Linear Algebra - Least Squares I only need the solution for 4 Please be careful,

Question

Linear Algebra - Least Squares

I only need the solution for 4

Please be careful, I have submitted this question twice so far and have gotten the wrong answer both times. I will not thumbs up an incorrect answer!

(1 pt) A high-altitude sounding balloon rises at an approximately constant rate. A balloon with GPS tracking equipment is launched and the following 4 data points are received: (102, 1350), (114, 1402), (126,1468), (138, 1536). The first coordinate is time in seconds since launch and the second coordinate is the height above sea level in meters. A. The best fit line can be found by solving the normal equations. In matrix form, the normal equations are AX-Y where X is the 2 by 1 column vector with first component equal to the y intercept and second component equal to the slope. Y is a 4 by 1 column vector that contains the heights. Determine the elements of the 4 x 2 matrix A in the above equation 102 114 126 138 B. As long as the columns of A are linearly independent,the matrix AT A is invertible and the unique least squares solution is given by X-[A X a ATY. Using a calculator or computer, compute X and use your solution to answer the following 1) The launch altitude of the weather balloon was about 815 2) The ascent rate of the balloon is about 5.2 3) The residual (predicted - observed) for the point (126, 1468) is 2.2 4) If the balloon bursts at time 5353 seconds after launch, then the least squares predicted peak altitude is 875 (5.2 538 meters above sea level meters above sea level meters per second. meters

Explanation / Answer

A = [1 102;1 114;1 126; 1 138];

y=[1350;1402;1468;1536];

X = (inv(A'*A))*(A'*y);

A =

1 102
1 114
1 126
1 138

y =

1350
1402
1468
1536

X =

815.0000
5.2000

hence height ^ = 815 + 5.2 * t

1) when t = 0 , height = 815

2) ascent = 5.2

3) residual for t = 126 is

y^ - y

= (815 + 5.2*126) - 1468

= 2.2

4)for t = 5353

y^ = 815 +5.2 * 5353 = 28650.6

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