5

Consider the linear systemg-EShow that (45) -{ is the smallest possible length ofh least of squares solution to this linear system....

Question

Consider the linear systemg-EShow that (45) -{ is the smallest possible length ofh least of squares solution to this linear system.

Consider the linear system g-E Show that (45) -{ is the smallest possible length ofh least of squares solution to this linear system.



Answers

Let $A$ be an $n \times n$ invertible matrix. Show that the unique solution to the linear system $A \mathbf{x}=\mathbf{b},$ namely, $\mathbf{x}=A^{-1} \mathbf{b},$ is also the least squares solution for this system.

In this question, we are given the augmented matrix over here and were asked to write the solution represented by The Matrix. So we see that there's two variables X and why. So we see. Hear that in our second room we see that why is equals to minus pi? And from our first show, we see that excess equals to seven or solutions that is seven comma minus five, and that's the answer to the question.

Okay, so we're given normal equation X transpose X y is equals toe x transpose be where X is er designed metrics. Okay. And we need to show the solution is unique off this normal immigration if at least two data points are different. Okay, so let's start with X transpose x. So if you calculate x transpose X transpose will be one x 11 x two and it will go to the one xn And when we multiply this with one x 11 x 21 x three one x n So we get X transpose x this one times 11 times once when we get in right. And here, if this one is multiplied with accents, we get summation off X and also here Submission off X I when I start from one pill and similarly exile And this is X one squared plus x two squared plus x three squares. This is a mission off excise square. When I get an iPhone one toe end. Okay, so this is a X transpose six, and here we have our vector. Why? And let's zoom on multiplying this. We get some Victor V Okay, so yeah, we already had experience would be so unlimited. B dash. Okay, so now so four solution to be unique from material thes two columns must be independent. Okay, so this column to be independent, that is end. And this won't be independent of all. Let's assume this all xar same. And the columns are independent. So if all xar same we have and this ex adding and times gifts annex again. This is an ex, and this is and x squared. Okay, so our assumption is is you mean all exa? Same yet Solomon Independent? No. If all except same, you can see if you multiply column one. Okay, so column one times X call them too. And therefore XY one minus two is it costs zero. So we have to non zero terms X and one, which makes column one and column 20 Therefore, they're linearly dependent. And from that we can say residents, we can say that if all x values air same, then the columns are linearly dependent and therefore we don't have unique solution. And if any two x are different, then we have a unique solution. Think

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

So in this question, were given the augmented matrix here, and we see that there's two variables X and why in the Matrix and we asked to solve for X and y so we see from our first role that X is equals to minus two. And from our second row, we see that why is he close to four? Her solution set his minus two common for and this is the answer to the question.


