2. Consider the matrix equation k = Cu, where k1 u- kp PA4 ug) are q real unknow

Question

2. Consider the matrix equation k = Cu, where k1 u- kp PA4 ug) are q real unknowns and all else here is real, constant, and known. The relevant where [ui, u2, least-squares vector is If our matrix equation has a solution, then uLs is that solution. If our matrix equation does not have a solution, then us is the best estimate. (a) We seek to fit the ay data pairs 11.1 to the linear form y ao +air, where the a-coefficients are constants to be determined. That is, we want to solve the simultaneous equations for the unknowns ao and al. Express this set of equations in the matrix form k Cu (b) Calculate the least-squares vector. What is the equation of the bestit e? You may find the relation 8 -9 to be useful

Explanation / Answer

a. given data points

x y

-1 -5

1 1.1

2 4

so the equations become

ao - a1 = -5

ao + a1 = 1.1

ao + 2a1 = 4

putting them in Cu = k form

C = [1 -1]

[1 1]

[1 2]

u = [ao]

[a1]

k = [-5]

[1.1]

[4]

b. now, Cu = k

so u = C^-1*k

as inverse of a square matrix can be found

we drop the third rwo to arrive at the solution

ao = -1.95

a1 = 3.05

we put this in third equation to get ao + 2ai = 4.15 which is close to 4

to find the best fit we try various different values close to this solution to get the third equation valuke close to 4

so the solution becomes

a1 = 3.0071

ao = -1.9714

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