compute-the-least-squares-error-associated-with-the-least-squares-solution-found-in-exercise-3

Name: compute the least squares error associated with the least squares solution found in exercise 3
Uploaded: 2019-08-08
Duration: 1 min 13 s
Channel: Numerade
Description: Compute the least-squares error associated with the least-squares solution found in Exercise $3 .$

Question

Compute the least-squares error associated with the least-squares solution found in Exercise $3 .$

Angelo Rendina · Accepted Answer

we want to find the least squares error associated to the system. A x equal be where a and B are This matter is here as an exercise for so we have computer ready that the least square solution except is given by 11 So now we just need to compute the error which is given by the length of the vector X at minus me. So we actually just compute the problem between the messages and then subtract. So we have the lands off well, except it turns out to be 402 and then was trucked the vector me, which is 510 And this is the length of the HVAC Tora minus one minus one, too. Now the land off this vector, of course, given by the square root of the sum of the squares of its entries, so minus one squared plus minus one squared plus two squared, which is out to be six. So the error in this case is given by route off six

Alayna Abraham · Answer

in this problem, we are going to find the least squares parable a or another way to put this is finding the quadratic regression of the line, uh, using the point that we&#x27;ve been given. So that means that our answer has to be in this form of y equals x squared plus B x plus c. And to solve this, we are going to need our formula on the formula will be a transposed times a on the whole thing is going to get in first times transposed again and that and transposed will be multiplied by being. Now what are a and B We have to create these matrixes. So the way we do that is by plugging in for our X and Y into this equation right here. So let&#x27;s take our 1st 1 So when we plug in one R y, value will be 10. So it&#x27;s 10 equal Teoh a plus B plus C and then our 2nd 1 would be 20 equals four a plus two b plus C on our third point will be 10 equals to nine a plus three b plus c. So our matrix are there are matrix A will be the coefficients of at this point, we all use a different color pen. Um, so are a will be the coefficients of this point and r b will be the cof these three values right here. So let&#x27;s set up the Matrix. So it will be one month 14 to 1, 931 times X And that&#x27;s what we&#x27;re going to try and solve. That will be our ABC, uh, values in this part. So we&#x27;re trying to find the coefficients, and that&#x27;s equal to 10 2010. And this is okay and this is be so we&#x27;re gonna use formula now. Thistle. So let&#x27;s sell for the different points. Let&#x27;s do a transposed. And that just means making our rose into columns and our columns in two rows. So we will be taking this and making it into this column right here and etcetera, etcetera. And I&#x27;m sure once I write it out, it will become clear. So our first column is now 111 2nd column is 4 to 1, and 1/3 column is 931 And because we have that, we can take a t on the multiply it by r a which again me have right here and make sure that the biggest rule and the most important rule when you&#x27;re doing matrix multiplication is to make sure your M times end is equal for each one. So M is the column and is your rope. So whatever you&#x27;re multiplying, they have to have the same number of M and the same number of end. So in 80 we have three for n and for A we have three years wear good. Three equals three so we can multiply. And once we multiply out, you will get 98th 36 14 for first row, 36 14 6 for a single 40 63 Right, There you go. And after we do that, we have to take the inverse of the whole thing. So I&#x27;m just gonna right over here, say some room and when you do that, you will get 1.5 negative 65 for your first row. Get a 6 24.5 and negative 21 for you said zero five negative 21 and 19 for here, there and to make it easier if you don&#x27;t have a calculator, What&#x27;s there? I would use the Gauss Jorden elimination method. So that, of course, just means I&#x27;m just gonna use variables air here. A B C d e a d age making sure it follows into the identity matrix. So solving for the identity matrix and the numbers you end of here would be your inverse. So that&#x27;s another thing to do if you don&#x27;t have a calculator and you don&#x27;t have, let&#x27;s say, Matt Lab or something doing the matrixes for you. So after we find this in first, let&#x27;s do the second part of it. Clear our equation. So that&#x27;s a transposed times be. And so we already have a transposed from above. That&#x27;s 11 line 4 to 3. Long times 10 2010 and again, Our Rows and columns lineup. They&#x27;re the same, number three and three again. It&#x27;s on their equal and we can solve and we will get 100 e. T. 80 and before I know this is a lot of steps. But once you get the map out, it feels really good. You&#x27;ll feel really accomplished, I promise. So after you do that, our last thing to do is multiply this endless together on and once you do that, so it&#x27;s quickly that will supply it out. And again. You notice that our number of columns match a number of rows so we can solve this. We can multiply it out, and once you do that, you will get negative. 40. Negative 20. He was kind of satisfying. So are a is equal to negative. 10. RV is people toe 40 on their seat is equal to negative 20 and we still have to put in our equation form. So why equals negative? 10 Exclude plus 40 x minus 20 and that&#x27;s answer.