Similar Solved Questions

5 answers
5. Solve the following DEs by the Laplace transform method:. (a) y" + 5y' + 4y = 0 y(0) =1, Y(0) = 0 (b) y" - 6y' + 9y = e A* y(O) =0, Y() =0
5. Solve the following DEs by the Laplace transform method:. (a) y" + 5y' + 4y = 0 y(0) =1, Y(0) = 0 (b) y" - 6y' + 9y = e A* y(O) =0, Y() =0...
5 answers
8. (10 Foints) Determne whether Sis & basis for the indicated vector space s-{5 %Hs 'Jls H' Mz c (" &]'& C 7 , ef8 .7 4[;,7'0104' ! (^ 10 ( 3 0 (1A€ 44 ( 4!C3'( , ~023{^4 C 4(C ~(5'1 (, < ( 1 " . . "' 1 1' 3 ' 4'{''tC"2 ( 4'3 ` < '>YJ { < | ,(B) $-{'-2'+4/'-4/'+2,54}
8. (10 Foints) Determne whether Sis & basis for the indicated vector space s-{5 %Hs 'Jls H' Mz c (" &]'& C 7 , ef8 .7 4[;,7 '0104' ! (^ 10 ( 3 0 (1 A€ 44 ( 4!C3 '( , ~02 3{^ 4 C 4(C ~(5 '1 (, < ( 1 " . . "' 1 1' 3 '...
5 answers
(10 prs) Find the equation of the plare containing the two lines below.21:*=3+5y =-6t_2 =12. + 22z:{=-49+1,y = 8s ~ 2,72 ~163
(10 prs) Find the equation of the plare containing the two lines below. 21: *=3+5 y =-6t_ 2 =12. + 2 2z: {=-49+1, y = 8s ~ 2, 72 ~163...
5 answers
QUESTI( 10 pointsSave AnswerProvide an appropriate response:The distribution of Master'$ degrees conferred by a university is listed in the tableMajor Frequenc; Mathemaics 216 English 207 Engineering 86 Business 176 Educaton 267What is the probability that a randomly selected student graduating with a Master's degree has major of Education? Round your answer to three decimal places 0.2800.3900.0040.720
QUESTI( 10 points Save Answer Provide an appropriate response: The distribution of Master'$ degrees conferred by a university is listed in the table Major Frequenc; Mathemaics 216 English 207 Engineering 86 Business 176 Educaton 267 What is the probability that a randomly selected student gradu...
5 answers
SiRNAs Long Short Long are double-stranded = thought be to RNA RNA RNA primarily derived from the following precursor:
siRNAs Long Short Long are double-stranded = thought be to RNA RNA RNA primarily derived from the following precursor:...
1 answers
The reaction $\mathrm{A} \longrightarrow \mathrm{B}$ shown here follows first-order kinetics. Initially different amounts of A molecules are placed in three containers of equal volume at the same temperature. (a) What are the relative rates of the reaction in these three containers? (b) How would the relative rates be affected if the volume of each container were doubled? (c) What are the relative half-lives of the reactions in (i) to (iii)?
The reaction $\mathrm{A} \longrightarrow \mathrm{B}$ shown here follows first-order kinetics. Initially different amounts of A molecules are placed in three containers of equal volume at the same temperature. (a) What are the relative rates of the reaction in these three containers? (b) How would th...
5 answers
Graph y fx) = -3sin (2xr +-3
Graph y fx) = -3sin (2xr + -3...
5 answers
V16 1 82 ~ du
V16 1 82 ~ du...
4 answers
Given the following combustion equation of octane:C8H18(l) + 25/2 O2(g) ---> 8CO2(g) + 9 H2O(g) + 5075 kJFinda) How many litres CO2 ( At SATP) are released to the atmosphere per each liquid litre of combusted octane (assume complete combustion and pure fuel) (Density octane at 25 C : 703g/L , gases : VM SATP ~24 L/mol)b) How many kg CO2 are released to the atmosphere for each 1.00 TJ energy produced by octane combustion?
Given the following combustion equation of octane:C8H18(l) + 25/2 O2(g) ---> 8CO2(g) + 9 H2O(g) + 5075 kJFinda) How many litres CO2 ( At SATP) are released to the atmosphere per each liquid litre of combusted octane (assume complete combustion and pure fuel) (Density octane at 25 C : 703g/L , ga...
5 answers
Let v_and H= Span{V1 Vz V3} . Note that V3 3V1 4v2. Which of the following sets form basis for the subspace H; that is,-3which sets fomm an efficient spanning set containing no unnecessary vectors? {V1.Vz Vs} {V1.V2} {V,V3B, C and DB only Aonly B and C
Let v_ and H= Span{V1 Vz V3} . Note that V3 3V1 4v2. Which of the following sets form basis for the subspace H; that is, -3 which sets fomm an efficient spanning set containing no unnecessary vectors? {V1.Vz Vs} {V1.V2} {V,V3 B, C and D B only Aonly B and C...
5 answers
Trials in an experiment with a polygraph include 9999 results that include 2424 cases of wrong results Use a 0.05 significance level to test the claim that such polygraph results are incorrect more than 80% of the time. Identify the null hypothesis, alternative hypothesis, test statistic, p-value, conclusion about the null hypothesis, and final conclusion that addresses the original claim Show all calculations that lead to your results:
Trials in an experiment with a polygraph include 9999 results that include 2424 cases of wrong results Use a 0.05 significance level to test the claim that such polygraph results are incorrect more than 80% of the time. Identify the null hypothesis, alternative hypothesis, test statistic, p-value, c...
5 answers
(a) Find the term up to x'in the Maclaurin expansion of f (x)= In(cosx) (6) Use this series t0 find an approximation in terms of T for In2 [7 marks]
(a) Find the term up to x'in the Maclaurin expansion of f (x)= In(cosx) (6) Use this series t0 find an approximation in terms of T for In2 [7 marks]...

-- 0.023572--