4. (15 pts.) Let A E Rmxn, and let z be a solution to least squares problem min, Ax-bll2. Shovw that
4. (15 pts.) Let A E Rmxn, and let z be a solution to least squares problem min, Ax-bll2. Shovw that
(2) Let A∈Rm+n and let xˆ be a solution of the least squares problem Ax = b. Show that a vector y ∈ Rn will also be a solution if and only if y = xˆ+z, for some vector z ∈ N(A). [HINT: N(ATA) = N(A)].
Let
.
(a) Find the singular value decomposition of A.
(b) Find the least squares solution to the linear system
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. Find the least squares straight line for the data: UN - e Anne 2. Let A Apply Jacobi iteration 4 times, starting with rº), to produce estimates pl. 22). 23) and rd to the solution of the equation Ar
Problem #4: Let 0 2 A = 2 9 and b = 0 2 4 Find the least squares solution of the linear system Ax = b. Enter the components of the least squares solution x = [x y]? into the answer box below (in order), separated with a comma. Problem #4: Enter your answer symbolically, as in these examples Just Save Submit Problem #4 for Grading Problem #4 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark:
(1 point) Find the least-squares solution x* of the system [10] 0 1 x= 4 Loo] [-9 (1 point) Find the least-squares solution x* of the system [ 1 | -1 ( 5 -2] [ 10 ] 2 X= 2 . 4 ] (-36]
110 In Problem 5.49, a least squares inverse was found for the matrix A = ſi -1 -2 -2 4 3 [ 1 1 -3 17 -2. 11 (a) Use this least squares inverse to show that the system of equations Az = C is inconsistent, where c' = (2,1,5). (b) Find a least squares solution. (c) Compute the sum of squared errors for a least squares solution to this system of equations.
12 === 9 1 4 4. Find the least-squares solution of A7 = 5 where A = 0 1 and 6 = 1 0 Hint: Solve the normal equations AT AT = ATV.
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax=b. - 1 4 A= 1 8 b= -1 - 1 4 4 a. The orthogonal projection of b onto ColA is 6= (Simplify your answers.) b. A least-squares solution of Ax=b is x = (Simplify your answers.)
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax=b. -1 4 11 A= 18 b= - 1 -1 4 a. The orthogonal projection of b onto Col A is @ = (Simplify your answers.) b. A least-squares solution of Ax=b is x= (Simplify your answers.)