 10 because three squared plus one square it is 10 and take the square root. So you have square with 10 and 31 So the second I can vector will be one neck of three and this is after the normalization. We have one over square root too. Times one negative street. So the matrix p in this problem will be the spin off these two vectors. So take out that one over square root. 10 inside will be 31 on one and three. Sorry. 31 So that&#x27;s it for our problems. Problems. 10

Jack Chen · Answer

Okay, so Eunice proper and we have a phenomenal X one square, minus six x Y x two plus nine x two square. So there is a cross product turn here on this party naming, of course, bones to a quadratic form with man checks a equals toe minus three minus three nights. So we want to find out a transformation such that there&#x27;s no on cross pro doctor ERM under this transformation. So we&#x27;re looking for the Agon value and I backed off magic say the characteristic equation is a determined off a mine. Islam die cost zero. Which is why mine, Islam, that Pam&#x27;s nine miners Lambda minus 90 causes Europe. Salam DeWine causes zero in the lamb that we close to 10 for Lambda one Any miners? Lambda I I&#x27;m no one times I equals to yourself. So the corresponding Eigen vector will be won over route off 10 times 31 And the lamb that week was to turn the corresponding I connect her. It was the one of a root of 10 one minus three. So the metrics p we&#x27;re looking for will be won over route off 10 31 one, minus three. So with this transformation, X equals two p times white. And why just equals Teoh? Why one times y one y two. So with this transformation, this quadratic form equals to, um why transpose times peaches posed as a times p times Why? So this matrix multiplication we will be and a diagonal magics with dragon value ozone diagonal entrance. So we have 10 times y to a square. There&#x27;s no cross pro doctor amused expression.

Runpeng Li · Answer

given a quadratic form that is a negative X one squared connected X one squared and minus two ex wives too ex one next to minus X two squared. And so again, first thing we need to do is to write out that makes six a which in this case, we checked the politicians or X one squared X two squared. They are both like once of this diagonal will be all naked ones and check the X one x to the Cory vision is connected to so we can separate light headedness has to sum up to negative one. So that&#x27;s negative. One negative one. So here&#x27;s our matrix. And, um, the next thing is to find out the wagon betters So do that. We used a determinant a minus Lambda chi. So for this matrix, it is just next one minus number. Uh, next one next one next one month Dum dum and we&#x27;re looking for the determined. So this will be no one Minds Lunda squared minus one is zero. So this means lambda iss Okay, um, the students that step by step so we know by the previous expression we know that Lambda plus One the best one squared is one. So that means loved a plus one is one. So we can take number one to be elected to. Or London too to be C zero. Yeah. So that means since we have, ah, negative value value and we also have the there I can value. So that means this matrix a has the court verdict form indefinite. Okay. And now we can write down Thea quadratic forms off the change of variables. That is, um, next to active to why one squared? Why one square? Okay. And so we take the take the Eigen value back to our matrix. So that is to consider the system a axe equals London acts. So by the first row, we have negative X one minus x two iss Here we first apply. Our first I could value, which is connected to is naked too. Oh, excellent. So what? This means it implies that x one. His ex, too, is equal to X to all right. And for the second leg of value, we have x one minus x two equal Cyril. So that means X one is negative x two. So the first idea in vector we have is 11 Well, this second I in Vector we have is one negative one. So in order to normalize it divided it. We divide this by square it up too. So this is the one over square with 21 one. And same for the second dog in Victor one. Negative. Okay, so, Bye. So the matrix p we&#x27;ll just be this band, uh, these two Aiken, Victor&#x27;s. So we have one over square root too. That&#x27;s a scaler. Kinds The matrix. 111 Next one. So that&#x27;s it.

Jack Chen · Answer

Yes. Program. We have a party. Nominal. Two eggs one square minus four x one x two minus x two square which corresponds toe are quadratic form with cemented metrics. A equals two to minus one minus two minus Tokyo. So we are looking for transformation. Ex seacoast of P Hamzawy. Where? Why? Just simply Because too Why one in the white You with this? We want to find out a matrix p such that with this information. Um, the quadratic form contains no, um cross product terms. So in order to figure out this metrics p, we&#x27;re looking for the wagon vectors off. Hey, so the characteristic equation is, um, to mine Islam that times minus one minus Lambda minus four equals zero. Salam DeWine causes three Lambda Tweet coastal minus two. So for Lambda One eight miners, Lunda one, I equals two minus one minus two minus two minus four. Corresponds to Adam that turvey one. You coast to one over. Route five. Tu minus one in the Fulham. That too a minus. Lambda Tau. I cause to four minus two minus 21 corresponds to. And now that I give vector B two week lows toe one overtook five 12 So the matrix p we are looking for we will be one of root of five minus two or minus out to minus one. The 12 So with this transformation, X equals two p times Why this quadratic phone will be x chestful post times a times X because two y chance post times p Just post times a times p times Why? So this matrix multiplication gives us three minus 200 which is a diagonal matrix. And in the end we have three y one square, minus two white. There&#x27;s no uncrossed productor Munir&#x27;s party. No me, no.