Karol Hajduk · Answer

Okay, so here we have. Oh, task to find three births, some secret. So this would be capital. Thanks. Experts wipe. See minus 92 0 Do you want fine numbers? Which some music, man. And, uh, some of the squares is the smallest minimized sum of squares. This number. So this will be X squared. Plus vice places and two minimize under strain. The same sequence tonight. Okay. So typical concerned Immunization can do this. Great, great, great Jason. Someone&#x27;s computer radiance, Sum of squares years thing Derivative. Next this function. So just one equation, Sly. She&#x27;s exorcist things toe Q Yes. Okay, so, uh, this 1st 3 equations us that toe XY Quest two ex acquits. Why so ex? Hi. Like this Train kids, us. Thanks x three. And that&#x27;s why. And that&#x27;s you, uh, c quote square trees, just like you to just okay,

Zachary Watson · Answer

Okay, so we want to find three real numbers. Who? Someone&#x27;s nine and their squares to a number. That&#x27;s Ismael is possibly so what we want to do Teoh to do this. We know that LF is gonna be equal to lander Times, the g in order to solve this problem. So we need to take the partial derivatives of each of these. So two x, I have plus two Why j a happ us to Z Okay, I was gonna equal Teoh one I landed I had plus Lander Jack was land OK, so two x is equal to lander to wise and we quits or lander and two z is gonna be equal to lander. Okay, so now for each of these, I&#x27;m just gonna solve for X y z So X is gonna be equal to Landover to why&#x27;s going todo land over to z is gonna leave until I have over two. And as you can clearly see, exit butcher wine is equal to Z. So let&#x27;s try and now plug these back into the G equation and saw for that. So land over to slander over to purse land over to most unequal tonight. So you really Atlanta over to equal tonight. That means Lander must be equal to to over three times, and I or there can cancel this out. So too times three and then lander is gonna be equal to six. Okay, so not at Lando&#x27;s equals. Six. We know that X Y and Z is equal to land over to, so actually quit. Otherwise he could Azi, which is equal to three. So that&#x27;s gonna be our real numbers. Some his line and someone who squares the smallest possible to you green three.

Kendall Eley · Answer

okay, When we&#x27;re trying to solve this problem, we want to find three numbers. Who some is nine will represent that it&#x27;s X plus y plus Z equals nine. The sum of their squares is to be minimized so x weird plus y squared plus z squared. Now we need to begin by taking our constraint problems and solving it for Z. We get Z equals nine minus x minus. Why? And now we&#x27;re going to make a problem that we want to minimize the function of x and y by replacing Z with that quantity. So we get X squared plus y squared plus quantity not minus six months. Why squared? In order to minimize, we need to find the derivatives and determine whether equal to zero. So the derivative with respect to X is two acts plus two times this same quantity to the negative one. And then the derivative with respect to why is to why, plus two times this in quantity times minus one. And now if we clean those up, we get that the 1st 1 becomes two X minus 18 plus two x plus two. Why? And the 2nd 1 becomes two. Why minus 18 plus two X plus two. Why? And since we&#x27;re trying to minimize, we&#x27;re gonna set each of these derivatives equal to zero. So when we simplify the 1st 1 we can make it into four X Plus two. Why equals 18 and the 2nd 1 becomes two X plus four. Why equals 18? Now we can solve these by elimination. I&#x27;m gonna multiply the top equation on negative too. And in doing so, I would get negative eight x minus for why? Equal to negative 36 which now, when we add, we get negative. Six X equals negative 18. So X equals three. Now we can go back to the second equation right here and substitute in and we get two times three plus four. Why? It was 18 and we consult that and get that why I was three and then to to determine our Z. We can go back up here to the very first equation that we saw, and we can find out that Z equals nine minus three minus three, which is three also. So it appears that the three numbers are 33 in three, but we need to verify that it is a minimal. In order to verify that it&#x27;s a minimum, we need to look at the second derivatives. The second derivative is simply the derivative of the first derivative. So for the first derivative with respect to X, when we take its second derivative, we simply get four. And when we take the derivative with respect to why excuse me, this drifter with respect to why of the second derivative, that&#x27;s this derivative right here. We also get four. And then we also need to look at the partial derivative which is the derivative of the first derivative of X. With respect to why, what we could do this first derivative of y with respect to X either way. But if I look at the first derivative of X, this one right here and take it derivative with respect to why everything goes away except the number two now, the test when we&#x27;ve got two variables, is we look at the second derivative with respect to acts times a second derivative with respect to why, minus the second derivative of X, with respect or why that quantity squared. So we get four times for minus four, which is 12 which is greater than zero. Therefore, that means that this is a minimal. That means that con cavity is upwards. So then our three numbers that we&#x27;re trying to do to find or 33 angry.

Problem

Compute the least-squares error associated with t…

Problem 7 Easy Difficulty

Compute the least-squares error associated with the least-squares solution found in Exercise $3 .$

Answer

2$\sqrt{5}$

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

Linear Algebra and Its Applications

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Kayleah T.

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2$\sqrt{5}$

Linear Algebra and Its Applications

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

Lily A.

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Kristen K.

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

Linear Algebra and Its Applications