Runpeng Li · Answer

home. 14. Um, we have the, uh, medical form that is three x one squared plus four x one next to so by this quadratic reform we have, we can write down the matrix A, that is. First, we have three for the first entry of the guy, you know. And second interest would be zero. All right. And three, three and zero. So, since the yet the coefficient of X one next to it is for so we can write down too. And two. So this is our matrix eight. So next thing next thing is to find out the confetti, so consider determined and off a minus. Some guy in this case will be three minus lambda two and two and negative Lunda. And that is zero. That means naked, naked London times three miners. Lambda minus four is zero. So by solving this quadratic equation, we finally found we kind of get the two Aiken vetters number one, which is for, and Linda too, which is a negative one. So, seize. Um, I noticed that the number one and number two are not both positive or both. Negative. So that means, um so that means whatever form is indefinite, right? And therefore, we can just write down the without informing for the change of bearable. We changed the, uh, axe to why? And to cans all tense about the cross product term. So that will be four of four times. Why? Ones where? Minus while you two square. All right. And now we need to find out the connectors. Chris, look at the first again I can value. So that&#x27;s four. So we have and see that the first rule of matrix a three x one plus to next to you cause we consider it I can do for you. That is X one. So that gives two x two you close x one and we can&#x27;t take Yeah again, Victor, to be one and two. Now, don&#x27;t forget to you normalize it. So that that is equivalent to one over square was five times 21 All right. And for the second again, value, which is next one. We have negative x one. And that gives, say, two x two equals negative or x one, and this is X two equals connected to X one. Therefore, our second again, Victor can be one negative too. Yeah, I normalize it. We have one over split five times. One negative too. All right. And hence our Matrix p will be this band off these two vectors for staff. One over square root five and you write down to one and the one next to, and that&#x27;s

Jack Chen · Answer

Okay, so in this question, we are given a phenomenal So this pollen nominal corresponds toe quadratic form with a symmetric matrix A equals two for minus two minus 24 And they were looking for information on turned this quadratic form into one with without any cross product room. So it basically want to eliminate this one. So we are looking for We are looking for metrics p such that X equals two Pete Holmes. Why? Where? Why equals two? Ah y one y two. So in order to figure out this metrics P, we need to figure out the agon value in Agra vector for the metric say so. The characteristic equation is their top. A minor slammed I, the determined off a equals 20 off a minus lambda lambda because zero So we have four miners. Lambda Square minus four equals zero. So we have to wagon May tomatoes. Lambda wine caused 26 and the lamb that we close to two for Lambda One a miner&#x27;s Lambda one I because to minus two minus two minus two minus still corresponding to Agon. Vector B one equals 21 over it. Off to one minus one in the lump that we coasted to a mine Islam, that why equals two Tu minus two minus 22 corresponding toe a second Eigen vector. You&#x27;ll be one of the root off two times 11 So the make six we&#x27;re looking for P equals two one of Rudolf two one minus one. The 11 So with this transformation X transpose times a times X equals to white transpose times p transpose times eight times p times Why, which becomes so the this matrix multiplication gives us, um 62 which is a diagonal magics and Indiana. We have six of why one square pressed too wide to square. Also, this, um, party nominee opened as o no across product er&#x27;s.

Problem

Classify the quadratic forms in Exercises $9-18 .…

Problem 10 Medium Difficulty

Answer

Hence the required new quadratic form is $Q(y)=-7 y_{1}^{2}+3 y_{2}^{2}$

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Algebra Educators

Anna Marie V.

Heather Z.

Alayna H.

Kayleah T.

Algebra Courses

Absolute Value - Example 1

Absolute Value - Example 2

Recommended Videos

Watch More Solved Questions in Chapter 7

Video Transcript

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

Hence the required new quadratic form is $Q(y)=-7 y_{1}^{2}+3 y_{2}^{2}$

Linear Algebra and Its Applications

Anna Marie V.

Heather Z.

Alayna H.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Anna Marie V.

Heather Z.

Alayna H.

Kayleah T.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications